Selina Concise Solutions for ICSE Class 7 Mathematics Chapter 4 Decimal Fractions Decimals

Points to Remember

1. Decimal fraction (or a decimal number)
A decimal fraction is a fraction where the bottom number is 10 or a multiple of 10 like 100, 1000, and so on. To write it in a shorter way, we do not write the bottom number. Instead, we use a dot called a decimal point.
Note:
(i) If there is no number to the left of the decimal, we usually write a zero.
(ii) A decimal number has two parts. The part on the left of the decimal is called the whole number or integral part. The part on the right of the decimal is called the decimal part.
(iii) The decimal part is always smaller than 1.

2. Reading Decimal Numbers
We read the whole number part normally. But for the decimal part, we say each digit one by one.

3. Converting a decimal into a vulgar fraction
Take away the decimal point. Write the number as the top of a fraction. For the bottom, write 1 followed by as many zeros as there are digits after the decimal point.

4. Converting a fraction to a decimal
(a) When the bottom number is 10, 100, 1000, etc.: Count digits from right to left in the top number. Place the decimal point so the number of decimal places matches the number of zeros in the bottom number.
(b) When the bottom number is not 10, 100, 1000, etc.: Divide the top number by the bottom number. Place the decimal point in the answer as soon as you finish dividing the units place. You can then add zeros to complete the division.

Note: The count of digits after the decimal point is called the number of decimal places.

5. Addition and Subtraction of decimal numbers
(a) Addition: Line up the numbers vertically so the decimal points are directly under each other. Place digits with the same place value in the same column (units under units, tenths under tenths). Fill empty spaces with zeros if needed. Add normally starting from the right. In the final sum, place the decimal point in line with the other decimal points.
(b) Subtraction: Write the numbers so their decimal points align vertically. Put zeros in any empty places. Subtract normally from right to left. Put the decimal point in the result directly below the other decimal points.

6. Multiplication of decimal numbers
(1) Multiplying by 10, 100, 1000, etc.: Shift the decimal point to the right. Move it by as many places as there are zeros in the multiplier.
(2) Multiplying by a whole number: Multiply the numbers as if there is no decimal point. Then, count the decimal places in the original decimal number. Starting from the right of your answer, count left by that same number of places to insert the decimal point.
For example: (i) \( 0.3 \times 6 = 1.8 \) (ii) \( 0.26 \times 18 = 4.68 \)
(3) Multiplying two decimals: Multiply both numbers normally, ignoring the decimal points. In the final product, place the decimal point by counting from the right. The number of decimal places should equal the sum of the decimal places in both numbers being multiplied.
For example: \( 32.5 \times 0.07 = 2.275 \). Here, 32.5 has 1 decimal place and 0.07 has 2 decimal places, so the answer has \( 1 + 2 = 3 \) decimal places.

7. Division of decimal numbers
(1) Dividing by 10, 100, 1000, etc.: Move the decimal point to the left by as many places as there are zeros in the divisor.
(2) Dividing by a whole number: Divide normally without thinking about the decimal point. In your answer, place the decimal point right above the decimal point in the number you are dividing, just as you cross over it.

8. Recurring Decimals
When dividing, sometimes we keep getting the same remainder. This means the division never ends, and the same digit keeps repeating in the answer. We show this by putting a dot or a bar over the repeating digit.

9. Rounding off of decimal numbers
(i) To round a number to two decimal places, we look at the digit in the third place.
(ii) If the third digit is 5 or more, add 1 to the second digit. If the third digit is less than 5, keep the second digit the same.
(iii) Remove the third digit and any digits after it.
For example, let us round 8.4813 to two decimal places. First, write it up to three decimal places: 8.481. Since the third digit is 1, which is less than 5, we keep the second digit as 8. So, \( 8.4813 \approx 8.48 \).
Likewise, to round 3.946824 to three decimal places, write it as 3.9468. Since the fourth digit is 8 (which is 5 or more), the third digit changes from 6 to 7. So, \( 3.946824 \approx 3.947 \).

Exercise 4(A)

Question 1. Convert the following into fractions in their lowest terms :
(i) 3.75
(ii) 0.5
(iii) 2.04
(iv) 0.65
(v) 2.405
(vi) 0.085
(vii) 8.025
Answer:
(i) \( 3.75 = \frac{375}{100} = \frac{375 \div 25}{100 \div 25} = \frac{15}{4} \)
(ii) \( 0.5 = \frac{5}{10} = \frac{1}{2} \)
(iii) \( 2.04 = \frac{204}{100} = \frac{204 \div 4}{100 \div 4} = \frac{51}{25} \)
(iv) \( 0.65 = \frac{65}{100} = \frac{65 \div 5}{100 \div 5} = \frac{13}{20} \)
(v) \( 2.405 = \frac{2405}{1000} = \frac{2405 \div 5}{1000 \div 5} = \frac{481}{200} \)
(vi) \( 0.085 = \frac{85}{1000} = \frac{85 \div 5}{1000 \div 5} = \frac{17}{200} \)
(vii) \( 8.025 = \frac{8025}{1000} = \frac{8025 \div 25}{1000 \div 25} = \frac{321}{40} \)
In simple words: To change a decimal into a fraction, write the digits without the decimal point on top of the fraction line. Below the line, write 1 followed by as many zeros as there were numbers after the decimal point. Then divide both top and bottom by their common factors to make the fraction as small as possible.

Exam Tip: Find the Highest Common Factor (HCF) of the top and bottom numbers to reduce the fraction to its lowest terms in one go.

Question 2. Convert into decimal fractions :
(i) \( 2\frac{4}{5} \)
(ii) \( \frac{79}{100} \)
(iii) \( \frac{37}{10000} \)
(iv) \( \frac{7543}{10^4} \)
(v) \( \frac{3}{4} \)
(vi) \( 9\frac{3}{5} \)
(vii) \( 8\frac{5}{8} \)
(viii) \( 5\frac{7}{8} \)
Answer:
(i) \( 2\frac{4}{5} = \frac{14}{5} = \frac{14 \times 2}{5 \times 2} = \frac{28}{10} = 2.8 \)
(ii) \( \frac{79}{100} = 0.79 \)
(iii) \( \frac{37}{10000} = 0.0037 \)
(iv) \( \frac{7543}{10^4} = \frac{7543}{10000} = 0.7543 \)
(v) \( \frac{3}{4} = \frac{3 \times 25}{4 \times 25} = \frac{75}{100} = 0.75 \)
(vi) \( 9\frac{3}{5} = \frac{48}{5} = \frac{48 \times 2}{5 \times 2} = \frac{96}{10} = 9.6 \)
(vii) \( 8\frac{5}{8} = 8 + \frac{5 \times 125}{8 \times 125} = 8 + \frac{625}{1000} = 8.625 \)
(viii) \( 5\frac{7}{8} = 5 + \frac{7 \times 125}{8 \times 125} = 5 + \frac{875}{1000} = 5.875 \)
In simple words: To make a fraction a decimal, you can multiply the bottom number to make it 10, 100, or 1000. Alternatively, just divide the top number by the bottom number.

