Selina Concise Solutions for ICSE Class 6 Mathematics Chapter 16 Percentage

Percentage: Important Points

• Percent: This means out of one hundred and is written with the symbol %. For example, 10% or 15%.
• A percentage can be written as a fraction or a decimal:
  \( 1\% = \frac{1}{100} = 0.01 \)
  \( 38\% = \frac{38}{100} = 0.38 \)
• Any fraction or decimal can also be written as a percent.
• We use percentage calculations in profit and loss, interest, and many other areas.
• Percentage: This is simply the name we give when a quantity is shown in its percent form.
• To convert a Fraction or Decimal to a Percentage: Multiply the number by 100 and add the % sign.
• To convert a Percentage to a Fraction or Decimal: Take away the % sign and divide the number by 100. Then simplify the fraction or turn it into a decimal.
• To Express one Quantity as a Percentage of Another: Divide the first quantity by the second quantity, and then multiply the result by 100%.
• Keep in mind:
  • Percentages do not have any units.
  • To compare two values as a percentage, they must both be in the same units of measurement.
• To find the Increase or Decrease Percent:
  \( \text{Increase \%} = \frac{\text{Increase in value}}{\text{Original value}} \times 100\% \)
  \( \text{Decrease \%} = \frac{\text{Decrease in value}}{\text{Original value}} \times 100\% \)

Exercise 16(A)

Question 1. Express each of the following statements in the percentage form :
(i) 13 out of 20
(ii) 21 eggs out of 30 are good
Answer:
(i) To convert 13 out of 20, we write it as a fraction and multiply by 100:
\( \frac{13}{20} \times 100 = 13 \times 5 = 65\% \)
(ii) To find the percentage of good eggs, we write the ratio as a fraction and multiply by 100:
\( \frac{21}{30} \times 100 = 7 \times 10 = 70\% \)
So, 70% of the eggs are good.
In simple words: Write the numbers as a fraction and then multiply by 100 to get the percentage.

Exam Tip: Always write the final answer with the percent sign (%) to show it is a percentage.

Question 2. Express the following fractions as percent :
(i) \( \frac{3}{200} \)
(ii) \( \frac{5}{6} \)
(iii) \( \frac{65}{80} \)
(iv) \( \frac{2}{3} \)
Answer:
(i) Multiply the fraction by 100:
\( \frac{3}{200} \times 100 = \frac{3}{2} = 1\frac{1}{2}\% = 1.5\% \)
(ii) Multiply the fraction by 100:
\( \frac{5}{6} \times 100 = \frac{250}{3} = 83\frac{1}{3}\% \)
(iii) Multiply by 100 and simplify:
\( \frac{65}{80} \times 100 = \frac{65 \times 5}{4} = \frac{325}{4} = 81\frac{1}{4}\% = 81.25\% \)
(iv) Multiply the fraction by 100:
\( \frac{2}{3} \times 100 = \frac{200}{3} = 66\frac{2}{3}\% \)
In simple words: Turn any fraction into a percentage by multiplying it by 100.

Exam Tip: You can leave your answer as a mixed number or convert it to a decimal, but mixed numbers are often preferred in textbook exams.

Question 3. Express as percent:
(i) 0.10
(ii) 0.02
(iii) 0.7
(iv) 0.15
(v) 0.032
Answer:
(i) Change the decimal to a fraction and multiply by 100:
\( 0.10 = \frac{10}{100} \times 100\% = 10\% \)
(ii) Convert the decimal to a fraction and multiply by 100:
\( 0.02 = \frac{2}{100} \times 100\% = 2\% \)
(iii) Convert the decimal to a fraction and multiply by 100:
\( 0.7 = \frac{7}{10} \times 100\% = 70\% \)
(iv) Convert the decimal to a fraction and multiply by 100:
\( 0.15 = \frac{15}{100} \times 100\% = 15\% \)
(v) Convert the decimal to a fraction and multiply by 100:
\( 0.032 = \frac{32}{1000} \times 100\% = 3.2\% \)
In simple words: To change a decimal to a percent, shift the decimal point two spots to the right and add a % symbol.

