Selina Concise Solutions for ICSE Class 7 Mathematics Chapter 8 Percent and Percentage

Points to Remember

1. The word "cent" refers to one hundred. Thus, per cent translates to out of a hundred, and we use the "%" symbol to represent it.

2. To write a standard statement as a percentage:

• (i) Change the statement into a fraction first.
• (ii) Turn this fraction into another equal fraction that has 100 as its bottom number (denominator).

So, to change any fraction or decimal into a percentage, simply multiply the value by 100.

3. To write one amount as a percentage of another amount:

• (i) Convert both amounts into the same unit if they are different.
• (ii) Create a fraction where the number you are comparing is on top (numerator) and the total comparison number is on the bottom (denominator).
• (iii) Multiply this fraction by 100 and put the percentage symbol (%) next to the result.

Exercise 8(A)

Question 1. Express each of the following as percent :
(i) \( \frac{3}{4} \)
(ii) \( \frac{2}{3} \)
(iii) 0.025
(iv) 0.125
(v) \( \frac{3}{8} \)
(vi) 0.25
Answer:
(i) To convert the fraction to a percentage, we multiply it by 100:
\( \frac{3}{4} \times 100\% = 3 \times 25\% = 75\% \)
(ii) We multiply the fraction by 100 to get the percentage:
\( \frac{2}{3} \times 100\% = \frac{200}{3}\% = 66\frac{2}{3}\% \)
(iii) First, write the decimal as a fraction, then multiply by 100:
\( 0.025 = \frac{25}{1000} \)
\( \frac{25}{1000} \times 100\% = \frac{25}{10}\% = 2.5\% \)
(iv) Turn the decimal into a fraction and multiply by 100:
\( 0.125 = \frac{125}{1000} \)
\( \frac{125}{1000} \times 100\% = \frac{125}{10}\% = 12.5\% \)
(v) Multiply the fraction by 100:
\( \frac{3}{8} \times 100\% = \frac{300}{8}\% = \frac{75}{2}\% = 37\frac{1}{2}\% \)
(vi) Turn the decimal into a fraction and multiply by 100:
\( 0.25 = \frac{25}{100} \)
\( \frac{25}{100} \times 100\% = 25\% \)
In simple words: To change a fraction or a decimal into a percentage, just multiply it by 100 and add the % sign at the end.

Exam Tip: Remember to write mixed fractions in their simplest form and always include the percent symbol (%) in your final answer.

Question 2. Express the following percentages as fractions and as decimal numbers :
(i) \( 7\frac{1}{2}\% \)
(ii) 2.50%
(iii) 0.02%
(iv) 175%
(v) 5%
(vi) 25%
Answer:
(i) Convert the mixed fraction to an improper fraction, then divide by 100 to remove the percentage sign:
As a fraction: \( 7\frac{1}{2}\% = \frac{15}{2}\% = \frac{15}{2 \times 100} = \frac{15}{200} = \frac{3}{40} \)
As a decimal: \( \frac{15}{200} = 0.075 \)
(ii) Divide by 100 to convert to a fraction and simplify:
As a fraction: \( 2.50\% = \frac{2.5}{100} = \frac{25}{1000} = \frac{1}{40} \)
As a decimal: \( \frac{2.5}{100} = 0.025 \)
(iii) Divide by 100 to remove the percentage:
As a fraction: \( 0.02\% = \frac{0.02}{100} = \frac{2}{100 \times 100} = \frac{2}{10000} = \frac{1}{5000} \)
As a decimal: \( 0.0002 \)
(iv) Divide by 100 to simplify:
As a fraction: \( 175\% = \frac{175}{100} = \frac{7}{4} = 1\frac{3}{4} \)
As a decimal: \( \frac{175}{100} = 1.75 \)
(v) Divide by 100:
As a fraction: \( 5\% = \frac{5}{100} = \frac{1}{20} \)
As a decimal: \( 0.05 \)
(vi) Divide by 100:
As a fraction: \( 25\% = \frac{25}{100} = \frac{1}{4} \)
As a decimal: \( 0.25 \)
In simple words: To change a percentage to a fraction or decimal, just divide the number by 100 and simplify it.

Exam Tip: To convert a percentage to a decimal quickly, just shift the decimal point two places to the left.

Question 3. What percent is :
(i) 16 hours of 2 days ?
(ii) 40 paisa of Rs. 2 ?
(iii) 25 cm of 4 metres
(iv) 600 gm of 5 kg ?
Answer:
(i) First, make sure the units are the same. Since 1 day has 24 hours, 2 days equal \( 2 \times 24 = 48 \) hours.
Now, find the percentage: \( \frac{16}{48} \times 100\% = \frac{1}{3} \times 100\% = \frac{100}{3}\% = 33\frac{1}{3}\% \)
(ii) Convert Rs. 2 into paisa. Since Rs. 1 is equal to 100 paisa, Rs. 2 is equal to \( 2 \times 100 = 200 \) paisa.
Now, calculate the percentage: \( \frac{40}{200} \times 100\% = 20\% \)
(iii) Convert 4 metres to centimetres. Since 1 metre equals 100 cm, 4 metres equals \( 4 \times 100 = 400 \) cm.
Find the percentage: \( \frac{25}{400} \times 100\% = \frac{25}{4}\% = 6\frac{1}{4}\% \)
(iv) Convert 5 kg to grams. Since 1 kg is 1000 grams, 5 kg equals \( 5 \times 1000 = 5000 \) grams.
Calculate the percentage: \( \frac{600}{5000} \times 100\% = 12\% \)
In simple words: First, change both numbers to the same unit. Then, divide the first number by the second number and multiply by 100.