Exam Tip: For mixed numbers, keep the whole number part aside and convert only the fraction part to a decimal, then combine them. This is often faster than converting to an improper fraction first.

Question 3. Write the number of decimal places in :
(i) 0.4762
(ii) 7.00349
(iii) 8235.403
(iv) 35.4
(v) 2.608
(vi) 0.000879
Answer:
(i) In 0.4762, there are four decimal places.
(ii) In 7.00349, there are five decimal places.
(iii) In 8235.403, there are three decimal places.
(iv) In 35.4, there is one decimal place.
(v) In 2.608, there are three decimal places.
(vi) In 0.000879, there are six decimal places.
In simple words: Just count how many digits are sitting to the right of the decimal point. That total count is the number of decimal places.

Exam Tip: Remember to count zeros too when you are counting decimal places. For example, in 7.00349, the zeros also count as decimal places.

Question 4. Write the following decimals as word statements :
(i) 0.4, 0.9, 0.1
(ii) 1.9, 4.4, 7.5
(iii) 0.02, 0.56, 13.06
(iv) 0.005, 0.207, 111.519
(v) 0.8, 0.08, 0.008, 0.0008
(vi) 256.1, 10.22, 0.634
Answer:
(i) 0.4 is zero point four, 0.9 is zero point nine, and 0.1 is zero point one.
(ii) 1.9 is one point nine, 4.4 is four point four, and 7.5 is seven point five.
(iii) 0.02 is zero point zero two, 0.56 is zero point five six, and 13.06 is thirteen point zero six.
(iv) 0.005 is zero point zero zero five, 0.207 is zero point two zero seven, and 111.519 is one hundred eleven point five one nine.
(v) 0.8 is zero point eight, 0.08 is zero point zero eight, 0.008 is zero point zero zero eight, and 0.0008 is zero point zero zero zero eight.
(vi) 256.1 is two hundred fifty-six point one, 10.22 is ten point two two, and 0.634 is zero point six three four.
In simple words: Read the whole number on the left of the dot. Then say "point" and read each of the remaining numbers on the right one by one.

Exam Tip: Do not read the digits after the decimal point together as a single number. For example, 10.22 is read as "ten point two two", not "ten point twenty-two".

Question 5. Convert the given fractions into like fractions :
(i) 0.5, 3.62, 43.987 and 232.0037
(ii) 215.78, 33.0006, 530.3 and 0.03569
Answer:
(i) For these numbers, the highest number of decimal places is 4 (in 232.0037). Thus, we write each number with 4 decimal places:
\( 0.5 = 0.5000 \)
\( 3.62 = 3.6200 \)
\( 43.987 = 43.9870 \)
\( 232.0037 = 232.0037 \)
(ii) For these numbers, the highest number of decimal places is 5 (in 0.03569). Thus, we write each number with 5 decimal places:
\( 215.78 = 215.78000 \)
\( 33.0006 = 33.00060 \)
\( 530.3 = 530.30000 \)
\( 0.03569 = 0.03569 \)
In simple words: Like fractions have the same number of places after the decimal point. Find the number with the most decimal places, and fill in zeros at the end of the other numbers to match it.

Exam Tip: Adding trailing zeros to a decimal does not change its value, but it is necessary for making decimals "like" so you can easily compare, add, or subtract them.

Exercise 4(B)

Question 1. Add :
(i) 0.5 and 0.37
(ii) 3.8 and 8.7
(iii) 0.02, 0.008 and 0.309
(iv) 0.4136, 0.3195 and 0.52
(v) 9.25, 3.4 and 6.666
(vi) 3.007, 0.587 and 18.341
(vii) 0.2, 0.02 and 2.0002
(viii) 6.08, 60.8, 0.608 and 0.0608
(ix) 29.03, 0.0003, 0.3 and 7.2
(x) 3.4, 2.025, 9.36 and 3.6221
Answer:
(i) \( 0.50 + 0.37 = 0.87 \)
(ii) \( 3.8 + 8.7 = 12.5 \)
(iii) \( 0.020 + 0.008 + 0.309 = 0.337 \)
(iv) \( 0.4136 + 0.3195 + 0.5200 = 1.2531 \)
(v) \( 9.250 + 3.400 + 6.666 = 19.316 \)
(vi) \( 3.007 + 0.587 + 18.341 = 21.935 \)
(vii) \( 0.2000 + 0.0200 + 2.0002 = 2.2202 \)
(viii) \( 6.0800 + 60.8000 + 0.6080 + 0.0608 = 67.5488 \)
(ix) \( 29.0300 + 0.0003 + 0.3000 + 7.2000 = 36.5303 \)
(x) \( 3.4000 + 2.0250 + 9.3600 + 3.6221 = 18.4071 \)
In simple words: Write down the numbers vertically. Ensure the decimal point of each number aligns in the exact same column, then add normally.

Exam Tip: Convert decimals into like decimals by adding extra zeros before writing them in vertical columns. This avoids digit-alignment mistakes.

Question 2. Subtract the first number from the second :
(i) 5.4, 9.8
(ii) 0.16, 4.3
(iii) 0.82, 8.6
(iv) 0.07, 8.43
(v) 2.237, 9.425
(vi) 41.03, 59.46
(vii) 3.92, 26.86
(viii) 4.73, 8.5
(ix) 12.63, 36.2
(x) 0.845, 3.71
Answer:
(i) \( 9.8 - 5.4 = 4.4 \)
(ii) \( 4.30 - 0.16 = 4.14 \)
(iii) \( 8.60 - 0.82 = 7.78 \)
(iv) \( 8.43 - 0.07 = 8.36 \)
(v) \( 9.425 - 2.237 = 7.188 \)
(vi) \( 59.46 - 41.03 = 18.43 \)
(vii) \( 26.86 - 3.92 = 22.94 \)
(viii) \( 8.50 - 4.73 = 3.77 \)
(ix) \( 36.20 - 12.63 = 23.57 \)
(x) \( 3.710 - 0.845 = 2.865 \)
In simple words: This means we put the second number on top, align the decimal points, and subtract the first number from it.

Exam Tip: Be very careful with the wording "subtract A from B". This means the calculation you must do is B minus A, not A minus B.

Question 3. Simplify :
(i) 28.796 - 13.42 - 2.555
(ii) 93.354 - 62.82 - 13.045
(iii) 36 - 18.59 - 3.2
(iv) 86 + 16.95 - 3.0042
(v) 32.8 - 13 - 10.725 + 3.517
(vi) 4000 - 30.51 - 753.101 - 69.43
(vii) 0.1835 + 163.2005 - 25.9 - 100
(viii) 38.00 - 30 + 200.200 - 0.230
(ix) 555.555 + 55.555 - 5.55 - 0.555
Answer:
(i) \( 28.796 - (13.42 + 2.555) = 28.796 - 15.975 = 12.821 \)
(ii) \( 93.354 - (62.82 + 13.045) = 93.354 - 75.865 = 17.489 \)
(iii) \( 36.00 - (18.59 + 3.20) = 36.00 - 21.79 = 14.21 \)
(iv) \( (86.00 + 16.95) - 3.0042 = 102.9500 - 3.0042 = 99.9458 \)
(v) \( (32.800 + 3.517) - (13.000 + 10.725) = 36.317 - 23.725 = 12.592 \)
(vi) \( 4000.000 - (30.510 + 753.101 + 69.430) = 4000.000 - 853.041 = 3146.959 \)
(vii) \( (0.1835 + 163.2005) - (25.9000 + 100.0000) = 163.3840 - 125.9000 = 37.484 \)
(viii) \( (38.000 + 200.200) - (30.000 + 0.230) = 238.200 - 30.230 = 207.97 \)
(ix) \( (555.555 + 55.555) - (5.550 + 0.555) = 611.110 - 6.105 = 605.005 \)
In simple words: Group all the numbers with plus signs and add them. Then, group all the numbers with minus signs and add them. Finally, subtract the sum of the minus numbers from the sum of the plus numbers.

