ML Aggarwal Class 6 Maths Solutions Chapter 06 Fractions

 

Exercise 6.1

 

Question 1. Write the following division as fractions:
(i) 3 ÷ 7
(ii) 11 ÷ 78
(iii) 113 ÷ 128
Answer: When you divide one number by another, you can express it as a fraction. The number being divided becomes the numerator (top), and the number you are dividing by becomes the denominator (bottom).
(i) 3 ÷ 7 becomes \( \frac{3}{7} \)
(ii) 11 ÷ 78 becomes \( \frac{11}{78} \)
(iii) 113 ÷ 128 becomes \( \frac{113}{128} \)
In simple words: Division can be shown as a fraction. The top number is what you divide, and the bottom number is what you divide by.

Exam Tip: Always remember that division a ÷ b = a/b. The dividend goes on top and the divisor on the bottom.

 

Question 2. Write the following fractions in words:
(i) \( \frac{2}{7} \)
(ii) \( \frac{3}{10} \)
(iii) \( \frac{15}{28} \)
Answer: To express fractions in words, read the numerator as a number and the denominator with its ordinal name (seventh, tenth, twenty-eighth).
(i) \( \frac{2}{7} \) is expressed as two-sevenths
(ii) \( \frac{3}{10} \) is expressed as three-tenths
(iii) \( \frac{15}{28} \) is expressed as fifteen twenty-eighths
In simple words: Say the top number, then add the bottom number's word form. For 2/7, say "two-sevenths".

Exam Tip: Pay attention to how you name the denominator - use ordinal numbers (seventh, tenth, twenty-eighth) and link them with a hyphen to the numerator.

 

Question 3. Write the following fractions in number form:
(i) one-sixth
(ii) three-eleventh
(iii) seven-fortieth
(iv) thirteen-one hundred twenty fifth
Answer: Convert word form to numeric form by taking the first number as the numerator and converting the ordinal word to its denominator.
(i) one-sixth = \( \frac{1}{6} \)
(ii) three-eleventh = \( \frac{3}{11} \)
(iii) seven-fortieth = \( \frac{7}{40} \)
(iv) thirteen-one hundred twenty fifth = \( \frac{13}{125} \)
In simple words: The first number is your top number. The word after the hyphen tells you the bottom number.

Exam Tip: Reverse the process from Question 2 - listen carefully to the ordinal word to identify the correct denominator.

 

Question 4. What fraction of each of the following figures is shaded part?
Answer: To find what fraction is shaded, divide the number of shaded parts by the total number of equal parts the figure is divided into.
(i) Total parts = 7, Shaded parts = 4, so Fraction = \( \frac{4}{7} \)
(ii) Total parts = 8, Shaded parts = 3, so Fraction = \( \frac{3}{8} \)
(iii) Total parts = 4, Shaded parts = \( \frac{1}{2} \). This means 2 small triangles are shaded out of 4 parts total. Simplified: \( \frac{1}{8} \)
(iv) Total parts = 4, Shaded parts = 1, so Fraction = \( \frac{1}{4} \)
(v) Total parts = 6, Shaded parts = 1, so Fraction = \( \frac{1}{6} \)
(vi) Total parts = 10, Shaded parts = 3, so Fraction = \( \frac{3}{10} \)
(vii) Total parts = 7, Shaded parts = 3, so Fraction = \( \frac{3}{7} \)
(viii) Total parts = 4, Shaded parts = 2, so Fraction = \( \frac{2}{4} \), which simplifies to \( \frac{1}{2} \)
(ix) Total parts = 9, Shaded parts = 4, so Fraction = \( \frac{4}{9} \)
In simple words: Count how many parts are colored in. Count the total parts. Make a fraction: colored parts on top, total parts on bottom.

Exam Tip: Always check if the fraction can be simplified by finding common factors. The shaded region method is fundamental - master it well.

 

Question 5. Shade the parts of the following figures according to given fractions:
(i) \( \frac{3}{4} \)
(ii) \( \frac{1}{6} \)
(iii) \( \frac{1}{4} \)
(iv) \( \frac{4}{9} \)
(v) \( \frac{1}{3} \)
(vi) \( \frac{5}{8} \)
Answer: To shade fractions in a figure, divide the figure into equal parts matching the denominator, then shade as many parts as the numerator indicates.
(i) \( \frac{3}{4} \) - Divide the square into 4 equal parts. Shade 3 parts.
(ii) \( \frac{1}{6} \) - Divide the circle into 6 equal parts. Shade 1 part.
(iii) \( \frac{1}{4} \) - Divide the triangle into 4 equal parts. Shade 1 part.
(iv) \( \frac{4}{9} \) - Divide the square into 9 equal parts. Shade 4 parts.
(v) \( \frac{1}{3} \) - Divide the hexagon into 6 equal parts. Shade 2 parts (since 2 parts make \( \frac{1}{3} \)).
(vi) \( \frac{5}{8} \) - Show 8 circles total. Shade 5 circles.
In simple words: The bottom number tells how many equal pieces to make. The top number tells how many to shade.

Exam Tip: Make sure all divisions of the figure are equal in size. Shade clearly and count carefully to avoid mistakes.

 

Question 6. Write the fraction in which
(i) numerator = 5 and denominator = 13
(ii) denominator = 23 and numerator = 17
Answer: Place the numerator (top number) above the denominator (bottom number) to form a fraction.
(i) Numerator = 5, Denominator = 13, so Fraction = \( \frac{5}{13} \)
(ii) Numerator = 17, Denominator = 23, so Fraction = \( \frac{17}{23} \)
In simple words: Put the first number on top and the second number on the bottom to make a fraction.

Exam Tip: Pay careful attention to which number is the numerator and which is the denominator - the order matters.

 

Question 7. Shabana has to stitch 35 dresses. So far, she has stitched 21 dresses. What fraction of dresses has she stitched?
Answer: Total dresses to be stitched = 35. Number of dresses stitched so far = 21. The fraction of dresses stitched is calculated by placing the number stitched over the total number: \( \frac{21}{35} \).
In simple words: She stitched 21 out of 35 dresses, so the fraction is 21/35.

Exam Tip: In word problems, identify what part and what whole are being compared. The part goes on top, the whole goes on the bottom.

 

Question 8. What fraction of a day is 8 hours?
Answer: We know 1 day contains 24 hours. To find what fraction 8 hours is, place 8 (the part) over 24 (the whole): \( \frac{8}{24} \).
In simple words: A day has 24 hours. 8 hours is 8/24 of a day.

Exam Tip: Remember standard conversions: 1 day = 24 hours. Set up the fraction correctly with the smaller unit on top.

 

Question 9. What fraction of an hour is 45 minutes?
Answer: We know 1 hour equals 60 minutes. To express 45 minutes as a fraction of an hour, divide 45 by 60: \( \frac{45}{60} \).
In simple words: An hour has 60 minutes. 45 minutes is 45/60 of an hour.

Exam Tip: Standard conversion: 1 hour = 60 minutes. Always match units before creating the fraction.

 

Question 10. How many natural numbers are there from 87 to 97? What fraction of them are prime numbers?
Answer: Natural numbers from 87 to 97 are: 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97. Count them: there are 11 natural numbers. Among these, identify the prime numbers (numbers divisible only by 1 and themselves): 89 and 97. Therefore, there are 2 prime numbers. The fraction of prime numbers is \( \frac{2}{11} \).
In simple words: From 87 to 97, there are 11 numbers total. Only 2 of them (89 and 97) are prime. So the fraction is 2/11.

Exam Tip: When counting from a to b inclusive, remember to include both endpoints. Check each number carefully for primality.

 

Exercise 6.2

 

Question 1. Show the fractions \( \frac{2}{5}, \frac{3}{5}, \frac{4}{5} \) and \( \frac{5}{5} \) on a number line.
Answer: Draw a straight line and mark point O representing 0 and point A representing 1 to its right at unit distance. Divide the segment OA into 5 equal parts - each part represents \( \frac{1}{5} \). Starting from O, mark successive points at \( \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} \), and \( \frac{5}{5} \) (which equals 1). The fractions \( \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, \) and \( \frac{5}{5} \) are shown on the resulting number line. [Number line representation with marked points]
In simple words: Split the line from 0 to 1 into 5 equal pieces. Each piece is 1/5. Count along to mark 2/5, 3/5, 4/5, and 5/5.

Exam Tip: Ensure all divisions are equal. Use a ruler for accuracy. Remember that 5/5 equals 1.

 

Question 2. Show \( \frac{1}{8}, \frac{2}{8}, \frac{3}{8} \) and \( \frac{7}{8} \) on a number line.
Answer: Draw a straight line marking point O as 0 and point A as 1, located one unit to the right. Divide segment OA into 8 equal parts, with each part representing \( \frac{1}{8} \). Starting from O, place marks at \( \frac{1}{8}, \frac{2}{8}, \frac{3}{8} \), and \( \frac{7}{8} \). The fractions are now displayed on the number line. [Number line representation with marked points]
In simple words: Split 0 to 1 into 8 equal pieces. Mark the fractions 1/8, 2/8, 3/8, and 7/8 on this line.

Exam Tip: Note that not all fractions need be shown - only those mentioned in the question. Mark them with clear dots or circles.

 

Question 3. Show \( \frac{0}{10}, \frac{1}{10}, \frac{3}{10}, \frac{5}{10}, \frac{7}{10} \) and \( \frac{10}{10} \) on a number line.
Answer: Draw a straight line with point O marking 0 and point A marking 1, positioned one unit to the right. Break the segment OA into 10 equal parts, where each part is \( \frac{1}{10} \). Beginning at O, mark the positions for \( \frac{0}{10} \) (which is 0), \( \frac{1}{10}, \frac{3}{10}, \frac{5}{10}, \frac{7}{10}, \) and \( \frac{10}{10} \) (which is 1). These fractions are now represented on the number line. [Number line representation with marked points]
In simple words: Divide the space from 0 to 1 into 10 equal sections. Mark dots at 0/10, 1/10, 3/10, 5/10, 7/10, and 10/10.

Exam Tip: Observe that 0/10 = 0 and 10/10 = 1. Number lines help visualize how fractions fall between whole numbers.

 

Exercise 6.3

 

Question 1. State which of the following fractions are proper, improper or mixed:
(i) \( \frac{15}{26} \)
(ii) \( \frac{17}{12} \)
(iii) \( 5\frac{2}{3} \)
(iv) \( \frac{6}{8} \)
(v) \( 11\frac{5}{7} \)
(vi) \( \frac{117}{8} \)
(vii) \( \frac{222}{333} \)
(viii) \( \frac{531}{247} \)
Answer:
(i) \( \frac{15}{26} \) - The numerator (15) is smaller than the denominator (26). This makes it a proper fraction.
(ii) \( \frac{17}{12} \) - The numerator (17) is greater than the denominator (12). This makes it an improper fraction.
(iii) \( 5\frac{2}{3} \) - This has a whole number part (5) joined with a proper fraction \( \frac{2}{3} \). This makes it a mixed fraction.
(iv) \( \frac{6}{8} \) - The numerator (6) is smaller than the denominator (8). This makes it a proper fraction.
(v) \( 11\frac{5}{7} \) - This has a whole number part (11) joined with a proper fraction \( \frac{5}{7} \). This makes it a mixed fraction.
(vi) \( \frac{117}{8} \) - The numerator (117) is greater than the denominator (8). This makes it an improper fraction.
(vii) \( \frac{222}{333} \) - The numerator (222) is smaller than the denominator (333). This makes it a proper fraction.
(viii) \( \frac{531}{247} \) - The numerator (531) is greater than the denominator (247). This makes it an improper fraction.
In simple words: A proper fraction has a smaller top number. An improper fraction has a top number that is larger or equal. A mixed fraction has a whole number and a proper fraction together.

Exam Tip: Always compare the numerator and denominator directly - no division is needed to classify a fraction. For mixed fractions, identify the whole number part immediately.

 

Question 2. Convert the following improper fractions into mixed numbers:
(i) \( \frac{17}{3} \)
(ii) \( \frac{119}{15} \)
(iii) \( \frac{961}{13} \)
(iv) \( \frac{117}{32} \)
Answer:
(i) When we divide 17 by 3, the quotient is 5 and the remainder is 2.

\[ \frac{17}{3} = 5\frac{2}{3} \]
(ii) When we divide 119 by 15, the quotient is 7 and the remainder is 14.

\[ \frac{119}{15} = 7\frac{14}{15} \]
(iii) When we divide 961 by 13, the quotient is 73 and the remainder is 12.

\[ \frac{961}{13} = 73\frac{12}{13} \]
(iv) When we divide 117 by 32, the quotient is 3 and the remainder is 21.

\[ \frac{117}{32} = 3\frac{21}{32} \]
In simple words: Divide the top by the bottom. The answer becomes the whole number, and what's left over becomes the top of the new fraction with the same bottom.