Exam Tip: Always double check that both quantities are in the exact same unit before you perform any division.

Question 4. Find the value of:
(i) 5% of Rs. 350
(ii) 10% of Rs. 400.40
(iii) 1% of Rs. 500
(iv) \( 12\frac{1}{2}\% \) of 80 kg
(v) \( \frac{5}{8}\% \) of Rs. 600
(vi) \( 33\frac{1}{3}\% \) of 27 m
Answer:
(i) Multiply the amount by the percentage written as a fraction:
\( \text{Value} = \text{Rs. } 350 \times \frac{5}{100} = \text{Rs. } \frac{1750}{100} = \text{Rs. } 17.50 \)
(ii) Calculate 10% by dividing by 10:
\( \text{Value} = \text{Rs. } 400.40 \times \frac{10}{100} = \text{Rs. } 40.04 \)
(iii) Multiply by \( \frac{1}{100} \):
\( \text{Value} = \text{Rs. } 500 \times \frac{1}{100} = \text{Rs. } 5 \)
(iv) Convert the mixed percentage to an improper fraction first:
\( 12\frac{1}{2}\% = \frac{25}{2}\% \)
\( \text{Value} = 80 \text{ kg} \times \frac{25}{2 \times 100} = \frac{80 \times 25}{200} = 10 \text{ kg} \)
(v) Multiply by the fractional percentage:
\( \text{Value} = \text{Rs. } 600 \times \frac{5}{8 \times 100} = \text{Rs. } \frac{30}{8} = \text{Rs. } 3.75 \)
(vi) Convert the mixed percentage to an improper fraction:
\( 33\frac{1}{3}\% = \frac{100}{3}\% \)
\( \text{Value} = 27 \text{ m} \times \frac{100}{3 \times 100} = 27 \times \frac{1}{3} = 9 \text{ m} \)
In simple words: To find the value, change the percentage into a fraction by dividing by 100, then multiply it by the given amount.

Exam Tip: Remember to write the proper unit (like Rs., kg, or m) along with your final calculated number to avoid losing marks.

Question 5. In a class of 60 children, 30% are girls. How many boys are there ?
Answer:
Given that the total number of children in the class is 60.
Since 30% of them are girls, we can calculate the number of girls as follows:
\( \text{Number of girls} = 30\% \text{ of } 60 = \frac{30}{100} \times 60 = 18 \)
To find the number of boys, we subtract the girls from the total number of students:
\( \text{Number of boys} = 60 - 18 = 42 \)
In simple words: Out of 60 children, 18 are girls. This leaves 42 boys in the class.

Exam Tip: Alternatively, since 30% are girls, 70% must be boys. Calculating 70% of 60 directly gives 42, which is a great way to verify your answer.

Question 6. In an election, two candidates A and B contested. A got 60% of the votes. The total votes polled were 8000. How many votes did each get ?
Answer:
The total number of cast votes is 8000.
Candidate A received 60% of these votes. We can compute A's share as:
\( \text{Votes received by A} = 60\% \text{ of } 8000 = \frac{60}{100} \times 8000 = 4800 \)
The remaining votes went to candidate B. We find B's votes by subtracting A's share from the total:
\( \text{Votes received by B} = 8000 - 4800 = 3200 \)
In simple words: Out of 8000 total votes, Candidate A got 4800 votes and Candidate B got 3200 votes.

Exam Tip: You can also find Candidate B's share by calculating 40% (100% - 60%) of 8000, which gives the same result of 3200 votes.

Question 7. A person saves 12% of his salary every month. If his salary is Rs. 2,500, find his expenditure.
Answer:
The person's monthly income is Rs. 2500.
He saves 12% of this monthly income. Let us calculate his savings:
\( \text{Savings} = 12\% \text{ of Rs. } 2500 = \frac{12}{100} \times 2500 = 300 \text{ Rs.} \)
To find his monthly expenses, subtract his savings from his total salary:
\( \text{Expenditure} = \text{Rs. } 2500 - \text{Rs. } 300 = \text{Rs. } 2200 \)
In simple words: The person earns Rs. 2500 and saves Rs. 300 of it. This means he spends Rs. 2200 every month.

Exam Tip: Since expenditure represents the remaining 88% of the salary, you can compute \( 88\% \text{ of } 2500 \) directly to get Rs. 2200.

Question 8. Seeta got 75% marks out of a total of 800. How many marks did she lose ?
Answer:
The maximum possible marks are 800.
Seeta achieved a score of 75%. Let us calculate the marks she scored:
\( \text{Marks obtained} = 75\% \text{ of } 800 = \frac{75}{100} \times 800 = 600 \)
To find how many marks she missed out on, we subtract her score from the total marks:
\( \text{Marks lost} = 800 - 600 = 200 \)
In simple words: Seeta scored 600 marks out of 800. This means she lost 200 marks in total.

Exam Tip: Losing marks is equivalent to the remaining percentage. She lost 25% of the total marks, and \( 25\% \text{ of } 800 = 200 \).

Question 9. A shop worth Rs. 25,000 was insured for 95% of its value. How much would the owner get in case of any mishappening ?
Answer:
The shop has a total value of Rs. 25000.
Since it is insured for 95% of its original cost, the payout amount in case of an accident is:
\( \text{Insurance payout} = 95\% \text{ of Rs. } 25000 = \frac{95}{100} \times 25000 = 95 \times 250 = \text{Rs. } 23750 \)
In simple words: The shop is worth Rs. 25000. In case of any damage, the owner will receive Rs. 23750 from the insurance company.

Exam Tip: Be careful with basic multiplication here. Dropping zeros during cancellation is a common point where errors occur.