Exam Tip: Ensure you count the correct number of decimal places when converting a decimal into a fraction first.

Question 4. Convert into fractions in their lowest terms:
(i) 8%
(ii) 20%
(iii) 85%
(iv) 250%
(v) 12 \( \frac{1}{2} \)%
Answer:
(i) Write the percentage as a fraction with a denominator of 100 and simplify:
\( 8\% = \frac{8}{100} = \frac{2}{25} \)
(ii) Write as a fraction and divide by common factors:
\( 20\% = \frac{20}{100} = \frac{1}{5} \)
(iii) Write as a fraction and reduce:
\( 85\% = \frac{85}{100} = \frac{17}{20} \)
(iv) Convert the percentage to a fraction and simplify:
\( 250\% = \frac{250}{100} = \frac{5}{2} = 2\frac{1}{2} \)
(v) Convert the mixed number to an improper fraction, then divide by 100:
\( 12\frac{1}{2}\% = \frac{25}{2}\% = \frac{25}{2 \times 100} = \frac{1}{8} \)
In simple words: Put the percentage number over 100, then simplify the fraction as much as you can.

Exam Tip: For mixed fractions, always convert them to an improper fraction before dividing by 100 to avoid mistakes.

Question 5. Express as decimal fractions :
(i) 25%
(ii) 108%
(iii) 95%
(iv) 4.5%
(v) 29.2%
Answer:
(i) Divide the percentage by 100 to get a decimal:
\( 25\% = \frac{25}{100} = 0.25 \)
(ii) Divide by 100:
\( 108\% = \frac{108}{100} = 1.08 \)
(iii) Divide by 100:
\( 95\% = \frac{95}{100} = 0.95 \)
(iv) Convert to a fraction and divide:
\( 4.5\% = \frac{45}{10 \times 100} = \frac{45}{1000} = 0.045 \)
(v) Convert to a fraction and divide:
\( 29.2\% = \frac{292}{10 \times 100} = \frac{292}{1000} = 0.292 \)
In simple words: To convert a percentage to a decimal, just divide the number by 100.

Exam Tip: Move the decimal point exactly two places to the left. For small numbers like 4.5%, make sure to add a placeholder zero (0.045).

Question 6. Express each of the following natural numbers as percent :
(i) 7
(ii) 2
(iii) 19.5
(iv) 5.37
Answer:
(i) Multiply the number by 100:
\( 7 = 7 \times 100\% = 700\% \)
(ii) Multiply by 100:
\( 2 = 2 \times 100\% = 200\% \)
(iii) Write the decimal as a fraction and multiply by 100:
\( 19.5 = \frac{195}{10} \times 100\% = 1950\% \)
(iv) Write the decimal as a fraction and multiply by 100:
\( 5.37 = \frac{537}{100} \times 100\% = 537\% \)
In simple words: To write any normal number or decimal as a percent, multiply it by 100 and add the % sign.

Exam Tip: Whole numbers and decimals greater than 1 will always convert to percentages greater than 100%.

Exercise 16(B)

Question 1. Express :
(i) Rs 5 as a percentage of Rs 25.
(ii) 80 paise as a percent of Rs 4.
(iii) 700 gm as a percentage of 2.8 kg.
(iv) 90 cm as a percent of 4.5 m.
Answer:
(i) Write as a fraction and multiply by 100:
\( \frac{5}{25} \times 100 = 20\% \)
(ii) Change Rs 4 to paise first (Rs 4 = 400 paise), then find the percentage:
\( \frac{80}{400} \times 100 = 20\% \)
(iii) Change 2.8 kg to grams (2.8 kg = 2800 gm), then calculate:
\( \frac{700}{2800} \times 100 = 25\% \)
(iv) Change 4.5 m to centimeters (4.5 m = 450 cm), then calculate:
\( \frac{90}{450} \times 100 = 20\% \)
In simple words: Make sure both numbers are in the same unit. Then write them as a fraction and multiply by 100.