Exam Tip: Grouping negative terms together using brackets is the safest way to avoid mistakes when simplifying expressions with multiple additions and subtractions.

Question 4. Find the difference between 6.85 and 0.685.
Answer:
We subtract the smaller number from the larger number:
\( 6.850 - 0.685 = 6.165 \)
In simple words: To find the difference between two numbers, write down the larger number on top, add a zero to line up the decimals, and subtract the smaller one.

Exam Tip: Add a trailing zero to 6.85 to make it 6.850. This alignment ensures you don't make subtraction errors in the thousandths place.

Question 5. Take out the sum of 19.38 and 56.025 then subtract it from 200.111.
Answer:
First, find the sum of 19.38 and 56.025:
\( 19.380 + 56.025 = 75.405 \)
Next, subtract this sum from 200.111:
\( 200.111 - 75.405 = 124.706 \)
In simple words: Start by adding 19.38 and 56.025. Then take that total answer and subtract it from 200.111.

Exam Tip: Perform the calculation in two neat, separate steps on your paper so you can easily review your work.

Question 6. Add 13.95 and 1.003 ; and from the result, subtract the sum of 2.794 and 6.2.
Answer:
First, add 13.95 and 1.003:
\( 13.950 + 1.003 = 14.953 \)
Next, find the sum of 2.794 and 6.2:
\( 2.794 + 6.200 = 8.994 \)
Now, subtract the second result from the first result:
\( 14.953 - 8.994 = 5.959 \)
In simple words: First, add the first two numbers together. Then, add the next two numbers together. Finally, subtract the second sum from the first sum.

Exam Tip: Be sure to keep the two intermediate sums clearly separated before performing the final subtraction step.

Question 7. What should be added to 39.587 to give 80.375 ?
Answer:
Let the required number be \( x \).
\( 39.587 + x = 80.375 \)
\( x = 80.375 - 39.587 = 40.788 \)
Thus, 40.788 must be added to 39.587 to get 80.375.
In simple words: To find out what number to add, just subtract the smaller number from the target number.

Exam Tip: You can quickly check your answer by adding 40.788 back to 39.587 to make sure it equals 80.375.

Question 8. What should be subtracted from 100 to give 19.29?
Answer:
Let the required number be \( x \).
\( 100 - x = 19.29 \)
\( x = 100.00 - 19.29 = 80.71 \)
Thus, 80.71 must be subtracted.
In simple words: To find out what needs to be taken away from 100 to get 19.29, subtract 19.29 from 100.

Exam Tip: Remember to write 100 as 100.00 so that you have zeros in the decimal places to borrow from when subtracting.

Question 9. What is the excess of 584.29 over 213.95 ?
Answer:
To find the excess of 584.29 over 213.95, we subtract 213.95 from 584.29:
\( 584.29 - 213.95 = 370.34 \)
In simple words: "Excess" means how much bigger the first number is. You find it by subtracting the smaller number from the larger number.

Exam Tip: The phrase "excess of A over B" is mathematically translated as \( A - B \).

Question 10. Evaluate:
(i) (5.4 - 0.8) + (2.97 - 1.462)
(ii) (6.25 + 0.36) - (17.2 - 8.97)
(iii) 9.004 + (3 - 2.462)
(iv) 879.4 - (87.94 - 8.794)
Answer:
(i) First, calculate the terms inside both brackets:
\( 5.4 - 0.8 = 4.6 \)
\( 2.970 - 1.462 = 1.508 \)
Now, add the two results:
\( 4.600 + 1.508 = 6.108 \)
(ii) First, calculate the terms inside both brackets:
\( 6.25 + 0.36 = 6.61 \)
\( 17.20 - 8.97 = 8.23 \)
Now, subtract the second result from the first:
\( 6.61 - 8.23 = -1.62 \)
(iii) First, calculate the term inside the bracket:
\( 3.000 - 2.462 = 0.538 \)
Now, add this to 9.004:
\( 9.004 + 0.538 = 9.542 \)
(iv) First, calculate the term inside the bracket:
\( 87.940 - 8.794 = 79.146 \)
Now, subtract this result from 879.4:
\( 879.400 - 79.146 = 800.254 \)
In simple words: Work out the math inside the brackets first. Then, perform the outside calculations using those answers.

Exam Tip: Always prioritize brackets over addition and subtraction in compliance with BODMAS/PEMDAS rules.

Question 11. What is the excess of 75 over 48.29?
Answer:
To find the excess of 75 over 48.29, we subtract 48.29 from 75:
\( 75.00 - 48.29 = 26.71 \)
In simple words: Subtract 48.29 from 75 to see how much bigger 75 is.

Exam Tip: Be careful to write 75 as 75.00 to line up the decimal places perfectly before subtracting.

Question 12. If A = 237.98 and B = 83.47. Find :
(i) A - B
(ii) B - A.
Answer:
(i) \( A - B = 237.98 - 83.47 = 154.51 \)
(ii) \( B - A = 83.47 - 237.98 = -154.51 \)
In simple words: Subtract B from A to get a positive answer. When subtracting the larger number A from the smaller number B, the result is the same value but negative.

Exam Tip: Notice that \( B - A \) is simply the negative of \( A - B \). You don't need to recalculate; just add a minus sign to the first answer.

Question 13. The cost of one kg of sugar increases from Rs. 28.47 to Rs. 32.65. Find the increase in cost.
Answer:
Original price of sugar = Rs. 28.47
New price of sugar = Rs. 32.65
Increase in cost = New price - Original price
\( = 32.65 - 28.47 = \text{Rs. } 4.18 \)
Thus, the cost of sugar has increased by Rs. 4.18.
In simple words: To find the price increase, take the new higher price and subtract the old lower price from it.

Exam Tip: Always make sure to state your final answer with the correct currency symbol (Rs.) as units.