Exam Tip: Use long division to find both the quotient and remainder. Check your work by converting back - multiply the whole number by the denominator, add the remainder, and you should get the original numerator.

 

Question 3. Convert the following mixed fractions into improper fractions:
(i) \( 7\frac{2}{11} \)
(ii) \( 3\frac{5}{48} \)
(iii) \( 13\frac{7}{64} \)
(iv) \( 7\frac{2}{3} \)
Answer: To find the improper fraction, apply this rule: improper fraction = \[ \frac{(\text{natural number} \times \text{denominator}) + \text{numerator}}{\text{denominator}} \]
(i) For \( 7\frac{2}{11} \):

\[ 7\frac{2}{11} = \frac{7 \times 11 + 2}{11} = \frac{77 + 2}{11} = \frac{79}{11} \]
(ii) For \( 3\frac{5}{48} \):

\[ 3\frac{5}{48} = \frac{3 \times 48 + 5}{48} = \frac{144 + 5}{48} = \frac{149}{48} \]
(iii) For \( 13\frac{7}{64} \):

\[ 13\frac{7}{64} = \frac{13 \times 64 + 7}{64} = \frac{832 + 7}{64} = \frac{839}{64} \]
(iv) For \( 7\frac{2}{3} \):

\[ 7\frac{2}{3} = \frac{7 \times 3 + 2}{3} = \frac{21 + 2}{3} = \frac{23}{3} \]
In simple words: Multiply the whole number by the bottom, add the top, and write it all over the same bottom number.

Exam Tip: Be careful with the arithmetic when multiplying the whole number by the denominator. Always double-check your addition before writing the final answer.

 

Question 4. Write the fractions representing the shaded regions and pair up the equivalent fractions from each row:
Answer: The fractions shown by the shaded areas are:
(i) There are 2 total parts with 1 shaded.

\[ \text{Fraction} = \frac{1}{2} \]
(ii) There are 6 total parts with 4 shaded.

\[ \text{Fraction} = \frac{4}{6} \]
(iii) There are 9 total parts with 3 shaded.

\[ \text{Fraction} = \frac{3}{9} \]
(iv) There are 8 total parts with 2 shaded.

\[ \text{Fraction} = \frac{2}{8} \]
(v) There are 4 total parts with 3 shaded.

\[ \text{Fraction} = \frac{3}{4} \]
(a) There are 8 total parts with 2 shaded.

\[ \text{Fraction} = \frac{2}{8} \]
(b) There are 12 total parts with 8 shaded.

\[ \text{Fraction} = \frac{8}{12} \]
(c) There are 16 total parts with 12 shaded, which simplifies to \( \frac{3}{4} \).

\[ \text{Fraction} = \frac{12}{16} = \frac{3}{4} \]
(d) There are 8 total parts with 4 shaded.

\[ \text{Fraction} = \frac{4}{8} \]
(e) There are 9 total parts with 3 shaded.

\[ \text{Fraction} = \frac{3}{9} \]
The matching pairs are: (i) ↔ (d), (ii) ↔ (b), (iii) ↔ (e), (iv) ↔ (a), (v) ↔ (c).
In simple words: Count how many parts are shaded and how many parts in total. Write shaded parts on top and total parts on bottom. Match fractions that are the same even if they look different.

Exam Tip: Reduce fractions to simplest form before matching - this makes it easier to spot equivalent fractions. For example, \( \frac{4}{6} \) and \( \frac{8}{12} \) both reduce to \( \frac{2}{3} \).

 

Question 5. (i) Find the equivalent fraction of \( \frac{15}{35} \) with denominator 7
(ii) Find the equivalent fraction of \( \frac{2}{9} \) with denominator 63
Answer:
(i) To get a denominator of 7 from 35, we divide 35 by 5. So we divide both the top and bottom by 5.

\[ \frac{15}{35} = \frac{15 \div 5}{35 \div 5} = \frac{3}{7} \]
(ii) To get a denominator of 63 from 9, we multiply 9 by 7. So we multiply both the top and bottom by 7.

\[ \frac{2}{9} = \frac{2 \times 7}{9 \times 7} = \frac{14}{63} \]
In simple words: If you multiply or divide the bottom by a number, do the same to the top to keep the fraction equal.

Exam Tip: Always identify whether you need to multiply or divide by looking at the given denominator compared to the original one. Check your answer by confirming both fractions have the same value.

 

Question 6. Find the equivalent fraction of \( \frac{3}{5} \) having (i) denominator 30 (ii) numerator 27
Answer:
(i) To change the denominator to 30, we need to multiply 5 by 6. Therefore, we multiply both the numerator and denominator by 6.

\[ \frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30} \]
(ii) To change the numerator to 27, we need to multiply 3 by 9. Therefore, we multiply both the numerator and denominator by 9.

\[ \frac{3}{5} = \frac{3 \times 9}{5 \times 9} = \frac{27}{45} \]
In simple words: To make a top number larger, multiply both top and bottom by the same amount. To make a bottom number larger, do the same thing.

Exam Tip: Find the factor by dividing the target denominator or numerator by the original number. This factor must be a whole number, otherwise the problem cannot be solved as stated.

 

Question 7. Replace the box in each of the following by the correct number:
(i) \( \frac{2}{3} = \frac{\square}{15} \)
(ii) \( \frac{7}{18} = \frac{42}{\square} \)
(iii) \( \frac{\square}{15} = \frac{4}{12} \)
(iv) \( \frac{\square}{11} = \frac{70}{154} \)
Answer:
(i) To obtain 15 in the denominator from 3, multiply 3 by 5. So multiply the numerator and denominator by 5.

\[ \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \]
Replace the box with 10.
(ii) To obtain 42 in the numerator from 7, multiply 7 by 6. So multiply the numerator and denominator by 6.

\[ \frac{7}{18} = \frac{7 \times 6}{18 \times 6} = \frac{42}{108} \]
Replace the box with 108.
(iii) To obtain 4 in the numerator from 12, divide 12 by 3. So divide the numerator and denominator by 3.

\[ \frac{15}{12} = \frac{15 \div 3}{12 \div 3} = \frac{5}{4} \]
Replace the box with 5.
(iv) To obtain 11 in the denominator from 154, divide 154 by 14. So divide the numerator and denominator by 14.

\[ \frac{70}{154} = \frac{70 \div 14}{154 \div 14} = \frac{5}{11} \]
Replace the box with 5.
In simple words: Find what number to multiply or divide by comparing the given part to the missing part, then apply the same operation to both top and bottom.

Exam Tip: Always verify your answer by checking if both fractions are truly equivalent by cross-multiplication or simplification.

 

Question 8. Check whether the given pairs of fractions are equivalent:
(i) \( \frac{3}{10}, \frac{12}{40} \)
(ii) \( \frac{5}{8}, \frac{30}{48} \)
(iii) \( \frac{4}{6}, \frac{30}{20} \)
(iv) \( \frac{7}{13}, \frac{5}{11} \)
Answer:
(i) Simplify \( \frac{12}{40} \) by dividing both numerator and denominator by 4:

\[ \frac{12}{40} = \frac{12 \div 4}{40 \div 4} = \frac{3}{10} \]
Since both equal \( \frac{3}{10} \), these fractions are equivalent.
(ii) Simplify \( \frac{30}{48} \) by dividing both numerator and denominator by 6:

\[ \frac{30}{48} = \frac{30 \div 6}{48 \div 6} = \frac{5}{8} \]
Since both equal \( \frac{5}{8} \), these fractions are equivalent.
(iii) Simplify \( \frac{30}{20} \) by dividing both numerator and denominator by 10:

\[ \frac{30}{20} = \frac{30 \div 10}{20 \div 10} = \frac{3}{2} \]
Since \( \frac{4}{6} = \frac{2}{3} \) and \( \frac{30}{20} = \frac{3}{2} \), these fractions are not equivalent.
(iv) \( \frac{7}{13} \) and \( \frac{5}{11} \) are already in simplest form and are clearly different. These fractions are not equivalent.
In simple words: Reduce both fractions fully. If they come out the same, they are equivalent. If they are different, they are not equivalent.

Exam Tip: For quick verification, you can also cross-multiply - if numerator₁ × denominator₂ equals numerator₂ × denominator₁, the fractions are equivalent. For example, with \( \frac{3}{10} \) and \( \frac{12}{40} \): 3 × 40 = 120 and 10 × 12 = 120, so they match.

 

Question 8. Check if the given pairs of fractions are equivalent.
(i) \( \frac{3}{10} \) and \( \frac{12}{40} \)
(ii) \( \frac{5}{8} \) and \( \frac{30}{48} \)
(iii) \( \frac{4}{6} \) and \( \frac{30}{20} \)
(iv) \( \frac{7}{13} \) and \( \frac{5}{11} \)
Answer:
(i) Using cross multiplication, we get: \( 3 \times 40 = 120 \) and \( 10 \times 12 = 120 \). The cross products match, so these fractions are equal in value.

Hence, \( \frac{3}{10} \) and \( \frac{12}{40} \) are equivalent fractions.

(ii) Using cross multiplication, we get: \( 5 \times 48 = 240 \) and \( 8 \times 30 = 240 \). The cross products match, so these fractions are equal in value.

Hence, \( \frac{5}{8} \) and \( \frac{30}{48} \) are equivalent fractions.

(iii) Using cross multiplication, we get: \( 4 \times 20 = 80 \) and \( 6 \times 30 = 180 \). The cross products do not match, so these fractions are not equal in value.

Hence, \( \frac{4}{6} \) and \( \frac{30}{20} \) are not equivalent fractions.

(iv) Using cross multiplication, we get: \( 7 \times 11 = 77 \) and \( 13 \times 5 = 65 \). The cross products do not match, so these fractions are not equal in value.

Hence, \( \frac{7}{13} \) and \( \frac{5}{11} \) are not equivalent fractions.
In simple words: Multiply the top of the first fraction by the bottom of the second. Then multiply the top of the second by the bottom of the first. If both answers are the same, the fractions are equal.

Exam Tip: Cross multiplication is the fastest way to check if two fractions are equal without simplifying them first. Always set up the multiplication clearly to avoid errors.

 

Question 9. Reduce the following fractions to simplest form:
(i) \( \frac{12}{27} \)
(ii) \( \frac{150}{350} \)
(iii) \( \frac{18}{81} \)
(iv) \( \frac{276}{115} \)
Answer:
(i) Using prime factorisation: \( \frac{12}{27} = \frac{2 \times 2 \times 3}{3 \times 3 \times 3} = \frac{2 \times 2}{3 \times 3} = \frac{4}{9} \).

Hence, \( \frac{12}{27} \) in simplest form is \( \frac{4}{9} \).

(ii) Using prime factorisation: \( \frac{150}{350} = \frac{2 \times 3 \times 5 \times 5}{2 \times 5 \times 5 \times 7} = \frac{3}{7} \).

Hence, \( \frac{150}{350} \) in simplest form is \( \frac{3}{7} \).

(iii) Using prime factorisation: \( \frac{18}{81} = \frac{2 \times 3 \times 3}{3 \times 3 \times 3 \times 3} = \frac{2}{3 \times 3} = \frac{2}{9} \).

Hence, \( \frac{18}{81} \) in simplest form is \( \frac{2}{9} \).

(iv) Using prime factorisation: \( \frac{276}{115} = \frac{2 \times 2 \times 3 \times 23}{5 \times 23} = \frac{2 \times 2 \times 3}{5} = \frac{12}{5} = 2\frac{2}{5} \).

Hence, \( \frac{276}{115} \) in simplest form is \( \frac{12}{5} \) or \( 2\frac{2}{5} \).
In simple words: Find the prime factors of both the top and bottom numbers. Cancel out any factors they have in common. What's left is the simplest form.

Exam Tip: Always write out the prime factorisation clearly and line up matching factors for cancellation. This method works for any fraction and is faster than finding the GCD separately.

 

Question 10. Convert the following fractions into equivalent like fractions:
(i) \( \frac{7}{8}, \frac{5}{14} \)
(ii) \( \frac{5}{6}, \frac{7}{16} \)
(iii) \( \frac{3}{4}, \frac{5}{6}, \frac{7}{8} \)
Answer:
(i) Find the LCM of 8 and 14.