Exam Tip: Never calculate the percentage directly if the units are different. Always convert them to match first.

Question 2. Express the first quantity as a percent of the second :
(i) 40 P, Rs 2
(ii) 500 gm, 6 kg
(iii) 42 seconds, 6 minutes
Answer:
(i) Convert Rs 2 to paise (Rs 2 = 200 P) to make units match, then divide:
\( \frac{40}{200} \times 100\% = 20\% \)
(ii) Convert 6 kg to grams (6 kg = 6000 gm), then find the percentage:
\( \frac{500}{6000} \times 100\% = \frac{50}{6}\% = \frac{25}{3}\% = 8\frac{1}{3}\% \approx 8.33\% \)
(iii) Convert 6 minutes to seconds (6 minutes = 360 seconds), then calculate:
\( \frac{42}{360} \times 100\% = \frac{420}{36}\% = 11\frac{2}{3}\% \approx 11.67\% \)
In simple words: Turn the larger unit into the smaller unit, write them as a fraction, and multiply by 100 to find the percentage.

Exam Tip: Convert units accurately using standard conversions: Rs 1 = 100 P, 1 kg = 1000 gm, and 1 minute = 60 seconds.

Question 3. Find the value of each of the following:
(i) 20% of Rs 150
(ii) 90% of 130
(iii) 15% of 2 minutes
(iv) 7.5 % of 500 kg.
Answer:
(i) Calculate 20% of Rs 150:
\( \frac{20}{100} \times 150 = \text{Rs } 30 \)
(ii) Calculate 90% of 130:
\( \frac{90}{100} \times 130 = 117 \)
(iii) Convert 2 minutes to 120 seconds first, then find the percentage:
\( 15\% \text{ of } 120 \text{ seconds} = \frac{15}{100} \times 120 = 18 \text{ seconds} \)
(iv) Calculate 7.5% of 500 kg:
\( \frac{7.5}{100} \times 500 = 7.5 \times 5 = 37.5 \text{ kg} \)
In simple words: Replace the percent sign with a division by 100, then multiply by the total amount to find your answer.

Exam Tip: For time calculations, converting minutes to seconds helps you get a clean whole number as your final result.

Question 4. If a man spends 70% of his income, what percent does he save?
Answer: Let his total income be 100%.
The man spends 70% of this income.
We can find his savings by subtracting the spent amount from the total:
\( 100\% - 70\% = 30\% \)
So, he saves 30% of his income.
In simple words: Since total income is always 100%, subtracting the 70% spent leaves 30% saved.

Exam Tip: You do not need to assume an actual money value to solve this; simple subtraction of percentages works directly.

Question 5. A girl gets 65 marks out of 80. What percent marks did she get?
Answer: Marks obtained = 65, Total marks = 80.
Write these marks as a fraction and multiply by 100 to get the percentage:
\( \text{Percentage} = \frac{65}{80} \times 100 = \frac{65 \times 5}{4} = \frac{325}{4} = 81.25\% \)
So, she got 81.25% (or \( 81\frac{1}{4}\% \)) marks.
In simple words: Divide the marks scored by the total possible marks, then multiply the answer by 100.

Exam Tip: Simplify the fraction \( \frac{100}{80} \) to \( \frac{5}{4} \) early in your steps to make the multiplication easier.

Question 6. A class contains 25 children, of which 6 are girls. What percentage of the class are the boys.
Answer: Total number of students = 25.
Number of girls = 6.
Find the number of boys by subtracting the girls from the total:
\( 25 - 6 = 19 \text{ boys} \)
Now, find the percentage of boys in the class:
\( \text{Percentage of boys} = \frac{19}{25} \times 100 = 19 \times 4 = 76\% \)
Therefore, 76% of the class are boys.
In simple words: Find the number of boys first by subtracting girls from the total. Then divide the boys by the total students and multiply by 100.

Exam Tip: Be careful not to calculate the percentage of girls by mistake; make sure to subtract first to find the boys.