Question 10. A class has 30 boys and 25 girls. What is the percentage of boys in the class ?
Answer:
Number of boys in the class = 30
Number of girls in the class = 25
Total number of students = \( 30 + 25 = 55 \)
Now, calculate the percentage of boys out of the total strength:
\( \text{Percentage of boys} = \frac{30}{55} \times 100\% = \frac{6}{11} \times 100\% = \frac{600}{11}\% = 54\frac{6}{11}\% \)
In simple words: Out of 55 students in total, 30 are boys, which is about \( 54\frac{6}{11}\% \) of the whole class.

Exam Tip: Always calculate the total number of students first before trying to find the percentage of a specific group.

Question 11. Express :
(i) \( 3\frac{2}{5} \) as a percent
(ii) 0.0075 as percent
(iii) 3 : 20 as percent
(iv) 60 cm as percent of 1 m 25 cm
(v) 9 hours as a percent of 4 days.
Answer:
(i) First, convert the mixed fraction to an improper fraction:
\( 3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{17}{5} \)
Now multiply by 100 to get the percentage:
\( \frac{17}{5} \times 100\% = 17 \times 20\% = 340\% \)
(ii) To convert a decimal to a percentage, multiply by 100:
\( 0.0075 \times 100\% = 0.75\% \)
(iii) Write the ratio as a fraction and multiply by 100:
\( \frac{3}{20} \times 100\% = 3 \times 5\% = 15\% \)
(iv) Convert both quantities to centimetres:
\( 1 \text{ m } 25 \text{ cm} = 100 \text{ cm} + 25 \text{ cm} = 125 \text{ cm} \)
Now find the percentage:
\( \frac{60}{125} \times 100\% = \frac{12}{25} \times 100\% = 12 \times 4\% = 48\% \)
(v) Convert days to hours since 1 day equals 24 hours:
\( 4 \text{ days} = 4 \times 24 = 96 \text{ hours} \)
Now find the percentage:
\( \frac{9}{96} \times 100\% = \frac{3}{32} \times 100\% = \frac{300}{32}\% = \frac{75}{8}\% = 9\frac{3}{8}\% \)
In simple words: To change any number, ratio, or fraction into a percentage, just multiply it by 100 after making sure the units match.

Exam Tip: For ratios like \( a : b \), write them as \( \frac{a}{b} \) before multiplying by 100 to find the percentage.

Question 12.
(i) Find 2% of 2 hours 30 min.
(ii) What percent of 12 kg is 725 gm?
Answer:
(i) First, convert hours and minutes into minutes:
\( 2 \text{ hours } 30 \text{ min} = (2 \times 60) + 30 = 120 + 30 = 150 \text{ minutes} \)
Now, find 2% of this value:
\( 2\% \text{ of } 150 = \frac{2}{100} \times 150 = 3 \text{ minutes} \)
(ii) First, convert kilograms into grams so both have the same unit:
\( 12 \text{ kg} = 12 \times 1000 = 12000 \text{ grams} \)
Now, find what percent 725 grams is of 12000 grams:
\( \text{Percentage} = \frac{725}{12000} \times 100\% = \frac{725}{120}\% = \frac{145}{24}\% = 6\frac{1}{24}\% \)
In simple words: (i) Convert 2 hours 30 minutes to 150 minutes, then find 2% of it, which is 3 minutes. (ii) Since 12 kg is 12000 grams, 725 grams is about \( 6\frac{1}{24}\% \) of that total.

Exam Tip: Be sure to reduce fractions to their simplest improper or mixed fraction form for your final answer.

Exercise 8(B)

Question 1. Deepak bought a basket of mangoes containing 250 mangoes 12% of these were found to be rotten. Of the remaining, 10% got crushed. How many mangoes were in good condition ?
Answer:
The total number of mangoes in the basket is 250.
First, find how many mangoes are rotten (12% of 250):
\( \text{Rotten mangoes} = \frac{12}{100} \times 250 = 30 \)
Next, find the remaining mangoes that are not rotten:
\( \text{Remaining mangoes} = 250 - 30 = 220 \)
Out of these remaining mangoes, 10% were crushed:
\( \text{Crushed mangoes} = \frac{10}{100} \times 220 = 22 \)
Finally, find the number of mangoes in good condition by subtracting the crushed ones from the remaining:
\( \text{Good mangoes} = 220 - 22 = 198 \)
In simple words: Out of 250 mangoes, 30 are rotten, leaving 220. Then 22 of those get crushed, which leaves 198 mangoes in good condition.

Exam Tip: Pay close attention to whether a percentage applies to the "total" or the "remaining" amount. Here, the 10% is calculated from the remaining 220, not the original 250.

Question 2. In a Maths Quiz of 60 questions, Chandra got 90% correct answers and Ram got 80% correct answers. How many correct answers did each give ? What percent is Ram’s correct answers to Chandra’s correct answers ?
Answer:
Total questions in the quiz = 60
Calculate the number of correct answers given by Chandra (90% of 60):
\( \text{Chandra's correct answers} = \frac{90}{100} \times 60 = 54 \)
Calculate the number of correct answers given by Ram (80% of 60):
\( \text{Ram's correct answers} = \frac{80}{100} \times 60 = 48 \)
Now, find what percentage Ram's score is compared to Chandra's score:
\( \text{Percentage} = \frac{\text{Ram's score}}{\text{Chandra's score}} \times 100\% = \frac{48}{54} \times 100\% = \frac{8}{9} \times 100\% = \frac{800}{9}\% = 88\frac{8}{9}\% \)
In simple words: Chandra answered 54 questions correctly, and Ram answered 48 correctly. Ram's score is about \( 88\frac{8}{9}\% \) of Chandra's score.