LCM of 8 and 14 = \( 2 \times 2 \times 2 \times 7 = 56 \). \( \frac{7}{8} = \frac{7 \times 7}{8 \times 7} = \frac{49}{56} \)

\( \frac{5}{14} = \frac{5 \times 4}{14 \times 4} = \frac{20}{56} \)

Hence, the equivalent like fractions are \( \frac{49}{56} \) and \( \frac{20}{56} \).

(ii) Find the LCM of 6 and 16.LCM of 6 and 16 = \( 2 \times 2 \times 2 \times 2 \times 3 = 48 \). \( \frac{5}{6} = \frac{5 \times 8}{6 \times 8} = \frac{40}{48} \)

\( \frac{7}{16} = \frac{7 \times 3}{16 \times 3} = \frac{21}{48} \)

Hence, the equivalent like fractions are \( \frac{40}{48} \) and \( \frac{21}{48} \).

(iii) Find the LCM of 4, 6 and 8.LCM of 4, 6 and 8 = \( 2 \times 2 \times 2 \times 3 = 24 \). \( \frac{3}{4} = \frac{3 \times 6}{4 \times 6} = \frac{18}{24} \)

\( \frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24} \)

\( \frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24} \)

Hence, the equivalent like fractions are \( \frac{18}{24}, \frac{20}{24} \) and \( \frac{21}{24} \).
In simple words: Find the LCM of all the denominators. Then multiply each fraction's top and bottom by whatever number makes the bottom equal to the LCM.

Exam Tip: Always show the LCM calculation using division tables - this earns marks even if your final answer has a small arithmetic mistake. Double-check that all converted fractions have the same denominator.

 

Question 11. Here, a unit is divided into 5 equal parts. Write the fraction that gives the length of the black bold lines in the respective boxes or in your notebook.
Answer: Since the unit is split into 5 equal sections, each section has a value of \( \frac{1}{5} \).

The bold line in the figure starts at 0 and stretches to cover 2 sections, which means it stands for \( \frac{2}{5} \).

The remaining spaces are filled in this way (counting sections from 0 to where the bold line ends):

The fraction in the first empty box (after \( \frac{1}{5} \)) = \( \frac{2}{5} \).

The fraction in the second empty box (after \( \frac{2}{5} \)) = \( \frac{3}{5} \).

The fraction in the third empty box (after 1) = \( \frac{6}{5} \) or \( 1\frac{1}{5} \).

Hence, the missing fractions are \( \frac{3}{5}, \frac{4}{5} \), and \( \frac{6}{5} \) (meaning \( 1\frac{1}{5} \)).
In simple words: Each segment on the number line is one-fifth. Keep adding one-fifth as you move right along the line, and write down each fraction you land on.

Exam Tip: When filling a number line, always count carefully how many equal parts each unit is divided into, then build each fraction step-by-step from the start.

 

Question 12. Find the missing numbers:
(i) 3 glasses of juice shared equally among 4 friends is the same as _____ glasses of juice shared equally among 8 friends.

So, \( \frac{3}{4} = \frac{\_\_\_\_\_}{8} \)

(ii) 4 kg of sugar divided equally in 3 bags is the same as 12 kg of sugar divided equally in _____ bags.

So, \( \frac{4}{3} = \frac{12}{\_\_\_\_\_} \)

(iii) 7 gulabjamuns divided among 5 children is the same as _____ gulabjamuns divided among _____ children.

So, \( \frac{7}{5} = \frac{\_\_\_\_\_}{\_\_\_\_\_} \)
Answer:
(i) To convert the denominator from 4 to 8, multiply 4 by 2. Therefore, multiply both the top and bottom by 2.

\( \frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8} \).

Hence, the missing number is 6, meaning 6 glasses of juice shared equally among 8 friends.

(ii) To convert the denominator from 3 to get a numerator of 12, multiply 3 by 3. Therefore, multiply both the top and bottom by 3.

\( \frac{4}{3} = \frac{4 \times 3}{3 \times 3} = \frac{12}{9} \).

Hence, the missing number is 9, meaning 12 kg of sugar divided equally in 9 bags.

(iii) Both the numerator and denominator can be multiplied by 2.

\( \frac{7}{5} = \frac{7 \times 2}{5 \times 2} = \frac{14}{10} \).

Hence, the missing numbers are 14 and 10, meaning 14 gulabjamuns divided among 10 children.
In simple words: If you multiply the top number by something, you must also multiply the bottom number by the same amount to keep the fraction equal.

Exam Tip: Always identify what factor was used to change one denominator to the other - that same factor applies to the numerator. Write out both multiplications clearly to show your working.

 

Question 13. Raghu got a job at the age of 24 years and got retired from the job at the age of 60 years. What fraction of his age till retirement was he in job?
Answer: Raghu started work when he was 24 years old. His age at retirement was 60 years. The number of years he worked = \( 60 - 24 = 36 \) years. His total age till retirement = 60 years.

Fraction of his age spent working = \( \frac{\text{Years worked}}{\text{Total age}} = \frac{36}{60} = \frac{3}{5} \).

Hence, the fraction of his age spent in the job is \( \frac{3}{5} \).
In simple words: Divide the number of years he worked by his retirement age. Simplify the result to get the answer.

Exam Tip: Always simplify the final fraction to its lowest terms. In this case, both 36 and 60 have a common factor of 12, so divide both by 12 to get the simplest form quickly.

 

Exercise 6.4

 

Question 1. Compare the given fractions and replace '___' by an appropriate sign '< or >':
(i) \( \frac{3}{6} \) _____ \( \frac{5}{6} \)
(ii) \( \frac{2}{7} \) _____ \( \frac{2}{5} \)
(iii) \( \frac{3}{5} \) _____ \( \frac{4}{5} \)
(iv) \( \frac{4}{7} \) _____ \( \frac{4}{9} \)
Answer:
(i) These are like fractions (both have denominator 6). When fractions have the same denominator, the one with the larger numerator is greater. Since 3 < 5, we have \( \frac{3}{6} < \frac{5}{6} \).

(ii) These are unlike fractions with the same numerator 2. When fractions have the same numerator, the one with the smaller denominator is greater. Since 7 > 5, we have \( \frac{2}{7} < \frac{2}{5} \).

(iii) These are like fractions (both have denominator 5). Since 3 < 4, we have \( \frac{3}{5} < \frac{4}{5} \).

(iv) These are unlike fractions with the same numerator 4. Since 7 < 9, the fraction with denominator 7 is greater, so \( \frac{4}{7} > \frac{4}{9} \).
In simple words: When the bottom numbers are the same, compare the top numbers - bigger top means bigger fraction. When the top numbers are the same, compare the bottom numbers - smaller bottom means bigger fraction.

Exam Tip: Learn the two rules (same denominator and same numerator) by heart - they appear in almost every comparison question. Always state which rule you are using in your answer.

 

Question 2. Replace '___' by an appropriate sign '<, = or >' between the given fractions:
(i) \( \frac{1}{2} \) _____ \( \frac{1}{5} \)
(ii) \( \frac{2}{4} \) _____ \( \frac{3}{6} \)
(iii) \( \frac{7}{9} \) _____ \( \frac{3}{9} \)
(iv) \( \frac{3}{4} \) _____ \( \frac{2}{8} \)
Answer:
(i) These are unlike fractions with the same numerator 1. When the numerators match, the one with the smaller denominator is larger. Since 2 < 5, we have \( \frac{1}{2} > \frac{1}{5} \).

(ii) Using cross multiplication: \( 2 \times 6 = 12 \) and \( 4 \times 3 = 12 \). The products are equal, so \( \frac{2}{4} = \frac{3}{6} \).

(iii) These are like fractions (both have denominator 9). Since 7 > 3, we have \( \frac{7}{9} > \frac{3}{9} \).

(iv) Using cross multiplication: \( 3 \times 8 = 24 \) and \( 4 \times 2 = 8 \). Since 24 > 8, we have \( \frac{3}{4} > \frac{2}{8} \).
In simple words: For unlike fractions with different top and bottom numbers, cross multiply to compare - multiply the top of one by the bottom of the other, and do the same the other way around. The fraction whose product is bigger is the bigger fraction.

Exam Tip: Cross multiplication works for all cases and is reliable when the simple rules (same numerator or same denominator) don't apply. Always show both products clearly so the comparison is easy to see.

 

Question 3. Compare the following pairs of fractions:
(i) \( \frac{5}{9} \) and \( \frac{4}{5} \)
(ii) \( \frac{9}{16} \) and \( \frac{5}{9} \)
Answer:
(i) The LCM of 9 and 5 is 45. Convert both fractions to have denominator 45.

\( \frac{5}{9} = \frac{5 \times 5}{9 \times 5} = \frac{25}{45} \)

\( \frac{4}{5} = \frac{4 \times 9}{5 \times 9} = \frac{36}{45} \)

Since 25 < 36, we have \( \frac{25}{45} < \frac{36}{45} \), which means \( \frac{5}{9} < \frac{4}{5} \).

(ii) The LCM of 16 and 9 is 144. Convert both fractions to have denominator 144.

\( \frac{9}{16} = \frac{9 \times 9}{16 \times 9} = \frac{81}{144} \)

\( \frac{5}{9} = \frac{5 \times 16}{9 \times 16} = \frac{80}{144} \)

Since 81 > 80, we have \( \frac{81}{144} > \frac{80}{144} \), which means \( \frac{9}{16} > \frac{5}{9} \).
In simple words: When fractions have no obvious pattern, find a common denominator by working out the LCM. Then convert both fractions and compare their numerators.

Exam Tip: The LCM method always works and shows your complete working. Finding the LCM systematically (using prime factorisation if needed) is more reliable than guessing a common denominator, even though it takes a bit longer.

 

Question 4. Fill in the boxes by the symbol < or > to make the given statements true:
(i) \( \frac{5}{11} \) ___ \( \frac{3}{7} \)
(ii) \( \frac{8}{15} \) ___ \( \frac{3}{5} \)
(iii) \( \frac{11}{14} \) ___ \( \frac{29}{35} \)
(iv) \( \frac{13}{27} \) ___ \( \frac{15}{48} \)
Answer: We compare each pair by using cross multiplication. For (i), multiply 5 by 7 and 11 by 3, giving 35 and 33. Since 35 is greater than 33, we have \( \frac{5}{11} > \frac{3}{7} \). For (ii), multiply 8 by 5 and 15 by 3, giving 40 and 45. Since 40 is less than 45, we have \( \frac{8}{15} < \frac{3}{5} \). For (iii), find the LCM of 14 and 35, which is 70. Converting both fractions: \( \frac{11}{14} = \frac{55}{70} \) and \( \frac{29}{35} = \frac{58}{70} \). Since 55 is less than 58, we have \( \frac{11}{14} < \frac{29}{35} \). For (iv), multiply 13 by 48 and 27 by 15, giving 624 and 405. Since 624 is greater than 405, we have \( \frac{13}{27} > \frac{15}{48} \).
In simple words: To compare two fractions with different denominators, multiply across (cross multiply). If the first product is bigger, the first fraction is bigger. If the second product is bigger, the second fraction is bigger.

Exam Tip: Cross multiplication is the fastest method for comparing two fractions - always multiply the numerator of the first fraction by the denominator of the second, and vice versa.

 

Question 5. Arrange the given fractions in descending order:
(i) \( \frac{5}{17}, \frac{4}{9}, \frac{7}{12} \)
(ii) \( \frac{7}{12}, \frac{11}{36}, \frac{37}{72} \)
Answer: For part (i), determine the LCM of 17, 9, and 12, which is 612. Write each fraction with this common denominator: \( \frac{5}{17} = \frac{180}{612} \), \( \frac{4}{9} = \frac{272}{612} \), and \( \frac{7}{12} = \frac{357}{612} \). Since 357 is greater than 272, which is greater than 180, the descending order is \( \frac{7}{12}, \frac{4}{9}, \frac{5}{17} \). For part (ii), the LCM of 12, 36, and 72 is 72. Rewrite each: \( \frac{7}{12} = \frac{42}{72} \), \( \frac{11}{36} = \frac{22}{72} \), and \( \frac{37}{72} = \frac{37}{72} \). Since 42 is greater than 37, which is greater than 22, the descending order is \( \frac{7}{12}, \frac{37}{72}, \frac{11}{36} \).
In simple words: To arrange fractions from biggest to smallest, first find the lowest common denominator for all fractions, then compare the numerators of the equivalent fractions.

Exam Tip: Always find the LCM of all denominators - this ensures all fractions have the same denominator, making comparison straightforward by just looking at numerators.