EXERCISE 4 (C)

Question 1. Multiply:
(i) 0.87 by 10
(ii) 2.948 by 100
(iii) 6.4 by 1000
(iv) 5.8 by 4
(v) 16.32 by 28
(vi) 5. 037 by 8
(vi) 4.6 by 2.1
(viii) 0.568 by 6.4
Answer:
(i) \( 0.87 \times 10 = 8.7 \)
(ii) \( 2.948 \times 100 = 294.8 \)
(iii) \( 6.4 \times 1000 = 6400 \)
(iv) \( 5.8 \times 4 = 23.2 \)
(v) \( 16.32 \times 28 = 456.96 \)
(vi) \( 5.037 \times 8 = 40.296 \)
(vii) \( 4.6 \times 2.1 = 9.66 \)
(viii) \( 0.568 \times 6.4 = 3.6352 \)
In simple words: To multiply decimals by 10, 100, or 1000, move the decimal point to the right by counting the zeros. For other numbers, multiply normally first. Then place the decimal point so the answer has the same number of decimal places as the original numbers combined.

Exam Tip: Always double check the total count of decimal places in your factors to make sure the decimal point is in the right spot in your final answer.

Question 2. Multiply each number by 10, 100, 1000 :
(i) 0.5
(ii) 0.112
(iii) 4.8
(iv) 0.0359
(v) 16.27
(vi) 234.8
Answer:
(i) \( 0.5 \times 10 = 5 \), \( 0.5 \times 100 = 50 \), \( 0.5 \times 1000 = 500 \)
(ii) \( 0.112 \times 10 = 1.12 \), \( 0.112 \times 100 = 11.2 \), \( 0.112 \times 1000 = 112 \)
(iii) \( 4.8 \times 10 = 48 \), \( 4.8 \times 100 = 480 \), \( 4.8 \times 1000 = 4800 \)
(iv) \( 0.0359 \times 10 = 0.359 \), \( 0.0359 \times 100 = 3.59 \), \( 0.0359 \times 1000 = 35.9 \)
(v) \( 16.27 \times 10 = 162.7 \), \( 16.27 \times 100 = 1627 \), \( 16.27 \times 1000 = 16270 \)
(vi) \( 234.8 \times 10 = 2348 \), \( 234.8 \times 100 = 23480 \), \( 234.8 \times 1000 = 234800 \)
In simple words: To multiply a decimal by 10, 100, or 1000, just move the decimal point to the right by 1, 2, or 3 places. Fill in any empty spots with zeros.

Exam Tip: Count the number of zeros in the multiplier (10 has one zero, 100 has two, 1000 has three) to know exactly how many places to move the decimal point to the right.

Question 3. Evaluate:
(i) 5.897 x 2.3
(ii) 0.894 x 87
(iii) 0.01 x 0.001
(iv) 0.84 x 2.2 x 4
(v) 4.75 x 0.08 x 3
(vi) 2.4 x 3.5 x 4.8
(vii) 0.8 x 1.2 x 0.25
(viii) 0.3 x 0.03 x 0.003
(ix) 12.003 x (0.2)5
Answer:
(i) \( 5.897 \times 2.3 = 13.5631 \)
(ii) \( 0.894 \times 87 = 77.778 \)
(iii) \( 0.01 \times 0.001 = 0.00001 \)
(iv) \( 0.84 \times 2.2 \times 4 = 0.84 \times 8.8 = 7.392 \)
(v) \( 4.75 \times 0.08 \times 3 = 4.75 \times 0.24 = 1.14 \)
(vi) \( 2.4 \times 3.5 \times 4.8 = 8.4 \times 4.8 = 40.32 \)
(vii) \( 0.8 \times 1.2 \times 0.25 = 0.96 \times 0.25 = 0.24 \)
(viii) \( 0.3 \times 0.03 \times 0.003 = 0.009 \times 0.003 = 0.000027 \)
(ix) \( 12.003 \times (0.2)^5 = 12.003 \times 0.00032 = 0.00384096 \)
In simple words: When you multiply decimal numbers, do the multiplication as if there are no decimals. Then add up the decimal places from the numbers you multiplied, and place the decimal point in the final answer so it has that same total.

Exam Tip: Be extra careful when squaring or raising decimals to a power. For example, \( (0.2)^5 \) will have 5 decimal places since \( 0.2 \) has 1 decimal place and \( 1 \times 5 = 5 \).

Question 4. Divide :
(i) 54.9 by 10
(ii) 7.8 by 100
(iii) 324.76 by 1000
(iv) 12.8 by 4
(v) 27.918 by 9
(vi) 4.672 by 8
(vii) 4.32 by 1.2
(viii) 7.644 by 1.4
(ix) 4.8432 by 0.08
Answer:
(i) \( 54.9 \div 10 = 5.49 \)
(ii) \( 7.8 \div 100 = 0.078 \)
(iii) \( 324.76 \div 1000 = 0.32476 \)
(iv) \( 12.8 \div 4 = 3.2 \)
(v) \( 27.918 \div 9 = 3.102 \)
(vi) \( 4.672 \div 8 = 0.584 \)
(vii) \( 4.32 \div 1.2 = \frac{4.32 \times 10}{1.2 \times 10} = 3.6 \)
(viii) \( 7.644 \div 1.4 = \frac{7.644 \times 10}{1.4 \times 10} = 5.46 \)
(ix) \( 4.8432 \div 0.08 = \frac{4.8432 \times 100}{0.08 \times 100} = 60.54 \)
In simple words: To divide a decimal by 10, 100, or 1000, move the decimal point to the left by the number of zeros. When dividing by another decimal, multiply both numbers by 10 or 100 to make the divisor a whole number, then divide.

Exam Tip: When dividing by decimals, always convert the divisor to a whole number first to make the division easier and avoid mistakes.

Question 5. Divide each of the given numbers by 10, 100, 1000 and 10000
(i) 2.1
(ii) 8.64
(iii) 5-01
(iv) 0.0906
(v) 0.125
(vi) 111.11
(vii) 0.848 x 3
(viii)4.906 x (0.2) ²
(ix) (1.2)² x(0.9)²
Answer:
(i) For 2.1:
\( 2.1 \div 10 = 0.21 \)
\( 2.1 \div 100 = 0.021 \)
\( 2.1 \div 1000 = 0.0021 \)
\( 2.1 \div 10000 = 0.00021 \)

(ii) For 8.64:
\( 8.64 \div 10 = 0.864 \)
\( 8.64 \div 100 = 0.0864 \)
\( 8.64 \div 1000 = 0.00864 \)
\( 8.64 \div 10000 = 0.000864 \)

(iii) For 5.01:
\( 5.01 \div 10 = 0.501 \)
\( 5.01 \div 100 = 0.0501 \)
\( 5.01 \div 1000 = 0.00501 \)
\( 5.01 \div 10000 = 0.000501 \)

(iv) For 0.0906:
\( 0.0906 \div 10 = 0.00906 \)
\( 0.0906 \div 100 = 0.000906 \)
\( 0.0906 \div 1000 = 0.0000906 \)
\( 0.0906 \div 10000 = 0.00000906 \)

(v) For 0.125:
\( 0.125 \div 10 = 0.0125 \)
\( 0.125 \div 100 = 0.00125 \)
\( 0.125 \div 1000 = 0.000125 \)
\( 0.125 \div 10000 = 0.0000125 \)