Exam Tip: When comparing two scores as a percentage, the person being compared to (Chandra) goes in the denominator.

Question 3. In an examination, the maximum marks are 900. A student gets 33% of the maximum marks and fails by 45 marks. What is the passing mark ? Also, find the pass percentage.
Answer:
The maximum marks for the exam are 900.
First, let us calculate the marks obtained by the student (33% of 900):
\( \text{Marks obtained} = \frac{33}{100} \times 900 = 297 \)
Since the student failed by 45 marks, the passing mark is 45 marks higher than their score:
\( \text{Passing mark} = 297 + 45 = 342 \)
Next, calculate the pass percentage by comparing the passing mark to the maximum marks:
\( \text{Pass percentage} = \frac{342}{900} \times 100\% = 38\% \)
In simple words: The student got 297 marks but needed 45 more to pass, making the passing mark 342. This passing mark represents 38% of the total 900 marks.

Exam Tip: Remember to solve both parts of the question: the passing mark value (342) and the final passing percentage (38%).

Question 4. In a train, 15% people travel in first class, 35% travel in second class. The balance travel in the A.C. class ? Calculate the percentage of A.C. class travellers ?
Answer:
Let the total number of passengers on the train be 100%.
The percentage of passengers in first class = 15%
The percentage of passengers in second class = 35%
To find the percentage of passengers traveling in the A.C. class, we subtract the sum of the other classes from 100%:
\( \text{Percentage of A.C. class travellers} = 100\% - (15\% + 35\%) = 100\% - 50\% = 50\% \)
In simple words: If we add the first class (15%) and second class (35%) travelers, they make up 50% of the passengers. The other 50% must be traveling in the A.C. class.

Exam Tip: Since percentages always sum to 100%, you can easily solve problems like this by subtracting the known percentages from 100.

Question 5. A boy eats 25% of the cake and gives away 35% of it to his friends. What percent of the cake is still left with him ?
Answer:
Let us assume the entire cake represents 100%.
Percentage of the cake eaten by the boy = 25%
Percentage of the cake given to his friends = 35%
To find the portion of the cake still remaining with him, we subtract both portions from the whole:
\( \text{Remaining cake} = 100\% - (25\% + 35\%) = 100\% - 60\% = 40\% \)
In simple words: The boy used up 60% of the cake by eating some and giving some away. This means he still has 40% of the cake left.

Exam Tip: When no total amount is given, assuming a starting base of 100% makes the addition and subtraction straightforward.

Question 6. What is the percentage of vowels in the English alphabet ?
Answer:
There are 5 vowels in the English alphabet (A, E, I, O, and U).
The total number of letters in the English alphabet is 26.
To find the percentage of vowels, write it as a fraction of the total alphabet and multiply by 100:
\( \text{Percentage of vowels} = \frac{5}{26} \times 100\% = \frac{500}{26}\% = \frac{250}{13}\% = 19\frac{3}{13}\% \)
In simple words: Out of 26 letters, 5 are vowels. This means vowels make up about \( 19\frac{3}{13}\% \) of the whole alphabet.

Exam Tip: Make sure you know the fundamental counts (like 26 letters in the alphabet) as these facts are often expected but not provided in the question.

Question 7.
(i) \( 6\frac{1}{4}\% \) of what number is 375 ?
(ii) 0.2% of a number is 5. Find the number.
(iii) 30 is \( 16\frac{2}{3}\% \) of a number. Find the number.
Answer:
(i) Let the required number be \( x \).
We can write the equation as:
\( 6\frac{1}{4}\% \text{ of } x = 375 \)
Convert the percentage to a fraction: \( 6\frac{1}{4}\% = \frac{25}{4}\% = \frac{25}{4 \times 100} = \frac{25}{400} = \frac{1}{16} \)
Now, solve for \( x \):
\( \frac{1}{16} \times x = 375 \)
\( x = 375 \times 16 = 6000 \)
The number is 6000.
(ii) Let the required number be \( x \).
We can set up the equation:
\( 0.2\% \text{ of } x = 5 \)
Convert the percentage to a fraction: \( 0.2\% = \frac{0.2}{100} = \frac{2}{1000} = \frac{1}{500} \)
Solve for \( x \):
\( \frac{1}{500} \times x = 5 \)
\( x = 5 \times 500 = 2500 \)
The number is 2500.
(iii) Let the required number be \( x \).
We write the equation:
\( 16\frac{2}{3}\% \text{ of } x = 30 \)
Convert the percentage to a fraction: \( 16\frac{2}{3}\% = \frac{50}{3}\% = \frac{50}{3 \times 100} = \frac{50}{300} = \frac{1}{6} \)
Solve for \( x \):
\( \frac{1}{6} \times x = 30 \)
\( x = 30 \times 6 = 180 \)
The number is 180.
In simple words: Turn the percentage into a fraction first, set it up as an equation where the unknown number is \( x \), and solve to find the full value.

Exam Tip: Always write the equation clearly as "percentage of \( x = \text{given value} \)" to keep your algebraic steps organized.

Question 8. The money spent on the repairs of a house was 1% of its value. If the repair, costs Rs. 5,000, find the cost of the house.
Answer:
Let the value of the house be \( x \).
The amount spent on repairs is 1% of the house's total cost, which equals Rs. 5000.
We can write this as an equation:
\( 1\% \text{ of } x = 5000 \)
\( \frac{1}{100} \times x = 5000 \)
Solving for \( x \):
\( x = 5000 \times 100 = 500000 \)
So, the cost of the house is Rs. 5,00,000.
In simple words: Since 1% of the house's value is Rs. 5000, the full value (100%) of the house is 100 times that amount, which is Rs. 5,00,000.