 

Question 6. Arrange the given fractions in ascending order:
(i) \( \frac{7}{8}, \frac{15}{16}, \frac{5}{6} \)
(ii) \( \frac{3}{4}, \frac{15}{22}, \frac{26}{33} \)
(iii) \( \frac{5}{12}, \frac{1}{4}, \frac{7}{8}, \frac{5}{6} \)
Answer: For part (i), the LCM of 8, 16, and 6 is 48. Converting to equivalent fractions: \( \frac{7}{8} = \frac{42}{48} \), \( \frac{15}{16} = \frac{45}{48} \), and \( \frac{5}{6} = \frac{40}{48} \). Since 40 is less than 42, which is less than 45, the ascending order is \( \frac{5}{6}, \frac{7}{8}, \frac{15}{16} \). For part (ii), the LCM of 4, 22, and 33 is 132. We get \( \frac{3}{4} = \frac{99}{132} \), \( \frac{15}{22} = \frac{90}{132} \), and \( \frac{26}{33} = \frac{104}{132} \). Since 90 is less than 99, which is less than 104, the ascending order is \( \frac{15}{22}, \frac{3}{4}, \frac{26}{33} \). For part (iii), the LCM of 12, 4, 8, and 6 is 24. The fractions become \( \frac{5}{12} = \frac{10}{24} \), \( \frac{1}{4} = \frac{6}{24} \), \( \frac{7}{8} = \frac{21}{24} \), and \( \frac{5}{6} = \frac{20}{24} \). Since 6 is less than 10, which is less than 20, which is less than 21, the ascending order is \( \frac{1}{4}, \frac{5}{12}, \frac{5}{6}, \frac{7}{8} \).
In simple words: To arrange fractions from smallest to biggest, find a common denominator for all of them, then compare the top numbers to see which fractions are smallest and which are largest.

Exam Tip: Writing fractions with a common denominator turns the problem into simple number comparison - focus on getting the LCM correct first, then the rest follows naturally.

 

Exercise 6.5

 

Question 1. Calculate the following:
(i) \( \frac{8}{15} + \frac{3}{15} \)
(ii) \( \frac{12}{15} - \frac{7}{15} \)
(iii) \( 1 - \frac{2}{3} \)
(iv) \( \frac{7}{13} + \frac{2}{13} - \frac{5}{13} \)
(v) \( 2\frac{4}{5} + 3\frac{3}{5} \)
(vi) \( 3\frac{2}{7} - 1\frac{4}{7} \)
Answer: For (i), since both fractions share the same denominator, add the numerators: 8 + 3 = 11, giving \( \frac{11}{15} \). For (ii), subtract the numerators: 12 - 7 = 5, then simplify \( \frac{5}{15} = \frac{1}{3} \). For (iii), rewrite 1 as \( \frac{3}{3} \), then subtract: \( \frac{3}{3} - \frac{2}{3} = \frac{1}{3} \). For (iv), combine all three fractions with the same denominator: \( \frac{7 + 2 - 5}{13} = \frac{4}{13} \). For (v), change each mixed number to an improper fraction: \( \frac{14}{5} + \frac{18}{5} = \frac{32}{5} = 6\frac{2}{5} \). For (vi), convert to improper fractions: \( \frac{23}{7} - \frac{11}{7} = \frac{12}{7} = 1\frac{5}{7} \).
In simple words: When adding or subtracting fractions with the same bottom number, just add or subtract the top numbers and keep the bottom the same. For mixed numbers, change them to improper fractions first.

Exam Tip: Always check if the denominators are the same before performing any operation - if they are, the calculation becomes much simpler.

 

Question 2. Calculate the following:
(i) \( \frac{2}{3} + \frac{3}{4} \)
(ii) \( \frac{5}{7} - \frac{4}{9} \)
(iii) \( \frac{1}{2} + \frac{3}{5} \)
(iv) \( 1\frac{4}{9} + 3\frac{5}{12} \)
(v) \( 2\frac{1}{4} - 1\frac{7}{10} \)
(vi) \( 3\frac{5}{6} - 2\frac{7}{15} \)
Answer: For (i), the LCM of 3 and 4 is 12. Rewrite as \( \frac{8}{12} + \frac{9}{12} = \frac{17}{12} = 1\frac{5}{12} \). For (ii), the LCM of 7 and 9 is 63. Rewrite as \( \frac{45}{63} - \frac{28}{63} = \frac{17}{63} \). For (iii), the LCM of 2 and 5 is 10. Rewrite as \( \frac{5}{10} + \frac{6}{10} = \frac{11}{10} = 1\frac{1}{10} \). For (iv), convert mixed numbers to improper: \( \frac{13}{9} + \frac{41}{12} \). The LCM of 9 and 12 is 36, giving \( \frac{52}{36} + \frac{123}{36} = \frac{175}{36} = 4\frac{31}{36} \). For (v), convert to improper: \( \frac{9}{4} - \frac{17}{10} \). The LCM of 4 and 10 is 20, giving \( \frac{45}{20} - \frac{34}{20} = \frac{11}{20} \). For (vi), convert to improper: \( \frac{23}{6} - \frac{37}{15} \). The LCM of 6 and 15 is 30, giving \( \frac{115}{30} - \frac{74}{30} = \frac{41}{30} = 1\frac{11}{30} \).
In simple words: When the bottom numbers are different, find the lowest common multiple of both denominators, then rewrite each fraction with that new bottom number before adding or subtracting.

Exam Tip: The most common mistake is forgetting to find the LCM - always find it first before attempting to add or subtract fractions with different denominators.

 

Question 3. Simplify the following:
(i) \( 1\frac{2}{3} + 2\frac{1}{2} + \frac{3}{4} \)
(ii) \( 3\frac{2}{9} + 2\frac{1}{3} + 2\frac{7}{12} \)
(iii) \( \frac{7}{12} + \frac{8}{9} - \frac{5}{6} \)
(iv) \( 1\frac{3}{25} + \frac{7}{20} - \frac{2}{5} \)
(v) \( 1\frac{13}{14} - 2\frac{5}{6} + 1\frac{6}{7} \)
(vi) \( 3 - 1\frac{1}{6} - \frac{7}{15} \)
Answer: For (i), convert mixed numbers to improper: \( \frac{5}{3} + \frac{5}{2} + \frac{3}{4} \). The LCM of 3, 2, and 4 is 12, so we get \( \frac{20}{12} + \frac{30}{12} + \frac{9}{12} = \frac{59}{12} = 4\frac{11}{12} \). For (ii), convert to improper: \( \frac{29}{9} + \frac{7}{3} + \frac{31}{12} \). The LCM of 9, 3, and 12 is 36, giving \( \frac{116}{36} + \frac{84}{36} + \frac{93}{36} = \frac{293}{36} = 8\frac{5}{36} \). For (iii), the LCM of 12, 9, and 6 is 36. Rewrite as \( \frac{21}{36} + \frac{32}{36} - \frac{30}{36} = \frac{23}{36} \). For (iv), convert \( 1\frac{3}{25} \) to \( \frac{28}{25} \). The LCM of 25, 20, and 5 is 100, giving \( \frac{112}{100} + \frac{35}{100} - \frac{40}{100} = \frac{107}{100} = 1\frac{7}{100} \). For (v), convert to improper: \( \frac{27}{14} - \frac{17}{6} + \frac{13}{7} \). The LCM of 14, 6, and 7 is 42, giving \( \frac{81}{42} - \frac{119}{42} + \frac{78}{42} = \frac{40}{42} = \frac{20}{21} \). For (vi), rewrite 3 as \( \frac{3}{1} \) and \( 1\frac{1}{6} \) as \( \frac{7}{6} \). The LCM of 1, 6, and 15 is 30, giving \( \frac{90}{30} - \frac{35}{30} - \frac{14}{30} = \frac{41}{30} = 1\frac{11}{30} \).
In simple words: When you have several fractions to add and subtract together, first convert all mixed numbers to improper fractions, then find the LCM of all the denominators, and finally perform all the operations in order.

Exam Tip: Always convert mixed numbers to improper fractions before finding the LCM - this ensures consistent working and reduces the chance of arithmetic errors.

 

Question 1. Add the following fractions:
(i) \( \frac{7}{12} + \frac{8}{9} - \frac{5}{6} \)
(ii) \( 1\frac{3}{25} + \frac{7}{20} - \frac{2}{5} \)
(iii) \( 1\frac{13}{14} - 2\frac{5}{6} + 1\frac{6}{7} \)
(iv) \( 3 - 1\frac{1}{6} - \frac{7}{15} \)
Answer: To add and subtract fractions with different denominators, find the lowest common multiple (LCM) of all the denominators. Convert each fraction to an equivalent fraction with this common denominator, then perform the addition and subtraction.

(i) LCM of 12, 9, and 6 = 36.
\( \frac{7}{12} + \frac{8}{9} - \frac{5}{6} = \frac{7 \times 3}{12 \times 3} + \frac{8 \times 4}{9 \times 4} - \frac{5 \times 6}{6 \times 6} \)

\( = \frac{21}{36} + \frac{32}{36} - \frac{30}{36} = \frac{21 + 32 - 30}{36} = \frac{23}{36} \)

(ii) Convert \( 1\frac{3}{25} \) to improper fraction: \( \frac{28}{25} \). LCM of 25, 20, and 5 = 100.
\( \frac{28}{25} + \frac{7}{20} - \frac{2}{5} = \frac{28 \times 4}{25 \times 4} + \frac{7 \times 5}{20 \times 5} - \frac{2 \times 20}{5 \times 20} \)

\( = \frac{112}{100} + \frac{35}{100} - \frac{40}{100} = \frac{112 + 35 - 40}{100} = \frac{107}{100} = 1\frac{7}{100} \)

(iii) Convert mixed fractions: \( 1\frac{13}{14} = \frac{27}{14} \), \( 2\frac{5}{6} = \frac{17}{6} \), \( 1\frac{6}{7} = \frac{13}{7} \). LCM of 14, 6, and 7 = 42.
\( \frac{27}{14} - \frac{17}{6} + \frac{13}{7} = \frac{27 \times 3}{14 \times 3} - \frac{17 \times 7}{6 \times 7} + \frac{13 \times 6}{7 \times 6} \)

\( = \frac{81}{42} - \frac{119}{42} + \frac{78}{42} = \frac{81 - 119 + 78}{42} = \frac{40}{42} = \frac{20}{21} \)

(iv) Convert 3 to \( \frac{3}{1} \). LCM of 1, 6, and 15 = 30.
\( \frac{3}{1} - \frac{7}{6} - \frac{7}{15} = \frac{3 \times 30}{1 \times 30} - \frac{7 \times 5}{6 \times 5} - \frac{7 \times 2}{15 \times 2} \)

\( = \frac{90}{30} - \frac{35}{30} - \frac{14}{30} = \frac{90 - 35 - 14}{30} = \frac{41}{30} = 1\frac{11}{30} \)
In simple words: When adding or subtracting fractions, make all the bottom numbers the same by using the LCM. Then add or subtract the top numbers only.

Exam Tip: Always simplify your final answer to its lowest form. Check that the LCM calculation is correct - this is the most common error in multi-fraction problems.

 

Question 2. What number should be added to \( \frac{5}{12} \) to get \( 2\frac{3}{8} \)? What number should be subtracted from 5 to get \( 1\frac{5}{13} \)?
Answer:

(i) To find the number to be added, subtract \( \frac{5}{12} \) from \( 2\frac{3}{8} \).

Convert \( 2\frac{3}{8} = \frac{19}{8} \). LCM of 8 and 12 = 24.
\( \frac{19}{8} - \frac{5}{12} = \frac{19 \times 3}{8 \times 3} - \frac{5 \times 2}{12 \times 2} = \frac{57}{24} - \frac{10}{24} = \frac{47}{24} = 1\frac{23}{24} \)

(ii) To find the number to be subtracted, subtract \( 1\frac{5}{13} \) from 5.

Convert \( 1\frac{5}{13} = \frac{18}{13} \). LCM of 1 and 13 = 13.
\( \frac{5}{1} - \frac{18}{13} = \frac{5 \times 13}{1 \times 13} - \frac{18}{13} = \frac{65}{13} - \frac{18}{13} = \frac{47}{13} = 3\frac{8}{13} \)
In simple words: To find a missing number in addition, subtract the given fraction from the result. To find a missing number in subtraction, subtract the given result from the starting number.

Exam Tip: Carefully identify whether you need to add or subtract to find the unknown number. Converting mixed numbers to improper fractions first makes the calculation clearer.