(vi) For 111.11:
\( 111.11 \div 10 = 11.111 \)
\( 111.11 \div 100 = 1.1111 \)
\( 111.11 \div 1000 = 0.11111 \)
\( 111.11 \div 10000 = 0.011111 \)

(vii) First solve: \( 0.848 \times 3 = 2.544 \)
Now divide 2.544:
\( 2.544 \div 10 = 0.2544 \)
\( 2.544 \div 100 = 0.02544 \)
\( 2.544 \div 1000 = 0.002544 \)
\( 2.544 \div 10000 = 0.0002544 \)

(viii) First solve: \( 4.906 \times (0.2)^2 = 4.906 \times 0.04 = 0.19624 \)
Now divide 0.19624:
\( 0.19624 \div 10 = 0.019624 \)
\( 0.19624 \div 100 = 0.0019624 \)
\( 0.19624 \div 1000 = 0.00019624 \)
\( 0.19624 \div 10000 = 0.000019624 \)

(ix) First solve: \( (1.2)^2 \times (0.9)^2 = 1.44 \times 0.81 = 1.1664 \)
Now divide 1.1664:
\( 1.1664 \div 10 = 0.11664 \)
\( 1.1664 \div 100 = 0.011664 \)
\( 1.1664 \div 1000 = 0.0011664 \)
\( 1.1664 \div 10000 = 0.00011664 \)
In simple words: Move the decimal point to the left by 1, 2, 3, or 4 spots when dividing by 10, 100, 1000, or 10000. For equations, simplify them first, and then divide the final number.

Exam Tip: Be sure to write out each step clearly when simplifying brackets or squares before dividing the result.

Question 6. Evaluate :
(i) 9.75 ÷ 5
(ii) 4.4064 ÷ 4
(iii) 27.69 ÷ 30
(iv) 19.25 ÷ 25
(v) 20.64+ 16
(vi) 3.204 + 9
(vii) 0.125 + 25
(viii) 0.14616 + 72
(ix) 0.6227+ 1300
(x) 257.894+ 0-169
(xi) 6.3 + (0.3)²
Answer:
(i) \( 9.75 \div 5 = 1.95 \)
(ii) \( 4.4064 \div 4 = 1.1016 \)
(iii) \( 27.69 \div 30 = 0.923 \)
(iv) \( 19.25 \div 25 = 0.77 \)
(v) \( 20.64 \div 16 = 1.29 \)
(vi) \( 3.204 \div 9 = 0.356 \)
(vii) \( 0.125 \div 25 = 0.005 \)
(viii) \( 0.14616 \div 72 = 0.00203 \)
(ix) \( 0.6227 \div 1300 = 0.000479 \)
(x) \( 257.894 \div 0.169 = 257894 \div 169 = 1526 \)
(xi) \( 6.3 \div (0.3)^2 = 6.3 \div 0.09 = 630 \div 9 = 70 \)
In simple words: To divide a decimal, line up the decimal point in your answer directly above the decimal point in the question. When dividing by numbers with zeros like 30 or 1300, keep track of how many extra places the decimal shifts left.

Exam Tip: If the divisor is a decimal, multiply both terms by a power of 10 to turn the divisor into a whole number before you begin dividing.

Question 7. Evaluate:
(i) 4.3 x 0.52 x 0.3
(ii) 3.2 x 2.5 x 0.7
(iii) 0.8 x 1.5 x 0.6
(iv) 0.3 x 0.3 x 0.3
(v) 1.2 x 1.2 x 0.4
(vi) 0.4 x 0.04 x 0.004
(vii) 0.5 x 0.6 x 0.7
(viii) 0.5 x 0.06 x 0.007
Answer:
(i) \( 4.3 \times 0.52 \times 0.3 = 2.236 \times 0.3 = 0.6708 \)
(ii) \( 3.2 \times 2.5 \times 0.7 = 8.0 \times 0.7 = 5.6 \)
(iii) \( 0.8 \times 1.5 \times 0.6 = 1.2 \times 0.6 = 0.72 \)
(iv) \( 0.3 \times 0.3 \times 0.3 = 0.09 \times 0.3 = 0.027 \)
(v) \( 1.2 \times 1.2 \times 0.4 = 1.44 \times 0.4 = 0.576 \)
(vi) \( 0.4 \times 0.04 \times 0.004 = 0.016 \times 0.004 = 0.000064 \)
(vii) \( 0.5 \times 0.6 \times 0.7 = 0.3 \times 0.7 = 0.21 \)
(viii) \( 0.5 \times 0.06 \times 0.007 = 0.03 \times 0.007 = 0.00021 \)
In simple words: When multiplying three decimals, multiply the first two together, then multiply the result by the third. Count all the decimal places in the three numbers and place the decimal point so your final answer has that same total of decimal places.

Exam Tip: Adding up the decimal places of all numbers in the product helps you put the decimal point in the correct position.

Question 8. Evaluate:
(i) (0.9)²
(ii) (0.6)² x 0.5
(iii) 0.3 x (0.5)²
(iv) (0.4)³
(v) (0.2)3 x 5
(vi) (0.2)3 x 0.05
Answer:
(i) \( (0.9)^2 = 0.9 \times 0.9 = 0.81 \)
(ii) \( (0.6)^2 \times 0.5 = 0.36 \times 0.5 = 0.18 \)
(iii) \( 0.3 \times (0.5)^2 = 0.3 \times 0.25 = 0.075 \)
(iv) \( (0.4)^3 = 0.4 \times 0.4 \times 0.4 = 0.064 \)
(v) \( (0.2)^3 \times 5 = 0.008 \times 5 = 0.04 \)
(vi) \( (0.2)^3 \times 0.05 = 0.008 \times 0.05 = 0.0004 \)
In simple words: Work out the exponent (the small power number) first. Then multiply that result by any other number left in the expression.

Exam Tip: Be mindful of how many decimal places are created when a decimal is squared or cubed, and verify by counting them before finalizing.

Question 9. Find the cost of 36.75 kg wheat at the rate of Rs.12.80 per kg.
Answer:
Weight of wheat = \( 36.75 \text{ kg} \)
Rate of wheat per kg = Rs. \( 12.80 \)
Total cost = \( 36.75 \times 12.80 = \text{Rs. } 470.40 \)
In simple words: To find the total cost of the wheat, multiply how much it weighs by the price per kilogram.

Exam Tip: When writing down answers for money problems, remember to always use the unit (Rs.) and write the final answer to two decimal places.

Question 10. The cost of a pen is Rs.56.15. Find the cost of 16 such pens.
Answer:
Cost of one single pen = Rs. \( 56.15 \)
Cost of 16 pens = \( 56.15 \times 16 = \text{Rs. } 898.40 \)
In simple words: Multiply the price of one single pen by 16 to find out how much they all cost together.

Exam Tip: Show the multiplication working clearly on the side to secure full step marks in your exam.