Exam Tip: Be careful with the number of zeros when multiplying by 100 to ensure your final amount has the correct place values.

Question 9. In a school out of 300 students, 70% are girls and 30% are boys. If 30 girls leave and no new boy is admitted, what is the new percentage of girls in the school ?
Answer:
The total number of students in the school initially is 300.
Let us calculate the initial number of girls (70% of 300):
\( \text{Initial number of girls} = \frac{70}{100} \times 300 = 210 \)
Let us calculate the initial number of boys (30% of 300):
\( \text{Initial number of boys} = \frac{30}{100} \times 300 = 90 \)
When 30 girls leave the school, the new count of girls becomes:
\( \text{New number of girls} = 210 - 30 = 180 \)
Since no boys left or joined, the number of boys remains 90.
The new total number of students in the school is:
\( \text{New total students} = 180 + 90 = 270 \)
Now, find the new percentage of girls in the school:
\( \text{New percentage of girls} = \frac{180}{270} \times 100\% = \frac{2}{3} \times 100\% = \frac{200}{3}\% = 66\frac{2}{3}\% \)
In simple words: Initially there were 210 girls and 90 boys. After 30 girls left, we have 180 girls out of a new total of 270 students. This makes girls \( 66\frac{2}{3}\% \) of the school.

Exam Tip: Remember to calculate the new total number of students (270) instead of dividing the remaining girls by the original total (300).

Question 10. Kumar bought a transistor for Rs. 960. He paid \( 12 \frac{1}{2}\% \) cash money. The rest he agreed to pay in 12 equal monthly instalments. How much will he pay each month ?
Answer:
Cost of the transistor = Rs. 960
Cash paid down = \( 12\frac{1}{2}\% \) of Rs. 960
\( = \frac{25}{2 \times 100} \times 960 = \text{Rs. } 120 \)
Remaining balance = Rs. 960 - Rs. 120 = Rs. 840
Total number of monthly payments = 12
Amount to be paid each month:

\( \implies \text{Rs. } 840 \div 12 = \text{Rs. } 70 \)
In simple words: Kumar first pays Rs. 120 in cash. He has Rs. 840 left to pay. He splits this into 12 equal parts, so he pays Rs. 70 every month.

Exam Tip: First, change the mixed percentage \( 12\frac{1}{2}\% \) to the fraction \( \frac{25}{2}\% \) before solving to prevent mistakes during division.

Question 11. An ore contains 20% zinc. How many kg of ore will be required to get 45 kg of zinc ?
Answer:
Percentage of zinc in the mineral ore = 20%
Let us assume the total weight of the ore is \( x \) kg.
According to the problem, 20% of this total weight must equal 45 kg.
\( \therefore 20\% \text{ of } x = 45\text{ kg} \)

\( \implies \frac{20}{100} \times x = 45 \)

\( \implies \frac{x}{5} = 45 \)

\( \implies x = 45 \times 5 = 225 \)
Hence, the total weight of the ore needed is 225 kg.
In simple words: Since zinc makes up only one-fifth of the ore, you need 5 times more ore than the zinc you want. So, to get 45 kg of zinc, you need 225 kg of ore.

Exam Tip: Setting up an equation with a variable like \( x \) representing the unknown total quantity is a systematic way to solve percentage composition problems.

Exercise 8(C)

Question 1. The salary of a man is increased from Rs. 600 per month to Rs. 850 per month. Express the increase in salary as percent.
Answer:
The initial monthly income of the person = Rs. 600
The new monthly income after raise = Rs. 850
\( \therefore \) Increase in the monthly income = Rs. 850 - Rs. 600 = Rs. 250
Percentage of salary increase = \( \frac{250 \times 100}{600} \% \)

\( \implies \frac{125}{3}\% = 41\frac{2}{3}\% \)
In simple words: The man's pay went up by Rs. 250. To find the percentage, we compare this increase of Rs. 250 to his starting salary of Rs. 600.

Exam Tip: When calculating percentage increase or decrease, always use the original value in the denominator, not the new or increased value.

Question 2. Increase :
(i) 60 by 5%
(ii) 20 by 15%
(iii) 48 by \( 12\frac{1}{2}\% \)
(iv) 80 by 140%
(v) 1000 by 3.5%
Answer:
(i) Increase 60 by 5%
Rate of rise = 5%
Value of increase = 5% of 60 = \( \frac{5}{100} \times 60 = 3 \)
New increased value = 60 + 3 = 63

(ii) Increase 20 by 15%
Rate of rise = 15%
Value of increase = 15% of 20 = \( \frac{15}{100} \times 20 = 3 \)
New increased value = 20 + 3 = 23

(iii) Increase 48 by \( 12\frac{1}{2}\% \)
Percentage rise = \( 12\frac{1}{2}\% = \frac{25}{2}\% \)
Value of increase = \( \frac{25}{2}\% \text{ of } 48 \)

\( \implies \frac{25}{2 \times 100} \times 48 = 6 \)
New increased value = 48 + 6 = 54

(iv) Increase 80 by 140%
Percentage rise = 140%
Value of increase = 140% of 80 = \( \frac{140}{100} \times 80 = 112 \)
New increased value = 80 + 112 = 192

(v) Increase 1000 by 3.5%
Percentage rise = 3.5%
Value of increase = 3.5% of 1000 = \( \frac{3.5}{100} \times 1000 = 35 \)
New increased value = 1000 + 35 = 1035
In simple words: To increase a number, first find the given percentage of that number. Then, add this extra amount to the starting number.

Exam Tip: When a percentage is greater than 100%, like 140%, the increased amount will be larger than the starting number itself.