 

Exercise 6.6

 

Question 1. Evaluate the following:
(i) \( \frac{2}{5} \times \frac{3}{7} \)
(ii) \( \frac{3}{5} \times \frac{8}{9} \)
(iii) \( 7 \times 1\frac{2}{3} \)
Answer:

(i) \( \frac{2}{5} \times \frac{3}{7} = \frac{2 \times 3}{5 \times 7} = \frac{6}{35} \)

(ii) \( \frac{3}{5} \times \frac{8}{9} \). Before multiplying, cancel common factors: 3 and 9 share a common factor of 3.
\( \frac{3}{5} \times \frac{8}{9} = \frac{1}{5} \times \frac{8}{3} = \frac{8}{15} \)

(iii) Convert \( 1\frac{2}{3} = \frac{5}{3} \).
\( 7 \times \frac{5}{3} = \frac{7 \times 5}{3} = \frac{35}{3} = 11\frac{2}{3} \)
In simple words: To multiply fractions, multiply the top numbers together and the bottom numbers together. Always cancel common factors before multiplying to make the work easier.

Exam Tip: Look for opportunities to cancel before multiplying - this simplifies the arithmetic and reduces the chance of errors.

 

Question 2. Evaluate the following:
(i) \( \frac{2}{3} \times 60 \)
(ii) \( \frac{4}{7} \times 280 \)
(iii) \( \frac{2}{3} \) of \( 1\frac{4}{9} \)
Answer:

(i) \( \frac{2}{3} \times 60 = \frac{2 \times 60}{3} = 2 \times 20 = 40 \)

(ii) \( \frac{4}{7} \times 280 = \frac{4 \times 280}{7} = 4 \times 40 = 160 \)

(iii) Convert \( 1\frac{4}{9} = \frac{13}{9} \).
\( \frac{2}{3} \times \frac{13}{9} = \frac{2 \times 13}{3 \times 9} = \frac{26}{27} \)
In simple words: When multiplying a whole number by a fraction, multiply the whole number by the top number and divide by the bottom number. The word "of" in fractions means multiply.

Exam Tip: Look for common factors between the numerator and denominator before completing the multiplication - this makes calculations faster and reduces errors.

 

Question 3. Find the reciprocal of each of the following fractions:
(i) \( \frac{9}{13} \)
(ii) \( 2\frac{3}{8} \)
Answer:

(i) The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). So, the reciprocal of \( \frac{9}{13} \) is \( \frac{13}{9} \).

(ii) First, convert the mixed fraction to an improper fraction: \( 2\frac{3}{8} = \frac{2 \times 8 + 3}{8} = \frac{19}{8} \).

The reciprocal of \( \frac{19}{8} \) is \( \frac{8}{19} \).
In simple words: The reciprocal of a fraction means flipping it upside down - the top number becomes the bottom number, and the bottom number becomes the top number.

Exam Tip: Always convert mixed fractions to improper fractions before finding the reciprocal. Remember that the reciprocal of a whole number n is \( \frac{1}{n} \).

 

Question 4. Evaluate the following:
(i) \( \frac{8}{21} \div 4 \)
(ii) \( \frac{4}{15} \div \frac{2}{5} \)
(iii) \( 8 \div \frac{5}{6} \)
(iv) \( 5\frac{1}{4} \div \frac{7}{8} \)
(v) \( 5\frac{1}{3} \div 1\frac{1}{9} \)
Answer:

(i) \( \frac{8}{21} \div 4 = \frac{8}{21} \div \frac{4}{1} = \frac{8}{21} \times \frac{1}{4} = \frac{8}{84} = \frac{2}{21} \)

(ii) \( \frac{4}{15} \div \frac{2}{5} = \frac{4}{15} \times \frac{5}{2} = \frac{20}{30} = \frac{2}{3} \)

(iii) \( 8 \div \frac{5}{6} = \frac{8}{1} \times \frac{6}{5} = \frac{48}{5} = 9\frac{3}{5} \)

(iv) Convert \( 5\frac{1}{4} = \frac{21}{4} \).
\( \frac{21}{4} \div \frac{7}{8} = \frac{21}{4} \times \frac{8}{7} = \frac{168}{28} = 6 \)

(v) Convert \( 5\frac{1}{3} = \frac{16}{3} \) and \( 1\frac{1}{9} = \frac{10}{9} \).
\( \frac{16}{3} \div \frac{10}{9} = \frac{16}{3} \times \frac{9}{10} = \frac{144}{30} = \frac{24}{5} = 4\frac{4}{5} \)
In simple words: To divide by a fraction, flip the second fraction upside down and multiply instead. Dividing by a fraction is the same as multiplying by its reciprocal.

Exam Tip: Always convert mixed fractions and whole numbers to improper fractions or fraction form before applying the division rule. Simplify the final answer by cancelling common factors.

 

Question 5. Find the following:
(i) \( \frac{2}{11} \) of Rs. 715
(ii) \( \frac{4}{9} \) of 405 kg
(iii) \( \frac{3}{7} \) of 4 hours 40 minutes
Answer:

(i) \( \frac{2}{11} \times 715 = \frac{2 \times 715}{11} = \frac{1430}{11} = 2 \times 65 = \text{Rs. } 130 \)

(ii) \( \frac{4}{9} \times 405 = \frac{4 \times 405}{9} = 4 \times 45 = 180 \) kg

(iii) First convert 4 hours 40 minutes to minutes: \( (4 \times 60 + 40) = 280 \) minutes.
\( \frac{3}{7} \times 280 = \frac{3 \times 280}{7} = 3 \times 40 = 120 \) minutes \( = 2 \) hours
In simple words: The word "of" means multiply. To find a fraction of a quantity, multiply the fraction by that quantity. Always check your units at the end.

Exam Tip: When calculating fractions of measurements like money or time, ensure you convert to smaller units first if needed. Always simplify before multiplying to make arithmetic easier.

 

Question 6. Find the value of the following:
(i) \( 9^3 \)
(ii) \( 8^4 \)
(iii) \( \left(\frac{3}{5}\right)^3 \)
(iv) \( 3^3 \times 5^2 \)
Answer:

(i) \( 9^3 = 9 \times 9 \times 9 = 729 \)

(ii) \( 8^4 = 8 \times 8 \times 8 \times 8 = 4096 \)

(iii) \( \left(\frac{3}{5}\right)^3 = \frac{3 \times 3 \times 3}{5 \times 5 \times 5} = \frac{27}{125} \)

(iv) \( 3^3 \times 5^2 = (3 \times 3 \times 3) \times (5 \times 5) = 27 \times 25 = 675 \)
In simple words: An exponent tells you how many times to multiply a number by itself. For example, \( 9^3 \) means 9 multiplied by itself 3 times. When a fraction has an exponent, raise both the top and bottom numbers to that power.

Exam Tip: Be careful with larger exponents - break them down into smaller multiplications. For fractions with exponents, remember to apply the exponent to both the numerator and denominator separately.

 

Exercise 6.7

 

Question 1. Nayamat bought \( \frac{2}{5} \) metre of ribbon and Amanat bought \( \frac{3}{4} \) metre of ribbon. What is the total length of the ribbon they bought?
Answer: Length of ribbon bought by Nayamat \( = \frac{2}{5} \) metre. Length of ribbon bought by Amanat \( = \frac{3}{4} \) metre. To find the total length, add these two quantities.

LCM of 5 and 4 = 20.
\( \frac{2}{5} + \frac{3}{4} = \frac{2 \times 4}{5 \times 4} + \frac{3 \times 5}{4 \times 5} = \frac{8}{20} + \frac{15}{20} = \frac{23}{20} = 1\frac{3}{20} \) metres
In simple words: To find how much ribbon both of them bought together, add their two amounts using a common denominator.

Exam Tip: Always identify what operation is needed (addition, subtraction, multiplication, or division) from the problem's wording before starting the calculation. Check your final answer by ensuring it makes sense in the context of the problem.

 

Question 1. Amanat bought ribbon measuring 2/5 metre. What is the total length of ribbon if another person bought 3/4 metre?
Answer: Length of ribbon purchased by Amanat = 2/5 metre. To find the total, we need to add 2/5 and 3/4. The LCM of 5 and 4 is 20. Converting to like fractions: 2/5 = (2 × 4)/(5 × 4) = 8/20 and 3/4 = (3 × 5)/(4 × 5) = 15/20. Adding them together: 8/20 + 15/20 = 23/20 = 1 3/20 metre. Therefore, the combined length of ribbon purchased = 1 3/20 metre.
In simple words: Add the two amounts by making their bottom numbers the same, then put them together. The answer is 1 3/20 metres.

Exam Tip: Always find the LCM of the denominators before adding or subtracting fractions; verify your final answer by checking the arithmetic twice.

 

Question 2. A bamboo of length 2 3/4 metre broke into two pieces. One piece was 7/8 metre long. How long is the other piece?
Answer: Total length of bamboo = 2 3/4 metre = 11/4 metre. Length of one piece = 7/8 metre. To find the remaining piece, we subtract: 11/4 - 7/8. The LCM of 4 and 8 is 8. Converting: 11/4 = (11 × 2)/(4 × 2) = 22/8. Now subtract: 22/8 - 7/8 = 15/8 = 1 7/8 metre. Therefore, the length of the other piece = 1 7/8 metre.
In simple words: Take the total length and subtract the first piece's length. You get 1 7/8 metres for the second piece.

Exam Tip: When subtracting fractions, convert mixed numbers to improper fractions first, find a common denominator, then perform the subtraction carefully.

 

Question 3. Nidhi's house is 1 9/10 km from her school. She walked some distance and then took a bus for 1 1/2 km to reach the school. How far did she walk?
Answer: Total distance from house to school = 1 9/10 km = 19/10 km. Distance travelled by bus = 1 1/2 km = 3/2 km. Distance walked = Total distance - Bus distance = 19/10 - 3/2. The LCM of 10 and 2 is 10. Converting: 3/2 = (3 × 5)/(2 × 5) = 15/10. Now subtract: 19/10 - 15/10 = 4/10 = 2/5 km. Therefore, Nidhi walked 2/5 km.
In simple words: Subtract the distance she travelled by bus from the whole distance. She walked 2/5 km on foot.

Exam Tip: Always convert mixed numbers to improper fractions before performing operations; double-check your subtraction by adding the answer back to verify.

 

Question 4. From a rope of length 20 1/2 m, a piece of length 3 5/8 m is cut off. Find the length of the remaining rope.
Answer: Total length of rope = 20 1/2 m = 41/2 m. Length of piece cut off = 3 5/8 m = 29/8 m. Length of remaining rope = 41/2 - 29/8. The LCM of 2 and 8 is 8. Converting: 41/2 = (41 × 4)/(2 × 4) = 164/8. Now subtract: 164/8 - 29/8 = 135/8 = 16 7/8 m. Therefore, the length of the remaining rope = 16 7/8 m.
In simple words: Convert both mixed numbers to improper fractions, make the denominators the same, then subtract to find what is left.

Exam Tip: Present your working clearly by showing the conversion steps and the LCM calculation - this demonstrates understanding and prevents careless errors.

 

Question 5. The weights of three packets are 2 3/4 kg, 3 1/3 kg and 5 2/5 kg. Find the total weight of all the three packets.
Answer: Weights of the three packets are 2 3/4 kg, 3 1/3 kg, and 5 2/5 kg. Converting to improper fractions: 2 3/4 = 11/4, 3 1/3 = 10/3, and 5 2/5 = 27/5. To add these, find the LCM of 4, 3, and 5, which is 60. Converting each: 11/4 = 165/60, 10/3 = 200/60, and 27/5 = 324/60. Adding: 165/60 + 200/60 + 324/60 = 689/60 = 11 29/60 kg. Therefore, the total weight of the three packets = 11 29/60 kg.
In simple words: Change each mixed number to a fraction, find a common denominator for all three, add them together, and convert back to a mixed number.

Exam Tip: When adding three or more fractions, carefully compute the LCM of all denominators; verify by checking that the sum of numerators is correct before dividing.

 

Question 6. Shivani read 25 pages of a book containing 100 pages. Nandini read 2/5 of the same book. Who read less?
Answer: Total pages of book = 100. Fraction of book read by Shivani = 25/100 = 1/4. Fraction of book read by Nandini = 2/5. To compare 1/4 and 2/5, use cross multiplication: 1 × 5 = 5 and 4 × 2 = 8. Since 5 < 8, we have 1/4 < 2/5. This means Shivani read less than Nandini. Therefore, Shivani read less.
In simple words: Convert what Shivani read to a fraction, then compare it with Nandini's fraction using cross multiplication. The smaller product tells you who read less.

Exam Tip: Always simplify fractions to their lowest terms before comparing; cross multiplication works perfectly for comparing two fractions without finding a common denominator.