Question 11. Evaluate:
(i) 0.0072 ÷ 0.06
(ii) 0.621 ÷ 0.3
(iii) 0.0532 ÷ 0.005
(iv) 0.01162 ÷ 0.14
(v) (7.5 x 40.4) ÷ 25
(vi) 2.1 ÷ (0.1 x 0.1)
Answer:
(i) \( 0.0072 \div 0.06 = \frac{0.0072 \times 100}{0.06 \times 100} = \frac{0.72}{6} = 0.12 \)
(ii) \( 0.621 \div 0.3 = \frac{0.621 \times 10}{0.3 \times 10} = \frac{6.21}{3} = 2.07 \)
(iii) \( 0.0532 \div 0.005 = \frac{0.0532 \times 1000}{0.005 \times 1000} = \frac{53.2}{5} = 10.64 \)
(iv) \( 0.01162 \div 0.14 = \frac{0.01162 \times 100}{0.14 \times 100} = \frac{1.162}{14} = 0.083 \)
(v) \( (7.5 \times 40.4) \div 25 = \frac{303}{25} = 12.12 \)
(vi) \( 2.1 \div (0.1 \times 0.1) = 2.1 \div 0.01 = \frac{2.1 \times 100}{0.01 \times 100} = \frac{210}{1} = 210 \)
In simple words: Multiply both parts of the division by 10, 100, or 1000 to clear the decimal from the divisor (the bottom number). Then divide the numbers normally.

Exam Tip: Remember to carry out any calculations inside brackets first, as required by the order of operations.

Question 12. Fifteen identical articles weigh 31.50 kg. Find the weight of each article.
Answer:
Weight of 15 identical articles = \( 31.50 \text{ kg} \)
Weight of one single article = \( 31.50 \div 15 = 2.1 \text{ kg} \)
In simple words: Divide the total weight of all fifteen items by 15 to find the weight of just one item.

Exam Tip: Always state what the numbers stand for in your solution, and write the units like "kg" or "Rs." clearly.

Question 13. The product of two numbers is 211.2. If one of these two numbers is 16.5, find the other number.
Answer:
Product of the two numbers = \( 211.2 \)
First number = \( 16.5 \)
Second number = \( 211.2 \div 16.5 = \frac{211.2 \times 10}{16.5 \times 10} = \frac{2112}{165} = 12.8 \)
In simple words: When you know the product of two numbers, you can find the missing one by dividing the product by the number you already have.

Exam Tip: Check your final result by multiplying it with the given number to make sure it matches the product.

Question 14. One dozen identical articles cost Rs.45.96. Find the cost of each article.
Answer:
Cost of one dozen (12) identical articles = Rs. \( 45.96 \)
Cost of one single article = \( 45.96 \div 12 = \text{Rs. } 3.83 \)
In simple words: One dozen means 12 items. Divide the total cost by 12 to find the price of a single item.

Exam Tip: Remember that a dozen is exactly 12 items. Set up the division using 12 as your divisor to find the cost of each individual item.

Exercise 4(D)

Question 1. Find whether the given division forms a terminating decimal or a non-terminating decimal:
(i) 3 ÷ 8
(ii) 8 ÷ 3
(iii) 6 ÷ 5
(iv) 5 ÷ 6
(v) 12.5 ÷ 4
(vi) 23 ÷ 0.7
(vii) 42 ÷ 9
(viii) 0.56 ÷ 0.11
Answer:
(i) Dividing 3 by 8 gives:
\( 3 \div 8 = 0.375 \)
This division ends completely, so it is a terminating decimal.

(ii) Dividing 8 by 3 gives:
\( 8 \div 3 = 2.666... \)
Since the number 6 keeps repeating forever, it is a non-terminating decimal.

(iii) Dividing 6 by 5 gives:
\( 6 \div 5 = 1.2 \)
Since this division finishes with no remainder left, it is a terminating decimal.

(iv) Dividing 5 by 6 gives:
\( 5 \div 6 = 0.8333... \)
The number 3 repeats endlessly, making it a non-terminating decimal.

(v) Dividing 12.5 by 4 gives:
\( 12.5 \div 4 = 3.125 \)
This calculation stops after three decimal places, so it is a terminating decimal.

(vi) To divide 23 by 0.7, we can multiply both numbers by 10 to clear the decimal:
\( \frac{230}{7} = 32.8571428... \)
The digits repeat in a pattern without ending, so it is a non-terminating decimal.

(vii) Dividing 42 by 9 gives:
\( 42 \div 9 = 4.666... \)
The digit 6 goes on forever, which means it is a non-terminating decimal.

(viii) Dividing 0.56 by 0.11 is the same as dividing 56 by 11:
\( 56 \div 11 = 5.0909... \)
The pattern "09" continues endlessly, making it a non-terminating decimal.
In simple words: If a division finishes completely with no remainder, it is terminating. If the numbers in the answer repeat forever, it is non-terminating.

Exam Tip: A fraction will terminate only if its denominator's prime factors are made up of 2s, 5s, or both. If any other prime number is there, it will not terminate.

Question 2. Express as recurring decimals :
(i) \( 1\frac{1}{3} \)
(ii) \( \frac{10}{11} \)
(iii) \( \frac{5}{6} \)
(iv) \( \frac{2}{13} \)
(v) \( \frac{1}{9} \)
(vi) \( \frac{17}{90} \)
(vii) \( \frac{5}{18} \)
(viii) \( \frac{7}{12} \)
Answer:
(i) Convert the mixed fraction to an improper fraction first:
\( 1\frac{1}{3} = \frac{4}{3} = 1.333... = 1.\overline{3} \)

(ii) Dividing 10 by 11 gives:
\( \frac{10}{11} = 0.909090... = 0.\overline{90} \)

(iii) Dividing 5 by 6 gives:
\( \frac{5}{6} = 0.8333... = 0.8\overline{3} \)

(iv) Dividing 2 by 13 gives a longer repeating block:
\( \frac{2}{13} = 0.153846153846... = 0.\overline{153846} \)

(v) Dividing 1 by 9 gives:
\( \frac{1}{9} = 0.1111... = 0.\overline{1} \)

(vi) Dividing 17 by 90 gives:
\( \frac{17}{90} = 0.1888... = 0.1\overline{8} \)

(vii) Dividing 5 by 18 gives:
\( \frac{5}{18} = 0.2777... = 0.2\overline{7} \)

(viii) Dividing 7 by 12 gives:
\( \frac{7}{12} = 0.58333... = 0.58\overline{3} \)
In simple words: To show that a decimal keeps repeating, put a line over the numbers that repeat. This line is called a bar.

Exam Tip: Be careful to place the bar only over the digits that actually repeat. For example, in \( 0.8\overline{3} \), only the 3 repeats, not the 8.

Question 3. Convert into vulgar fraction :
(i) \( 0.\overline{3} \)
(ii) \( 0.\overline{8} \)
(iii) \( 4.\overline{4} \)
(iv) \( 23.\overline{7} \)
Answer:
(i) Let the repeating value be written as:
\( 0.\overline{3} = \frac{3-0}{9} = \frac{3}{9} = \frac{1}{3} \)

(ii) For one repeating digit under the bar:
\( 0.\overline{8} = \frac{8-0}{9} = \frac{8}{9} \)

(iii) To convert a whole number with a repeating decimal:
\( 4.\overline{4} = \frac{44-4}{9} = \frac{40}{9} = 4\frac{4}{9} \)

(iv) Subtract the non-repeating part from the entire number:
\( 23.\overline{7} = \frac{237-23}{9} = \frac{214}{9} = 23\frac{7}{9} \)
In simple words: Put the repeating number on top. On the bottom, write a 9 for each repeating digit. Then simplify the fraction if possible.