Question 3. Decrease :
(i) 80 by 20%
(ii) 300 by 10%
(iii) 50 by 12.5%
Answer:
(i) Decrease 80 by 20%
Percentage reduction = 20%
Value of decrease = 20% of 80 = \( \frac{20}{100} \times 80 = 16 \)
New decreased value = 80 - 16 = 64

(ii) Decrease 300 by 10%
Percentage reduction = 10%
Value of decrease = 10% of 300 = \( \frac{10}{100} \times 300 = 30 \)
New decreased value = 300 - 30 = 270

(iii) Decrease 50 by 12.5%
Percentage reduction = 12.5%
Value of decrease = 12.5% of 50

\( \implies \frac{125}{10 \times 100} \times 50 = 6.25 \)
New decreased value = 50 - 6.25 = 43.75
In simple words: To decrease a number, first calculate the percentage value. Then, subtract this calculated value from the starting number.

Exam Tip: Be careful with decimals during subtraction. For example, when subtracting 6.25 from 50, write 50 as 50.00 to align the decimal points correctly.

Question 4. What number :
(i) When increased by 10% becomes 88 ?
(ii) When increased by 15% becomes 230 ?
(iii) When decreased by 15% becomes 170 ?
(iv) When decreased by 40% becomes 480 ?
(v) When increased by 100% becomes 100 ?
(vi) When decreased by 50% becomes 50 ?
Answer:
(i) Let us assume the starting number is 100.
Increase = 10% of 100 = 10
New number after increase = 100 + 10 = 110
If the increased value is 110, the starting number is 100.
If the actual increased value is 88, then the starting number is:

\( \implies \frac{100}{110} \times 88 = 80 \)

(ii) Let us assume the starting number is 100.
Increase = 15% of 100 = 15
New number after increase = 100 + 15 = 115
If the increased value is 115, the starting number is 100.
If the actual increased value is 230, then the starting number is:

\( \implies \frac{100}{115} \times 230 = 200 \)

(iii) Let us assume the starting number is 100.
Decrease = 15% of 100 = 15
New number after decrease = 100 - 15 = 85
If the decreased value is 85, the starting number is 100.
If the actual decreased value is 170, then the starting number is:

\( \implies \frac{100}{85} \times 170 = 200 \)

(iv) Let us assume the starting number is 100.
Decrease = 40% of 100 = 40
New number after decrease = 100 - 40 = 60
If the decreased value is 60, the starting number is 100.
If the actual decreased value is 480, then the starting number is:

\( \implies \frac{100}{60} \times 480 = 800 \)

(v) Let us assume the starting number is 100.
Increase = 100% of 100 = 100
New number after increase = 100 + 100 = 200
If the increased value is 200, the starting number is 100.
If the actual increased value is 100, then the starting number is:

\( \implies \frac{100}{200} \times 100 = 50 \)

(vi) Let us assume the starting number is 100.
Decrease = 50% of 100 = 50
New number after decrease = 100 - 50 = 50
If the decreased value is 50, the starting number is 100.
If the actual decreased value is 50, then the starting number is:

\( \implies \frac{100}{50} \times 50 = 100 \)
In simple words: To find the starting number, we assume it is 100. We find what 100 becomes after the change, and then use unitary method to find the real number.

Exam Tip: You can also solve these using algebra by setting the starting number as \( x \). For example, \( x + 0.10x = 88 \) gives \( 1.10x = 88 \), so \( x = 80 \).

Question 5. The price of a car is lowered by 20% to Rs. 40,000. What was the original price ? Also, find the reduction in price.
Answer:
Let us assume the starting price of the vehicle = Rs. 100
Discount rate = 20%, which is Rs. 20
\( \dots \) Discounted price = Rs. 100 - Rs. 20 = Rs. 80
If the discounted price is Rs. 80, the starting price is Rs. 100.
When the actual discounted price is Rs. 40,000, the starting price will be:

\( \implies \frac{100 \times 40000}{80} = \text{Rs. } 50,000 \)
Therefore, the amount reduced in price = Rs. 50,000 - Rs. 40,000 = Rs. 10,000
In simple words: The car's price dropped by 20%, meaning the buyer paid 80% of the original cost. Since 80% is Rs. 40,000, the full price was Rs. 50,000, saving them Rs. 10,000.

Exam Tip: Always read the question carefully to see if you need to find just the original price or both the original price and the actual price reduction.

Question 6. If the price of an article is increased by 25%, The increase is Rs. 10. Find the new price.
Answer:
Let us assume the initial price of the item = Rs. 100
Percentage price rise = 25%, which means an increase of Rs. 25
\( \dots \) New price after the rise = Rs. 100 + Rs. 25 = Rs. 125
If the increase is Rs. 25, the new price is Rs. 125.
When the actual price rise is Rs. 10, the new price will be:

\( \implies \frac{125 \times 10}{25} = \text{Rs. } 50 \)
In simple words: A 25% increase is equal to Rs. 10. This means the starting price of the item was Rs. 40. Adding the Rs. 10 increase gives a new price of Rs. 50.

Exam Tip: Be careful to distinguish between the starting price and the new increased price. The question specifically asks for the new price.

Question 7. If the price of an article is reduced by 10%, the reduction is Rs. 40. What is the old price ?
Answer:
Let us assume the original cost of the item = Rs. 100
Reduction rate = 10%, which equals Rs. 10
If the reduction amount is Rs. 10, the original cost is Rs. 100.
When the actual reduction amount is Rs. 40, the original cost is:

\( \implies \frac{100 \times 40}{10} = \text{Rs. } 400 \)
In simple words: A 10% price drop is equal to Rs. 40. Since 10% is one-tenth of the total, the original price must be 10 times larger, which is Rs. 400.