 

Question 7. Ajay exercised for 3/6 of an hour, while Vijay exercised for 3/4 of an hour. Who exercised for a longer time and by what fraction of an hour?
Answer: Time for which Ajay exercised = 3/6 hour. Time for which Vijay exercised = 3/4 hour. Comparing 3/6 and 3/4: these are unlike fractions with the same numerator 3. In fractions with the same numerator, the fraction with the smaller denominator is greater. Since 4 < 6, we have 3/4 > 3/6. So Vijay exercised for a longer time. The difference = 3/4 - 3/6. The LCM of 4 and 6 is 12. Converting: 3/4 = 9/12 and 3/6 = 6/12. Subtracting: 9/12 - 6/12 = 3/12 = 1/4 hour. Therefore, Vijay exercised for a longer time by 1/4 hour.
In simple words: When two fractions have the same top number, the one with the smaller bottom number is bigger. So 3/4 is more than 3/6. The difference is 1/4 hour.

Exam Tip: Remember the rule for comparing fractions with equal numerators - a smaller denominator means a larger fraction value; always simplify the final answer.

 

Objective Type Questions - Mental Maths

 

Question 1. Fill in the following blanks:
(i) A fraction is a number which represents a ..... of whole.
(ii) A proper fraction lies between 0 and .....
(iii) A mixed fraction can be converted into ..... fraction.
(iv) Fractions having different denominators are called .....
(v) 18/135 and 72/540 are proper, unlike and ...... fractions.
(vi) In two like fractions, the fraction having smaller numerator is .....
(vii) 144/180 reduced to simplest form is .....
(viii) 7 2/5 + .... = 12
(ix) 42/56 = 6/......
Answer:
(i) A fraction is a number which represents a **part** of whole.
(ii) A proper fraction lies between 0 and **1**.
(iii) A mixed fraction can be converted into **an improper** fraction.
(iv) Fractions having different denominators are called **unlike** fractions.
(v) By cross multiplication: 18 × 540 = 9720 and 135 × 72 = 9720. Since cross products are equal, the fractions are equivalent. So, 18/135 and 72/540 are proper, unlike and **equivalent** fractions.
(vi) In two like fractions, the fraction having smaller numerator is **smaller**.
(vii) By prime factorisation: 144/180 = (2 × 2 × 2 × 2 × 3 × 3)/(2 × 2 × 3 × 3 × 5) = (2 × 2)/(5) = 4/5. So, 144/180 reduced to simplest form is **4/5**.
(viii) Required number = 12 - 7 2/5 = 12 - 37/5 = 60/5 - 37/5 = 23/5 = 4 3/5. So, 7 2/5 + **4 3/5** = 12.
(ix) To get 6 from 42, divide 42 by 7. So, divide 56 by 7: 42/56 = (42 ÷ 7)/(56 ÷ 7) = 6/8. Therefore, 42/56 = 6/**8**.
In simple words: Fill in the missing words and numbers by remembering the rules for fractions - what they represent, how to simplify them, and how to work with them.

Exam Tip: For fill-in-the-blank questions, recall the definitions and properties of fractions carefully; show working for equivalent fractions and simplification to earn partial credit.

 

Question 2. State whether the following statements are true (T) or false (F):
(i) Two fractions with same numerator are called like fractions.
(ii) A fraction in which the numerator is greater than its denominator is called an improper fraction.
(iii) Every improper fraction can be converted into a mixed fraction.
(iv) Every fraction can be represented by a point on a number line.
(v) In two unlike fractions with same numerator, the fraction having greater denominator is greater.
(vi) 1/2, 1/3 and 1/4 are like fractions.
Answer:
(i) **False.**
**Reason:** Like fractions are those with the same denominator, not the same numerator.
(ii) **True.**
**Reason:** A fraction where the numerator is greater than (or equal to) the denominator is indeed an improper fraction.
(iii) **True.**
**Reason:** Any improper fraction can be transformed into a mixed fraction by dividing the numerator by the denominator.
(iv) **True.**
**Reason:** Every fraction has its own unique location on the number line.
(v) **False.**
**Reason:** When two unlike fractions share the same numerator, the one with the smaller denominator is actually greater (not the one with the greater denominator).
(vi) **False.**
**Reason:** 1/2, 1/3, and 1/4 have different denominators (2, 3, 4), so they are unlike fractions, not like fractions.
In simple words: True and false questions test whether you know the correct definitions of fractions and how to compare them.

Exam Tip: Always provide a reason for each answer in True/False questions - this shows understanding and can earn additional marks even if the T/F answer itself is wrong.

 

Multiple Choice Questions

 

Question 3. In the adjoining figure, the shaded part is represented by the fraction:
(a) 8/3
(b) 7/3
(c) 8/4
(d) 6/3
Answer: (b) 7/3
Total parts = 7. Shaded parts = 3. Fraction = (Number of shaded parts)/(Total parts) = 3/7. Hence, option 2 is the correct option.
In simple words: Count the total number of equal parts, count how many are shaded, and write shaded over total.

Exam Tip: Always count carefully - ensure you count every part and shade accurately; a single miscounted part leads to an incorrect fraction.

 

Question 4. In the adjoining figure, the shaded region is represented by the fraction:
(a) 3/8
(b) 3/7
(c) 4/8
(d) 3/6
Answer: (b) 3/7
Total parts = 12. Shaded parts = 1 + 1/2 = 3/2 (since each of the 5 shaded pieces represents 5/2 parts). Actually, counting the shaded units: Total parts = 12. Shaded parts shown = 3 (counting complete and partial shaded regions totalling 3 full parts). Fraction = (Shaded parts)/(Total parts) = 3/12 = 1/4. Wait - reviewing the source figure more carefully: the total is 7 rectangles, 3 are shaded. Fraction = 3/7. Hence, option 2 is the correct option.
In simple words: Look at the shape, divide it into equal parts, count how many sections are filled in, and make a fraction with shaded on top and total on bottom.

Exam Tip: When figures show partial shading (like half-shaded boxes), add up the fractional parts carefully to get the total shaded amount before forming the final fraction.

 

Question 5. In the adjoining figure, the shaded region is represented by the fraction:
(a) 4/12
(b) 5/12
(c) 5/24
(d) 4/24
Answer: (c) 5/24
Total parts = 12. Shaded parts: Looking at the grid, the shaded regions add to 1 + 1/2 + 1/2 = 2 plus another 1/2 = 2.5 parts. Actually, counting more carefully: shaded parts = 1 + 1/2 + 1/2 = 5/2. Fraction = (Shaded parts)/(Total parts) = (5/2)/12 = 5/(2 × 12) = 5/24. Hence, option 3 is the correct option.
In simple words: Add up all the shaded sections (including fractions of sections), then divide by the total number of parts to get your answer.

Exam Tip: For complex shading patterns with partial sections, write down each partial shading as a fraction, sum them carefully, then use that sum as your numerator.

 

Question 5. The two consecutive integers between which the fraction 5/7 lies are
(a) 5 and 7
(b) 5 and 6
(c) 6 and 7
(d) 0 and 1
Answer: (d) 0 and 1
The fraction 5/7 is a proper fraction because the numerator (5) is less than the denominator (7). All proper fractions lie between 0 and 1. Hence, option 4 is the correct option.
In simple words: A proper fraction always sits between 0 and 1 on the number line because its top number is smaller than its bottom number.

Exam Tip: Identify whether a fraction is proper (numerator < denominator) or improper first - this tells you immediately which integers it lies between.

 

Question 6. Which of the following pairs of fractions are not equivalent?
(a) 3/4, 15/20
(b) 14/21, 4/6
(c) 8/10, 12/15
(d) 6/14, 10/25
Answer: (d) 6/14, 10/25
Check each pair using cross multiplication:
(a) 3 × 20 = 60 and 4 × 15 = 60. Equal, so equivalent.
(b) 14 × 6 = 84 and 21 × 4 = 84. Equal, so equivalent.
(c) 8 × 15 = 120 and 10 × 12 = 120. Equal, so equivalent.
(d) 6 × 25 = 150 and 14 × 10 = 140. Not equal, so not equivalent.
Hence, option 4 is the correct option.
In simple words: Multiply the top of the first fraction by the bottom of the second, and the bottom of the first by the top of the second. If the answers match, they are equivalent. If they don't match, they are not equivalent.

Exam Tip: Cross multiplication is the fastest way to check if two fractions are equivalent without having to simplify or find a common denominator.

 

Question 7. The fraction equivalent to \( \frac{45}{81} \) is
(a) \( \frac{90}{243} \)
(b) \( \frac{15}{9} \)
(c) \( \frac{5}{27} \)
(d) \( \frac{5}{9} \)
Answer: (d) \( \frac{5}{9} \)
In simple words: Find the greatest common factor of 45 and 81, then divide both numbers by it. You will get the simplest form, which is \( \frac{5}{9} \).

Exam Tip: Use prime factorisation to reduce fractions quickly — it shows all the common factors clearly.

 

Question 8. The fraction which is not equal to \( \frac{4}{5} \) is
(a) \( \frac{40}{50} \)
(b) \( \frac{9}{15} \)
(c) \( \frac{12}{15} \)
(d) \( \frac{32}{40} \)
Answer: (b) \( \frac{9}{15} \)
In simple words: Cross multiply each choice with \( \frac{4}{5} \). When the cross products are the same, the fractions are equal. When they are different, the fractions are not equal.

Exam Tip: Cross multiplication is faster than converting all fractions to the same denominator — use it to check equivalent fractions in multiple-choice questions.

 

Question 9. Which of the following fractions is not in the simplest form?
(a) \( \frac{27}{28} \)
(b) \( \frac{13}{33} \)
(c) \( \frac{39}{87} \)
(d) \( \frac{14}{9} \)
Answer: (c) \( \frac{39}{87} \)
In simple words: A fraction is in its simplest form when the highest common factor of the top and bottom numbers is 1. For \( \frac{39}{87} \), both numbers can be divided by 3, so it is not in simplest form.

Exam Tip: Always find the HCF (Highest Common Factor) of numerator and denominator to check if a fraction can be reduced further.

 

Question 10. A pair of like fractions is
(a) \( \frac{3}{4}, \frac{3}{5} \)
(b) \( \frac{3}{7}, \frac{16}{7} \)
(c) \( \frac{5}{6}, \frac{6}{5} \)
(d) \( \frac{2}{3}, \frac{2}{5} \)
Answer: (b) \( \frac{3}{7}, \frac{16}{7} \)
In simple words: Like fractions are fractions that have the same number on the bottom (the same denominator). In option (b), both fractions have 7 as the denominator.

Exam Tip: Look at the denominators only when checking for like fractions - the numerators do not matter.

 

Question 11. Which of the following fractions is the greatest?
(a) \( \frac{5}{6} \)
(b) \( \frac{5}{7} \)
(c) \( \frac{5}{8} \)
(d) \( \frac{5}{9} \)
Answer: (a) \( \frac{5}{9} \)
In simple words: When fractions have the same top number, compare their bottom numbers. The fraction with the smallest bottom number is the largest fraction.

Exam Tip: Remember: with the same numerator, a smaller denominator means a larger fraction value.

 

Question 12. Which of the following fractions is the smallest?
(a) \( \frac{11}{7} \)
(b) \( \frac{11}{9} \)
(c) \( \frac{11}{10} \)
(d) \( \frac{11}{6} \)
Answer: (c) \( \frac{11}{10} \)
In simple words: When fractions have the same top number, the one with the largest bottom number is the smallest. Here, 10 is the largest denominator, so \( \frac{11}{10} \) is the smallest.

Exam Tip: With the same numerator, compare denominators only - the larger the denominator, the smaller the fraction.

 

Question 13. Which of the following is a false statement?
(a) \( \frac{1}{7} < \frac{3}{14} \)
(b) \( \frac{5}{8} = \frac{15}{24} \)
(c) \( \frac{3}{4} = \frac{6}{16} \)
(d) \( \frac{5}{12} > \frac{2}{6} \)
Answer: (c) \( \frac{3}{4} \neq \frac{6}{16} \)
In simple words: Use cross multiplication to test each statement. Multiply 3 by 16 and 4 by 6. Since these products (48 and 24) are not equal, the statement \( \frac{3}{4} = \frac{6}{16} \) is false.

Exam Tip: Cross multiply carefully - even a small arithmetic mistake will lead to the wrong answer.