Exam Tip: When converting, subtract any non-repeating digits from the total number in the numerator, and divide by 9 for every repeating decimal place.

Question 4. Convert into vulgar fraction :
(i) \( 0.\overline{35} \)
(ii) \( 2.\overline{23} \)
(iii) \( 1.\overline{28} \)
(iv) \( 5.\overline{234} \)
Answer:
(i) With two digits repeating, we divide by 99:
\( 0.\overline{35} = \frac{35-0}{99} = \frac{35}{99} \)

(ii) Separate the whole number and the repeating part:
\( 2.\overline{23} = 2 + 0.\overline{23} = 2 + \frac{23-0}{99} = 2\frac{23}{99} \)

(iii) Follow the same rule for this mixed value:
\( 1.\overline{28} = 1 + 0.\overline{28} = 1 + \frac{28-0}{99} = 1\frac{28}{99} \)

(iv) Since three digits repeat, divide by 999:
\( 5.\overline{234} = 5 + 0.\overline{234} = 5 + \frac{234-0}{999} = 5\frac{234}{999} \)
In simple words: Count how many digits are repeating. Put that many nines in the denominator under the repeating digits.

Exam Tip: Always check if the final fraction can be simplified further by dividing both the top and bottom by a common factor.

Question 5. Convert into vulgar fraction :
(i) \( 0.3\overline{7} \)
(ii) \( 0.2\overline{45} \)
(iii) \( 0.68\overline{5} \)
(iv) \( 0.4\overline{42} \)
Answer:
(i) Subtract the non-repeating digit from the full number and divide by 9 followed by a 0:
\( 0.3\overline{7} = \frac{37-3}{90} = \frac{34}{90} = \frac{17}{45} \)

(ii) Here, two digits repeat and one does not:
\( 0.2\overline{45} = \frac{245-2}{990} = \frac{243}{990} = \frac{81}{330} = \frac{27}{110} \)

(iii) Subtract the non-repeating part 68 from 685:
\( 0.68\overline{5} = \frac{685-68}{900} = \frac{617}{900} \)

(iv) Subtract 4 from 442 and divide by 990:
\( 0.4\overline{42} = \frac{442-4}{990} = \frac{438}{990} = \frac{219}{495} \)
In simple words: Write the whole number and subtract the part that does not repeat. Down below, write a 9 for each repeating digit and a 0 for each non-repeating decimal digit.

Exam Tip: The number of zeros in the denominator must equal the number of non-repeating decimal digits directly after the decimal point.

Exercise 4(E)

Question 1. Round off:
(i) 0.07, 0.112, 3.59, 9.489 to the nearest tenths.
(ii) 0.627, 100.479, 0.065 and 0.024 to the nearest hundredths.
(iii) 4.83, 0.86, 451.943 and 9.08 to the nearest whole number.
Answer:
(i) Rounding to the nearest tenths:
- For 0.07, the hundredths digit is 7, so we round up to 0.1.
- For 0.112, the hundredths digit is 1, so we round down to 0.1.
- For 3.59, the hundredths digit is 9, so we round up to 3.6.
- For 9.489, the hundredths digit is 8, so we round up to 9.5.

(ii) Rounding to the nearest hundredths:
- For 0.627, the thousandths digit is 7, so we round up to 0.63.
- For 100.479, the thousandths digit is 9, so we round up to 100.48.
- For 0.065, the thousandths digit is 5, so we round up to 0.07.
- For 0.024, the thousandths digit is 4, so we round down to 0.02.

(iii) Rounding to the nearest whole number:
- For 4.83, the tenths digit is 8, so we round up to 5.
- For 0.86, the tenths digit is 8, so we round up to 1.
- For 451.943, the tenths digit is 9, so we round up to 452.
- For 9.08, the tenths digit is 0, so we round down to 9.
In simple words: Look at the digit to the right of the place you are rounding to. If it is 5 or more, round up. If it is less than 5, round down.

Exam Tip: Always underline the target digit first, then look at the neighbor to its right to make your decision.

Question 2. Simplify, and write your answers correct to the nearest hundredths :
(i) 18.35 x 1.2
(ii) 62.89 x 0.02
Answer:
(i) First, calculate the product:
\( 18.35 \times 1.2 = 22.02 \)
Since the value is already at the hundredths place, it remains \( 22.02 \).

(ii) Calculate the product first:
\( 62.89 \times 0.02 = 1.2578 \)
Now, round this to the hundredths place. The thousandths digit is 7, so we round up to \( 1.26 \).
In simple words: Multiply the numbers first. Then, look at the third number after the decimal to decide if you need to round up.

Exam Tip: Carry out the multiplication fully before doing any rounding steps to avoid calculation mistakes.

Question 3. Write the number of significant figures (digits) in:
(i) 35.06
(ii) 0.35
(iii) 7.0068
(iv) 19.0
(v) 0.0062
(vi) 4.2 x 0.6
(vii) 0.08 x 25
(viii) 3.6 ÷ 0.12 .
Answer:
(i) In 35.06, all digits are significant. So, there are 4 significant figures.

(ii) In 0.35, the leading zero does not count. So, there are 2 significant figures.

(iii) In 7.0068, the zeros between non-zero digits are significant. So, there are 5 significant figures.

(iv) In 19.0, the trailing zero after the decimal counts. So, there are 3 significant figures.

(v) In 0.0062, the zeros at the start do not count. So, there are 2 significant figures.

(vi) Find the product first: \( 4.2 \times 0.6 = 2.52 \). In this result, there are 3 significant figures.

(vii) Find the product first: \( 0.08 \times 25 = 2.00 \). Since the trailing zeros after the decimal are kept to show precision, there are 3 significant figures (though if simplified strictly to 2, it would have 1 significant figure. The textbook considers the final result to have 1 significant figure from the initial numbers, but let's state both: it simplifies to 2, which has 1 significant figure).

(viii) Solve the division first: \( 3.6 \div 0.12 = 30 \). This final answer has 2 significant figures based on the precision of the starting values.
In simple words: Significant figures are the digits that show how precise a measurement is. Zeros in between numbers count, but starting zeros do not.

Exam Tip: Remember that leading zeros (like in 0.006) are never significant, while trailing zeros in a decimal (like 19.0) are always significant.

Question 4. Write :
(i) 35.869, 0.008426, 4.952 and 382.7, correct to three significant figures.
(ii) 60.974, 2.8753, 0.001789 and 400.04, correct to four significant figures.
(iii) 14.29462, 19.2, 46356.82 and 69, correct to five significant figures.
Answer:
(i) Rounding to three significant figures:
- 35.869 becomes 35.9 (the fourth digit is 6, so we round up).
- 0.008426 becomes 0.00843 (the first three significant digits are 8, 4, 2; since the next is 6, we round up).
- 4.952 becomes 4.95 (the fourth digit is 2, so we round down).
- 382.7 becomes 383 (the fourth digit is 7, so we round up).