Exam Tip: Since 10% of the old price is Rs. 40, you can quickly double-check your answer by verifying that \( 10\% \text{ of } 400 = 40 \).

Question 8. The price of a chair is reduced by 25%. What is the ratio of:
(i) Change in price to the old price.
(ii) Old price to the new price.
Answer:
Let us assume the starting cost of the chair = Rs. 100
Price drop = 25% of Rs. 100 = Rs. 25
\( \dots \) The discounted price = Rs. 100 - Rs. 25 = Rs. 75

(i) Ratio of the price difference to the starting cost:

\( \implies 25 : 100 = 1 : 4 \) (by dividing both terms by 25)

(ii) Ratio of the starting cost to the discounted cost:

\( \implies 100 : 75 = 4 : 3 \) (by dividing both terms by 25)
In simple words: A 25% discount means the price drops by Rs. 25 on a Rs. 100 chair, leaving the new price at Rs. 75. The ratio of the price drop to the old price is 1 to 4, and the ratio of the old price to the new price is 4 to 3.

Exam Tip: Write ratios in their simplest form by dividing both sides by their highest common factor.

Question 9. If x is 20% less than y, find :
(i) \( \frac{x}{y} \)
(ii) \( \frac{y-x}{y} \)
(iii) \( \frac{x}{y-x} \)
Answer:
Let us assume that \( y = 100 \).
Since \( x \) is 20% smaller than \( y \), the difference is 20.
Therefore, \( x = 100 - 20 = 80 \).

(i) Finding \( \frac{x}{y} \):

\( \implies \frac{80}{100} = \frac{4}{5} \) (by simplifying by 20)

(ii) Finding \( \frac{y-x}{y} \):

\( \implies \frac{100 - 80}{100} = \frac{20}{100} = \frac{1}{5} \) (by simplifying by 20)

(iii) Finding \( \frac{x}{y-x} \):

\( \implies \frac{80}{100 - 80} = \frac{80}{20} = 4 \) (by simplifying by 20)
In simple words: If \( y \) is 100, then \( x \) is 80. We use these two numbers to find the fraction values requested in each part of the question.

Exam Tip: Choosing 100 for the base variable \( y \) is a smart shortcut that makes percentage comparisons very simple and fast to solve.

Question 10. If x is 30% more than y; find :
(i) \( \frac{x}{y} \)
(ii) \( \frac{y+x}{x} \)
(iii) \( \frac{y}{y-x} \)
Answer:
Let \( y = a \).
Since \( x \) is 30% larger than \( y \), we have:
\( x = a \times \frac{100+30}{100} = \frac{13}{10}a \)

(i) Finding \( \frac{x}{y} \):

\( \implies \frac{\frac{13}{10}a}{a} = \frac{13}{10} \)

(ii) Finding \( \frac{y+x}{x} \):

\( \implies \frac{a + \frac{13}{10}a}{\frac{13}{10}a} = \frac{\frac{23}{10}a}{\frac{13}{10}a} = \frac{23}{13} \)

(iii) Finding \( \frac{y}{y-x} \):

\( \implies \frac{a}{a - \frac{13}{10}a} = \frac{a}{-\frac{3}{10}a} = -\frac{10}{3} \)
In simple words: If we assume \( y \) is 10, then \( x \) is 13 because it is 30% larger. Putting these values into the fractions gives us our final answers.

Exam Tip: Pay close attention to negative signs. In part (iii), since \( x \) is larger than \( y \), \( y - x \) results in a negative value, making the final fraction negative.

Question 11. The weight of a machine is 40 kg. By mistake it was weighed as 40.8 kg. Find the error percent.
Answer:
The correct mass of the machine = 40 kg
The incorrect recorded mass = 40.8 kg
\( \dots \) Mistake in calculation = 40.8 kg - 40 kg = 0.8 kg
Percentage of error:

\( \implies \frac{0.8}{40} \times 100\% = \frac{8 \times 100}{10 \times 40}\% = 2\% \)
In simple words: The scale showed 0.8 kg more than it should have. We find what percentage 0.8 kg is of the real weight (40 kg), which gives us a 2% error.

Exam Tip: The percentage error is always calculated with respect to the actual, correct value, not the incorrect reading.

Question 12. From a cask, containing 450 litres of petrol, 8% of the petrol was lost by leakage and evaporation. How many litres of petrol was left in the cask ?
Answer:
Total quantity of fuel initially in the container = 450 litres
Percentage of fuel lost = 8%
Quantity of fuel lost to leaks and evaporation = 8% of 450 litres

\( \implies \frac{8}{100} \times 450 = 36\text{ litres} \)
Therefore, the amount of fuel remaining in the container = 450 litres - 36 litres = 414 litres
In simple words: Out of 450 litres, 8% (which is 36 litres) was lost. Subtracting this loss from the starting amount leaves 414 litres of petrol.

Exam Tip: Always read the final question sentence. If it asks for the amount left, you must subtract the lost amount from the total.

Question 13. An alloy consists of 13 parts of copper, 7 parts of zinc and 5 parts of nickel. What is the percentage of each metal in the alloy?
Answer:
Parts of copper = 13
Parts of zinc = 7
Parts of nickel = 5
Total number of parts in the mixture = 13 + 7 + 5 = 25

Percentage of copper in the mixture:

\( \implies \frac{13}{25} \times 100\% = 52\% \)

Percentage of zinc in the mixture:

\( \implies \frac{7}{25} \times 100\% = 28\% \)

Percentage of nickel in the mixture:

\( \implies \frac{5}{25} \times 100\% = 20\% \)
In simple words: The entire mixture is divided into 25 equal parts. We find what percentage each metal's parts make of the total 25 parts.