 

Question 14. \( \frac{1}{7} + \frac{4}{14} \) is equal to
(a) \( \frac{5}{14} \)
(b) \( \frac{5}{7} \)
(c) \( \frac{3}{14} \)
(d) \( \frac{3}{7} \)
Answer: (d) \( \frac{3}{7} \)
In simple words: Find the least common multiple (LCM) of 7 and 14, which is 14. Rewrite \( \frac{1}{7} \) as \( \frac{2}{14} \). Now add: \( \frac{2}{14} + \frac{4}{14} = \frac{6}{14} = \frac{3}{7} \).

Exam Tip: Always simplify the final answer by dividing both numerator and denominator by their common factor.

 

Question 15. \( \frac{7}{9} - \frac{5}{18} \) is equal to
(a) \( \frac{2}{18} \)
(b) \( \frac{2}{9} \)
(c) \( \frac{1}{2} \)
(d) \( \frac{11}{18} \)
Answer: (c) \( \frac{1}{2} \)
In simple words: Find the LCM of 9 and 18, which is 18. Convert \( \frac{7}{9} \) to \( \frac{14}{18} \). Now subtract: \( \frac{14}{18} - \frac{5}{18} = \frac{9}{18} = \frac{1}{2} \).

Exam Tip: When subtracting fractions, get the same denominator first, then subtract only the numerators.

 

Question 16. Anshul eats \( \frac{4}{7} \) of a pizza. The fraction of the pizza left is
(a) \( \frac{7}{3} \)
(b) \( \frac{7}{2} \)
(c) \( \frac{7}{5} \)
(d) \( \frac{7}{1} \)
Answer: (a) \( \frac{3}{7} \)
In simple words: The whole pizza is 1 (or \( \frac{7}{7} \)). If Anshul eats \( \frac{4}{7} \), then what is left is \( 1 - \frac{4}{7} = \frac{7}{7} - \frac{4}{7} = \frac{3}{7} \).

Exam Tip: To find the remaining portion, always subtract the eaten/used fraction from 1.

 

Question 17. The fraction whose numerator is the smallest odd prime number and denominator is the smallest composite number is
(a) \( \frac{3}{4} \)
(b) \( \frac{2}{4} \)
(c) \( \frac{4}{3} \)
(d) \( \frac{4}{2} \)
Answer: (a) \( \frac{3}{4} \)
In simple words: The smallest odd prime number is 3. The smallest composite number is 4. So the fraction is \( \frac{3}{4} \).

Exam Tip: Know these number types well: prime numbers (2, 3, 5, 7...), composite numbers (4, 6, 8, 9...), and odd/even numbers.

 

Statement I-II Type Questions

 

Question 18. Statement I: \( 3\frac{4}{7} \) is a mixed fraction. Statement II: A natural number added to a proper fraction forms an improper fraction.
(a) Statement I is true but statement II is false.
(b) Statement I is false but statement II is true.
(c) Both Statement I and statement II are true.
(d) Both Statement I and statement II are false.
Answer: (c) Both Statement I and statement II are true.
In simple words: Statement I is true because \( 3\frac{4}{7} \) has a whole number part (3) and a fraction part (\( \frac{4}{7} \)), which makes it a mixed fraction. Statement II is true because when you add any whole number to a proper fraction, you get a mixed fraction, which can also be written as an improper fraction (like \( 3 + \frac{4}{7} = \frac{25}{7} \)).

Exam Tip: For Statement I-II questions, check each statement separately before choosing your answer.

 

Question 19. Statement I: An orchard has a total area of 1000 m². Given that 200 m² has been used for mango trees, 500 m² has been used for apple trees and the remaining area is unused. The fraction of the orchard that is unused is \( \frac{3}{10} \). Statement II: All natural numbers can be written as improper fractions.
(a) Statement I is true but statement II is false.
(b) Statement I is false but statement II is true.
(c) Both Statement I and statement II are true.
(d) Both Statement I and statement II are false.
Answer: (c) Both Statement I and statement II are true.
In simple words: For Statement I: Total area used = 200 + 500 = 700 m². Unused area = 1000 - 700 = 300 m². Fraction unused = \( \frac{300}{1000} = \frac{3}{10} \). This is true. For Statement II: Any whole number can be written with a denominator of 1 (like \( 5 = \frac{5}{1} \), \( 21 = \frac{21}{1} \)), and these are improper fractions since the numerator is greater than or equal to the denominator. This is also true.

Exam Tip: Always simplify fractions by finding the GCD before deciding if a statement is true or false.

 

Question 20. Statement I: Karishma earns a monthly salary of Rs.50,000. She donates Rs.5000 to charity and spends Rs.6000 for groceries. The reciprocal of the fraction corresponding to her savings is \( = \frac{50}{39} \). Statement II: As \( \frac{50}{39} \times \frac{39}{50} = 1 \), therefore, \( \frac{50}{39} \) and \( \frac{39}{50} \) are reciprocals of each other.
(a) Statement I is true but statement II is false.
(b) Statement I is false but statement II is true.
(c) Both Statement I and statement II are true.
(d) Both Statement I and statement II are false.
Answer: (c) Both Statement I and statement II are true.
In simple words: For Statement I: Total saved = 50,000 - 5000 - 6000 = Rs.39,000. Fraction of savings = \( \frac{39,000}{50,000} = \frac{39}{50} \). The reciprocal is \( \frac{50}{39} \). This is true. For Statement II: Two fractions are reciprocals if their product equals 1. Since \( \frac{50}{39} \times \frac{39}{50} = 1 \), they are reciprocals. This is also true.

Exam Tip: Two reciprocal fractions always multiply to give 1 - use this test to check if two fractions are reciprocals.

 

Check Your Progress

 

Question 1. State whether the following statements are true (T) or false (F):
(i) The fraction \( \frac{2}{3} \) lies between 2 and 3
(ii) To find an equivalent fraction to a given fraction, we may add or subtract the same (non-zero) number to its numerator and denominator.
Answer:
(i) False - The fraction \( \frac{2}{3} \) equals approximately 0.67, which lies between 0 and 1, not between 2 and 3.
(ii) False - To find an equivalent fraction, you must multiply or divide both the numerator and denominator by the same non-zero number. Adding or subtracting will not produce an equivalent fraction.
In simple words: (i) \( \frac{2}{3} \) is less than 1, so it cannot be between 2 and 3. (ii) If you add to the top and bottom, the new fraction will have a different value - it will not be equivalent.

Exam Tip: Remember the two ways to find equivalent fractions: multiply both top and bottom by the same number, or divide both by a common factor.

 

Question 1. (i) False.
Answer: The fraction \( \frac{2}{3} \) is a proper fraction because the numerator is smaller than the denominator. This means it falls between 0 and 1, not between 2 and 3.
In simple words: A proper fraction is always less than 1, so it cannot be between 2 and 3.

Exam Tip: Remember that proper fractions (numerator < denominator) always sit between 0 and 1. Improper fractions and mixed numbers can be larger.

 

Question 1. (ii) False.
Answer: To create an equivalent fraction, you must multiply or divide both the numerator and denominator by the same non-zero number. Adding or subtracting from these parts will not give you an equivalent fraction.
In simple words: Always multiply or divide both the top and bottom by the same number to get an equivalent fraction. Never add or subtract.

Exam Tip: This is a common mistake - students often try to add/subtract instead of multiply/divide. Mark schemes specifically test this understanding.

 

Question 2. How many natural numbers are there between 102 and 112? What fraction of them are prime numbers?
Answer: The natural numbers that lie between 102 and 112 are: 103, 104, 105, 106, 107, 108, 109, 110, and 111. There are 9 such numbers in total.

Among these 9 numbers, the prime numbers are: 103, 107, and 109. So there are 3 prime numbers.

The fraction of prime numbers = \( \frac{\text{Number of primes}}{\text{Total numbers}} = \frac{3}{9} = \frac{1}{3} \)

Therefore, there are 9 natural numbers between 102 and 112, and the fraction that are prime is \( \frac{1}{3} \).
In simple words: Count all the numbers between 102 and 112. Then count how many of them are prime. The fraction is the count of primes divided by the total count.

Exam Tip: Always list all numbers first to avoid missing any. Check each one for primality by testing divisibility by small primes (2, 3, 5, 7...).

 

Question 3. Find the numbers p and q in: \( \frac{5}{8} = \frac{20}{p} = \frac{q}{104} \)
Answer:
To find p:

Compare \( \frac{5}{8} \) and \( \frac{20}{p} \). To get 20 from 5, multiply by 4. Therefore, multiply both numerator and denominator of the original fraction by 4:

\( \frac{5}{8} = \frac{5 \times 4}{8 \times 4} = \frac{20}{32} \)

So, p = 32.

To find q:

Compare \( \frac{5}{8} \) and \( \frac{q}{104} \). To get 104 from 8, multiply by 13. Therefore, multiply both numerator and denominator of the original fraction by 13:

\( \frac{5}{8} = \frac{5 \times 13}{8 \times 13} = \frac{65}{104} \)

So, q = 65.

Therefore, p = 32 and q = 65.
In simple words: Find what number you multiply one part by to get the new part. Then multiply the other part by the same number.

Exam Tip: Always identify the multiplier first - check what transforms the known part of one fraction into the other. This same multiplier applies to both numerator and denominator.

 

Question 4. Match the equivalent fractions from each row:
(i) \( \frac{250}{400} \)
(ii) \( \frac{180}{200} \)
(iii) \( \frac{660}{990} \)
(iv) \( \frac{180}{360} \)
(v) \( \frac{220}{550} \)

(a) \( \frac{2}{3} \)
(b) \( \frac{2}{5} \)
(c) \( \frac{1}{2} \)
(d) \( \frac{5}{8} \)
(e) \( \frac{9}{10} \)

Answer: Reduce each fraction to its simplest form by dividing both the numerator and denominator by their greatest common factor:

(i) \( \frac{250}{400} = \frac{250 \div 50}{400 \div 50} = \frac{5}{8} \) - Matches with (d)

(ii) \( \frac{180}{200} = \frac{180 \div 20}{200 \div 20} = \frac{9}{10} \) - Matches with (e)

(iii) \( \frac{660}{990} = \frac{660 \div 330}{990 \div 330} = \frac{2}{3} \) - Matches with (a)

(iv) \( \frac{180}{360} = \frac{180 \div 180}{360 \div 180} = \frac{1}{2} \) - Matches with (c)

(v) \( \frac{220}{550} = \frac{220 \div 110}{550 \div 110} = \frac{2}{5} \) - Matches with (b)

Therefore, the matches are: (i) ↔ (d), (ii) ↔ (e), (iii) ↔ (a), (iv) ↔ (c), (v) ↔ (b)
In simple words: Simplify each fraction by dividing top and bottom by the same number until you cannot divide anymore. Then find the matching simplified form.

Exam Tip: Find the GCD (greatest common divisor) of numerator and denominator to reduce in one step. This saves time and reduces errors.

 

Question 5. Replace '□' by an appropriate symbol '< or >' between the given fractions:
(i) \( \frac{5}{6} \text{ □ } \frac{13}{15} \)
(ii) \( \frac{4}{5} \text{ □ } \frac{7}{9} \)
(iii) \( \frac{11}{12} \text{ □ } \frac{13}{14} \)

Answer:
(i) Compare \( \frac{5}{6} \) and \( \frac{13}{15} \) using cross multiplication:

\( 5 \times 15 = 75 \) and \( 6 \times 13 = 78 \)

Since 75 < 78, we have \( \frac{5}{6} < \frac{13}{15} \)

Therefore, \( \frac{5}{6} < \frac{13}{15} \)

(ii) Compare \( \frac{4}{5} \) and \( \frac{7}{9} \) using cross multiplication:

\( 4 \times 9 = 36 \) and \( 5 \times 7 = 35 \)

Since 36 > 35, we have \( \frac{4}{5} > \frac{7}{9} \)

Therefore, \( \frac{4}{5} > \frac{7}{9} \)

(iii) Compare \( \frac{11}{12} \) and \( \frac{13}{14} \) using cross multiplication:

\( 11 \times 14 = 154 \) and \( 12 \times 13 = 156 \)

Since 154 < 156, we have \( \frac{11}{12} < \frac{13}{14} \)

Therefore, \( \frac{11}{12} < \frac{13}{14} \)
In simple words: Multiply the numerator of the first fraction by the denominator of the second, and the denominator of the first by the numerator of the second. The larger product goes with the larger fraction.

Exam Tip: Cross multiplication is the fastest method for comparing two fractions without needing a common denominator. Always multiply in the same pattern: first numerator × second denominator, then first denominator × second numerator.

 

Question 6. Arrange the following fractions in descending order: \( \frac{7}{30}, \frac{13}{15}, \frac{9}{10}, \frac{3}{5} \)
Answer: To compare these fractions, convert them all to equivalent fractions with a common denominator.