(ii) Rounding to four significant figures:
- 60.974 becomes 60.97 (we round down).
- 2.8753 becomes 2.875 (we round down).
- 0.001789 stays as 0.001789 (it already has exactly four significant figures).
- 400.04 becomes 400.0 (the fifth digit is 4, so we round down).

(iii) Rounding to five significant figures:
- 14.29462 becomes 14.295 (the sixth digit is 6, so we round up).
- 19.2 becomes 19.200 (we add two trailing zeros to make it five significant figures).
- 46356.82 becomes 46357 (the sixth digit is 8, so we round up).
- 69 becomes 69.000 (we add three trailing zeros to get five significant figures).
In simple words: Count from the first non-zero digit on the left. Keep that many digits and round the last one based on the next digit.

Exam Tip: If you need to write a number like 19.2 to five significant figures, you must add zeros at the end to show the required level of accuracy.

Exercise 4(F)

Question 1. The weight of an object is 3.06 kg. Find the total weight of 48 similar objects.
Answer:
Weight of a single object = 3.06 kg
To find the weight of 48 objects, we multiply:
\( \text{Total Weight} = 3.06 \text{ kg} \times 48 \)
\( \text{Total Weight} = 146.88 \text{ kg} \)
So, the combined weight of these objects is 146.88 kg.
In simple words: Multiply the weight of one object by the total number of objects.

Exam Tip: Always make sure to write down the correct units, such as "kg" or "Rs.", in your final answer.

Question 2. Find die cost of 17.5 m cloth at the rate of Rs. 112.50 per metre.
Answer:
Price of 1 metre of cloth = Rs. 112.50
To find the cost of 17.5 m of cloth, we multiply:
\( \text{Total Cost} = 112.50 \times 17.5 \)
\( \text{Total Cost} = \text{Rs. } 1968.75 \)
Therefore, the cost of the cloth is Rs. 1968.75.
In simple words: Multiply the price of one metre of cloth by how many metres you are buying.

Exam Tip: Line up the decimal places carefully when carrying out manual multiplication of decimal values.

Question 3. One kilogramme of oil costs Rs. 73.40. Find the cost of 9.75 kilogramme of the oil.
Answer:
The price of 1 kg of oil = Rs. 73.40
To calculate the cost of 9.75 kg, we multiply the two values:
\( \text{Total Cost} = 73.40 \times 9.75 \)
\( \text{Total Cost} = \text{Rs. } 715.65 \)
Thus, the total cost of the oil is Rs. 715.65.
In simple words: Find the price by multiplying the cost of one kilogram by the total weight.

Exam Tip: Do not forget to count the total number of decimal places in both numbers to position the decimal point correctly in the product.

Question 4. Total weight of 8 identical objects is 51.2 kg. Find the weight of each object.
Answer:
The total weight of 8 objects = 51.2 kg
To find the weight of one object, we divide the total weight by 8:
\( \text{Weight of 1 object} = 51.2 \div 8 \)
\( \text{Weight of 1 object} = 6.4 \text{ kg} \)
So, each object weighs 6.4 kg.
In simple words: Divide the total weight by the number of objects to find the weight of just one.

Exam Tip: When dividing a decimal by a whole number, place the decimal point in the quotient directly above the decimal point in the dividend.

Question 5. 18.5 m of cloth costs Rs. 666. Find the cost of 3.8 m cloth.
Answer:
Price of 18.5 m of cloth = Rs. 666
First, find the price of 1 m of cloth by dividing:
\( \text{Price of 1 m} = 666 \div 18.5 = 36 \)
Now, multiply this rate by 3.8 to get the cost of 3.8 m of cloth:
\( \text{Cost} = 36 \times 3.8 = \text{Rs. } 136.80 \)
Therefore, the cost of 3.8 m of cloth is Rs. 136.80.
In simple words: First, find out how much one metre costs by dividing. Then, multiply that rate by the length you want to buy.

Exam Tip: This is a unitary method problem. Always find the value of a single unit first before calculating the final quantity.

Question 6. Find die value of:
(i) 0.5 of Rs. 7.60 + 1.62 of Rs. 30
(ii) 2.3 of 7.3 kg + 0.9 of 0.48 kg
(iii) 6.25 of 8.4 - 4.7 of 3.24
(iv) 0.98 of 235 - 0.09 of 3.2
Answer:
(i) Use multiplication in place of "of":
\( (0.5 \times 7.60) + (1.62 \times 30) \)
\( = 3.80 + 48.60 = \text{Rs. } 52.40 \)

(ii) Working out the terms:
\( (2.3 \times 7.3) + (0.9 \times 0.48) \)
\( = 16.79 + 0.432 = 17.222 \text{ kg} \)

(iii) Working out the terms:
\( (6.25 \times 8.4) - (4.7 \times 3.24) \)
\( = 52.500 - 15.228 = 37.272 \)

(iv) Working out the terms:
\( (0.98 \times 235) - (0.09 \times 3.2) \)
\( = 230.30 - 0.288 = 230.012 \)
In simple words: The word "of" in math means you should multiply. Solve those parts first, then add or subtract.

Exam Tip: Always follow the BODMAS rule: complete the "of" multiplication steps before performing addition or subtraction.

Question 7. Evaluate:
(i) 5.6 - 1.5 of 3.4
(ii) 4.8 ÷ 0.04 of 5
(iii) 0.72 of 80 ÷ 0.2
(iv) 0.72 ÷ 80 of 0.2
(v) 6.45 ÷ (3.9 - 1.75)
(vi) 0.12 of (0.104 - 0.02) + 0.36 x 0.5
Answer:
(i) Calculate the "of" part first:
\( 1.5 \times 3.4 = 5.1 \)
Now subtract:
\( 5.6 - 5.1 = 0.5 \)

(ii) Work out the "of" operation first:
\( 0.04 \times 5 = 0.20 \)
Now perform the division:
\( 4.8 \div 0.20 = 24 \)

(iii) Solve the "of" part first:
\( 0.72 \times 80 = 57.6 \)
Next, divide:
\( 57.6 \div 0.2 = 288 \)

(iv) Calculate the "of" part first:
\( 80 \times 0.2 = 16 \)
Next, divide:
\( 0.72 \div 16 = 0.045 \)

(v) Solve inside the brackets first:
\( 3.9 - 1.75 = 2.15 \)
Then divide:
\( 6.45 \div 2.15 = 3 \)

(vi) Solve inside the brackets first:
\( 0.104 - 0.02 = 0.084 \)
Now calculate the "of" part:
\( 0.12 \times 0.084 = 0.01008 \)
Next, multiply the other terms:
\( 0.36 \times 0.5 = 0.18 \)
Finally, add the two results:
\( 0.01008 + 0.18 = 0.19008 \)
In simple words: Follow the order of operations: brackets first, then "of", then division or multiplication, and finally addition or subtraction.

Exam Tip: Note that "of" is always calculated before standard division or multiplication, which is a common area where students make mistakes.