Exam Tip: You can verify your final percentages by adding them up: \( 52\% + 28\% + 20\% = 100\% \). They must always sum to 100%.

Question 14. In an examination, first division marks are 60%. A student secures 538 marks and misses the first division by 2 marks. Find the total marks of the examination.
Answer:
Minimum percentage required to get first division = 60%
Marks obtained by the candidate = 538
Since the candidate fell short of first division by 2 marks:
\( \dots \) Required marks for first division = 538 + 2 = 540
Let the maximum possible marks of the test be \( T \).
According to the given conditions:

\( \implies 60\% \text{ of } T = 540 \)

\( \implies \frac{60}{100} \times T = 540 \)

\( \implies T = \frac{540 \times 100}{60} = 900 \)
Hence, the total maximum marks of the exam are 900.
In simple words: The student needed 540 marks to reach the 60% limit. Since 60% of the total marks is 540, the total exam paper was out of 900 marks.

Exam Tip: Set up the equation using a variable like \( T \) for the total marks, then isolate \( T \) by moving the fraction to the other side.

Question 15. Out of 1200 pupils in a school, 900 are boys and the rest are girls. If 20% of the boys and 30% of the girls wear spectacles, find :
(i) how many pupils in all, wear spectacles ?
(ii) what percent of the total number of pupils wear spectacles ?
Answer:
Total strength of the school = 1200 students
Number of male students = 900
\( \dots \) Number of female students = 1200 - 900 = 300

Number of boys wearing glasses = 20% of 900

\( \implies \frac{20}{100} \times 900 = 180 \)

Number of girls wearing glasses = 30% of 300

\( \implies \frac{30}{100} \times 300 = 90 \)

(i) Total count of students wearing glasses = 180 + 90 = 270

(ii) Percentage of the school population wearing glasses:

\( \implies \frac{270}{1200} \times 100\% = \frac{270}{12}\% = 22.5\% \)
In simple words: There are 900 boys and 300 girls. 180 boys and 90 girls wear glasses, making 270 students in total. This means 22.5% of all students wear glasses.

Exam Tip: Find the actual numbers of boys and girls wearing spectacles separately before adding them up to get the total.

Question 16. Out of 25 identical bulbs, 17 are red, 3 are black and the remaining are yellow. Find the difference between the numbers of red and yellow bulbs and express this difference as percent.
Answer:
Total quantity of lightbulbs = 25
Quantity of red lightbulbs = 17
Quantity of black lightbulbs = 3
Combined count of red and black lightbulbs = 17 + 3 = 20
\( \dots \) Quantity of yellow lightbulbs = 25 - 20 = 5

Difference in count between red and yellow lightbulbs = 17 - 5 = 12
Percentage of this difference compared to total lightbulbs:

\( \implies \frac{12}{25} \times 100\% = 48\% \)
In simple words: Out of 25 bulbs, 17 are red and 5 are yellow. The difference between them is 12 bulbs. These 12 bulbs represent 48% of the total collection of 25 bulbs.

Exam Tip: Always base your final percentage calculation on the total number of items (25), not just the sum of the red and yellow ones.

Question 17. A number first increases by 20% and then decreases by 20%. Find the percentage increase or decrease on the whole.
Answer:
Let us assume the starting number is 100.
First Step (Increase):
Increase amount = 20% of 100 = 20
Value after this increase = 100 + 20 = 120

Second Step (Decrease):
Decrease amount = 20% of 120 = 24
Value after this decrease = 120 - 24 = 96

Comparing the final value to the starting value:
Net drop on the whole = 100 - 96 = 4
Percentage reduction on the whole:

\( \implies \frac{4}{100} \times 100\% = 4\% \)
In simple words: If you start with 100, a 20% rise brings it to 120. Then, a 20% drop on 120 lowers it by 24, ending at 96. This is a total loss of 4%.

Exam Tip: Note that a 20% increase followed by a 20% decrease does not bring you back to the original number, because the second percentage is calculated on a larger base.

Question 18. A number is first decreased by 40% and then again decreased by 60%. Find the percentage increase or decrease on the whole.
Answer:
Let us assume the starting number is 100.
First Step (Decrease):
Decrease amount = 40% of 100 = 40
Value after this decrease = 100 - 40 = 60

Second Step (Decrease):
Decrease amount = 60% of 60 = 36
Value after this second decrease = 60 - 36 = 24

Comparing the final value to the starting value:
Net reduction on the whole = 100 - 24 = 76
Percentage decrease on the whole:

\( \implies \frac{76}{100} \times 100\% = 76\% \)
In simple words: Starting with 100, a 40% cut leaves you with 60. Then, taking away another 60% from that 60 leaves you with just 24. This means your total drop is 76%.

Exam Tip: Consecutive decreases are applied one after the other. Make sure to calculate the second decrease on the already reduced value, not on the starting number.

Question 19. If 150% of a number is 750, find 60% of this number.
Answer:
Let us assume the unknown number is \( x \).
According to the problem, 150% of \( x \) is 750:

\( \implies \frac{150}{100} \times x = 750 \)

\( \implies x = \frac{750 \times 100}{150} = 500 \)
So, the number we are looking for is 500.

Now, we calculate 60% of 500:

\( \implies 500 \times \frac{60}{100} = 300 \)
In simple words: If one and a half times a number is 750, then the number itself must be 500. Finding 60% of 500 gives us 300.

Exam Tip: You can also solve this in one step by multiplying 750 by the ratio of the percentages: \( 750 \times \frac{60}{150} = 300 \).