Find the LCM of the denominators 30, 15, 10, and 5:

LCM = 30

Convert each fraction:

\( \frac{7}{30} = \frac{7}{30} \)

\( \frac{13}{15} = \frac{13 \times 2}{15 \times 2} = \frac{26}{30} \)

\( \frac{9}{10} = \frac{9 \times 3}{10 \times 3} = \frac{27}{30} \)

\( \frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30} \)

Since 27 > 26 > 18 > 7, we have:

\( \frac{27}{30} > \frac{26}{30} > \frac{18}{30} > \frac{7}{30} \)

Therefore, \( \frac{9}{10} > \frac{13}{15} > \frac{3}{5} > \frac{7}{30} \)
In simple words: Find a common denominator for all fractions. Then compare the numerators. The bigger numerator means the bigger fraction.

Exam Tip: Using LCM of denominators makes the comparison straightforward - you only need to look at numerators after conversion. Always arrange in the direction requested (ascending or descending).

 

Question 7. Simplify: \( 2\frac{1}{2} - 3\frac{1}{4} + 5\frac{5}{6} \)
Answer: First, convert mixed numbers to improper fractions:

\( 2\frac{1}{2} = \frac{5}{2} \)

\( 3\frac{1}{4} = \frac{13}{4} \)

\( 5\frac{5}{6} = \frac{35}{6} \)

So the expression becomes: \( \frac{5}{2} - \frac{13}{4} + \frac{35}{6} \)

Find the LCM of denominators 2, 4, and 6, which is 12.

Convert to equivalent fractions with denominator 12:

\( \frac{5}{2} = \frac{5 \times 6}{2 \times 6} = \frac{30}{12} \)

\( \frac{13}{4} = \frac{13 \times 3}{4 \times 3} = \frac{39}{12} \)

\( \frac{35}{6} = \frac{35 \times 2}{6 \times 2} = \frac{70}{12} \)

Now combine: \( \frac{30}{12} - \frac{39}{12} + \frac{70}{12} = \frac{30 - 39 + 70}{12} = \frac{61}{12} \)

Convert back to a mixed number: \( \frac{61}{12} = 5\frac{1}{12} \)

Therefore, \( 2\frac{1}{2} - 3\frac{1}{4} + 5\frac{5}{6} = 5\frac{1}{12} \)
In simple words: Change mixed numbers to improper fractions. Find a common denominator. Combine the numerators while keeping the denominator the same. Convert back to a mixed number if needed.

Exam Tip: Always convert mixed numbers first. Use the LCM for the common denominator to keep numbers manageable. Double-check your arithmetic when combining positive and negative terms.

 

Question 8. (i) Evaluate: \( \frac{3}{5} \times 180 \)
Answer: \( \frac{3}{5} \times 180 = \frac{3 \times 180}{5} = \frac{540}{5} = 108 \)

Therefore, \( \frac{3}{5} \times 180 = 108 \)
In simple words: Multiply the numerator by 180, then divide by the denominator. You can also divide 180 by 5 first if it divides evenly.

Exam Tip: Look for opportunities to simplify before multiplying - divide the whole number by the denominator if possible to make calculations easier.

 

Question 8. (ii) Evaluate: \( \frac{3}{7} \text{ of } 5\frac{5}{6} \)
Answer: First, convert the mixed number to an improper fraction:

\( 5\frac{5}{6} = \frac{35}{6} \)

"Of" means multiplication, so:

\( \frac{3}{7} \times \frac{35}{6} = \frac{3 \times 35}{7 \times 6} = \frac{105}{42} \)

Simplify by dividing both numerator and denominator by their GCD (21):

\( \frac{105}{42} = \frac{105 \div 21}{42 \div 21} = \frac{5}{2} = 2\frac{1}{2} \)

Therefore, \( \frac{3}{7} \text{ of } 5\frac{5}{6} = 2\frac{1}{2} \)
In simple words: Change mixed numbers to improper fractions. Multiply the two fractions by multiplying numerators together and denominators together. Simplify the result.

Exam Tip: The word "of" always means multiply in fraction problems. Cancel common factors between any numerator and denominator before multiplying to simplify calculations.

 

Question 8. (iii) Evaluate: \( \frac{5}{18} \div \frac{2}{3} \)
Answer: To divide fractions, multiply by the reciprocal of the second fraction:

\( \frac{5}{18} \div \frac{2}{3} = \frac{5}{18} \times \frac{3}{2} = \frac{5 \times 3}{18 \times 2} = \frac{15}{36} \)

Simplify by dividing both by their GCD (3):

\( \frac{15}{36} = \frac{15 \div 3}{36 \div 3} = \frac{5}{12} \)

Therefore, \( \frac{5}{18} \div \frac{2}{3} = \frac{5}{12} \)
In simple words: Flip the second fraction upside down. Then multiply the two fractions the normal way. Simplify the answer.

Exam Tip: Remember "KCF" - Keep the first fraction, Change the operation to multiply, Flip the second fraction. This makes division straightforward.

 

Question 9. Asha and Samuel have bookshelves of the same size partly filled with books. Asha's shelf is \( \frac{5}{6} \) full and Samuel's shelf is \( \frac{3}{5} \) full. Whose bookshelf is more full and by what fraction?
Answer: Asha's shelf is \( \frac{5}{6} \) full.

Samuel's shelf is \( \frac{3}{5} \) full.

To determine whose shelf is fuller, compare these fractions using cross multiplication:

\( 5 \times 5 = 25 \) and \( 6 \times 3 = 18 \)

Since 25 > 18, we have \( \frac{5}{6} > \frac{3}{5} \)

Therefore, Asha's bookshelf is more full.

To find by how much, calculate the difference using LCM of 6 and 5, which is 30:

\( \frac{5}{6} - \frac{3}{5} = \frac{5 \times 5}{6 \times 5} - \frac{3 \times 6}{5 \times 6} = \frac{25}{30} - \frac{18}{30} = \frac{25 - 18}{30} = \frac{7}{30} \)

Therefore, Asha's bookshelf is more full by \( \frac{7}{30} \)
In simple words: Compare the two fractions to see which is bigger. Then subtract the smaller from the larger to find the difference.

Exam Tip: Always state which person/object has more first, then calculate the exact difference. Some questions only ask for one part - read carefully.

 

Question 10. A farmer uses four out of five equal strips of his land for wheat crop and \( \frac{1}{7} \) of his land for cereal crop. What fraction of his land is available for other crops?
Answer: Fraction of land used for wheat crop = \( \frac{4}{5} \)

Fraction of land used for cereal crop = \( \frac{1}{7} \)

Total fraction of land used for crops = \( \frac{4}{5} + \frac{1}{7} \)

Find the LCM of 5 and 7, which is 35:

\( \frac{4}{5} + \frac{1}{7} = \frac{4 \times 7}{5 \times 7} + \frac{1 \times 5}{7 \times 5} = \frac{28}{35} + \frac{5}{35} = \frac{28 + 5}{35} = \frac{33}{35} \)

Fraction of land available for other crops = \( 1 - \frac{33}{35} = \frac{35}{35} - \frac{33}{35} = \frac{35 - 33}{35} = \frac{2}{35} \)

Therefore, the fraction of land available for other crops is \( \frac{2}{35} \)
In simple words: Add up the fractions used for the two crops. Subtract this total from 1 (the whole land) to find what remains for other crops.

Exam Tip: When finding "leftover" fractions, always subtract from the whole (1). The sum of all parts must equal 1.

 

Question 11. A rectangle is divided into a certain number of equal parts. If 16 of the parts so formed represents the fraction \( \frac{2}{5} \), find the number of parts in which the rectangle has been divided.
Answer: Let the total number of parts be x.

Given that 16 parts represent the fraction \( \frac{2}{5} \), we can write:

\( \frac{16}{x} = \frac{2}{5} \)

Cross multiply:

\( 16 \times 5 = 2 \times x \)

\( 80 = 2x \)

\( x = \frac{80}{2} = 40 \)

Therefore, the rectangle is divided into 40 equal parts.
In simple words: Set up an equation where 16 parts equal \( \frac{2}{5} \) of the total. Solve for the total by cross multiplication.

Exam Tip: Always set the ratio correctly - the "parts" value goes in the numerator and the "total" in the denominator, matching the given fraction.

 

Question 12. (i) Find the value of \( 5^5 \)
Answer: \( 5^5 = 5 \times 5 \times 5 \times 5 \times 5 = 3125 \)

Therefore, \( 5^5 = 3125 \)
In simple words: Multiply 5 by itself 5 times.

Exam Tip: Break down large exponent calculations into steps - calculate \( 5^2 = 25 \), then \( 5^3 = 125 \), etc., building up to avoid errors.

 

Question 12. (ii) Find the value of \( \left(\frac{3}{4}\right)^4 \)
Answer: \( \left(\frac{3}{4}\right)^4 = \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} = \frac{3 \times 3 \times 3 \times 3}{4 \times 4 \times 4 \times 4} = \frac{81}{256} \)

Therefore, \( \left(\frac{3}{4}\right)^4 = \frac{81}{256} \)
In simple words: Raise both the numerator and denominator to the power separately. Calculate \( 3^4 = 81 \) and \( 4^4 = 256 \).

Exam Tip: For fractional bases raised to a power, apply the exponent to both numerator and denominator independently. Always check if the final answer can be simplified.

 

Question 13. Write all proper fractions whose sum of numerator and denominator is 12.
Answer: A proper fraction has a numerator that is smaller than the denominator.

We need pairs where numerator + denominator = 12 and numerator < denominator.

If the sum is 12 and the numerator must be less than the denominator, then the numerator must be less than 6.

The possible fractions are:

If numerator = 1, denominator = 11 - gives \( \frac{1}{11} \)

If numerator = 2, denominator = 10 - gives \( \frac{2}{10} \)

If numerator = 3, denominator = 9 - gives \( \frac{3}{9} \)

If numerator = 4, denominator = 8 - gives \( \frac{4}{8} \)

If numerator = 5, denominator = 7 - gives \( \frac{5}{7} \)

Therefore, the proper fractions are: \( \frac{1}{11}, \frac{2}{10}, \frac{3}{9}, \frac{4}{8}, \frac{5}{7} \)
In simple words: Find all pairs of numbers that add to 12. Keep only the pairs where the first number (numerator) is smaller. These form the fractions you need.

Exam Tip: Always verify that numerator < denominator for each fraction you write. Systematic listing (starting from numerator = 1) ensures you find all solutions without repeating any.

 

Question 14. Find the value of x if \( 4\frac{x}{12} + 1\frac{1}{3} = 5\frac{3}{4} \)
Answer: First, rearrange the equation to isolate the term with x:

\( 4\frac{x}{12} = 5\frac{3}{4} - 1\frac{1}{3} \)

Convert mixed numbers to improper fractions:

\( 5\frac{3}{4} = \frac{23}{4} \) and \( 1\frac{1}{3} = \frac{4}{3} \)

Calculate the right side. Find LCM of 4 and 3, which is 12:

\( \frac{23}{4} - \frac{4}{3} = \frac{23 \times 3}{4 \times 3} - \frac{4 \times 4}{3 \times 4} = \frac{69}{12} - \frac{16}{12} = \frac{69 - 16}{12} = \frac{53}{12} \)

So: \( 4\frac{x}{12} = \frac{53}{12} \)

Convert the left side: \( 4\frac{x}{12} = \frac{48 + x}{12} \)

Therefore: \( \frac{48 + x}{12} = \frac{53}{12} \)

Since denominators are equal, the numerators must be equal:

\( 48 + x = 53 \)

\( x = 53 - 48 = 5 \)

Therefore, x = 5
In simple words: Move everything except the x term to one side. Convert mixed numbers to improper fractions. Solve by comparing numerators when denominators are the same.

Exam Tip: Always convert mixed numbers to improper fractions before manipulating. Keep denominators consistent by using LCM to avoid calculation errors.

 

Question 15. The adjoining figure represents the preferences of the students during breakfast in a hostel mess. If the total number of students in the mess is 540, then with reference to the given figure, answer the following questions:
Answer: [Note: The actual figure and specific sub-questions are not provided in the source material. To complete this answer, the figure showing student breakfast preferences and the specific sub-questions would be needed. This typically involves pie charts or bar charts with fractional sections representing different food preferences, and students would be asked to calculate the number of students preferring each option by multiplying the total (540) by the corresponding fraction.]
In simple words: Look at the figure to find what fraction each preference represents. Multiply that fraction by 540 to find how many students prefer each option.

Exam Tip: When working with figures and totals, always multiply the fraction shown by the given total. Double-check that all your answers add up to 540 at the end.