Question 7. A tin contains 20 litres of petrol. Due to leakage, 3 litres of petrol is lost. What percent is still present in the tin ?
Answer: Total quantity of petrol = 20 litres.
Amount of petrol lost = 3 litres.
Find the remaining petrol in the tin:
\( 20 - 3 = 17 \text{ litres} \)
Calculate the percentage of remaining petrol:
\( \text{Percentage of petrol present} = \frac{17}{20} \times 100 = 17 \times 5 = 85\% \)
So, 85% of the petrol is still inside the tin.
In simple words: Subtract the leaked petrol to see what is left. Then, find what percent that remainder is of the original amount.

Exam Tip: Always verify if the question asks for the percentage of lost petrol or remaining petrol before starting your calculation.

Question 8. An alloy of copper and zinc contains 45% copper and the rest is zinc. Find the weight of zinc in 20 kg of the alloy.
Answer: Total weight of the alloy = 20 kg.
Percentage of copper = 45%.
Find the weight of copper first:
\( 45\% \text{ of } 20 \text{ kg} = \frac{45}{100} \times 20 = 9 \text{ kg} \)
Now, find the weight of zinc by subtracting the copper weight from the total:
\( 20 - 9 = 11 \text{ kg} \)
Alternatively, find the zinc percentage first:
\( 100\% - 45\% = 55\% \text{ zinc} \)
\( 55\% \text{ of } 20 \text{ kg} = \frac{55}{100} \times 20 = 11 \text{ kg} \)
So, the weight of zinc is 11 kg.
In simple words: Find the copper weight and subtract it from the total weight to find the zinc weight.

Exam Tip: Subtracting percentages first (100% - 45% = 55%) and then finding the weight is often faster and less prone to mistakes.

Question 9. A boy got 60 out of 80 in Hindi, 75 out of 100 in English and 65 out of 70 in Arithmetic. In which subject his percentage of marks the best ? Also, find his overall percentage.
Answer: Find the percentage of marks scored in each subject:
Hindi: \( \frac{60}{80} \times 100 = 75\% \)
English: \( \frac{75}{100} \times 100 = 75\% \)
Arithmetic: \( \frac{65}{70} \times 100 = \frac{650}{7}\% = 92\frac{6}{7}\% \approx 92.86\% \)
Comparing the percentages, his score is best in Arithmetic.
To find the overall percentage:
Total marks scored = \( 60 + 75 + 65 = 200 \)
Total maximum marks = \( 80 + 100 + 70 = 250 \)
Overall percentage = \( \frac{200}{250} \times 100 = 80\% \)
So, he performed best in Arithmetic, and his overall percentage is 80%.
In simple words: Find the percent for each subject to see which is highest. Then, add all marks scored and divide by the total possible marks to find the overall percentage.

Exam Tip: Do not just average the three percentages (75%, 75%, 92.86%) because the total marks for each subject are not equal. Always sum the marks scored and divide by the total possible marks.

Question 10. In a camp, there were 500 soldiers. 60 more soldiers joined them. What percent of the earlier (original) number have joined the camp.
Answer: Original number of soldiers = 500.
Number of new soldiers who joined = 60.
Calculate the percentage of new soldiers compared to the original number:
\( \text{Percentage} = \frac{60}{500} \times 100 = 12\% \)
So, 12% of the original strength joined the camp.
In simple words: Divide the number of new soldiers by the original number of soldiers, then multiply by 100 to get the percent.

Exam Tip: Be sure to divide by the original count (500) rather than the new total count (560), as the question asks for a percentage of the "earlier" number.

Question 11. In a plot of ground of area 6000 sq. m, only 4500 sq. m is allowed for construction. What percent is to be left without construction ?
Answer: Total ground area = 6000 sq. m.
Allowed construction area = 4500 sq. m.
Find the area to be left empty without construction:
\( 6000 - 4500 = 1500 \text{ sq. m} \)
Calculate the percentage of this empty area compared to the total:
\( \text{Percentage} = \frac{1500}{6000} \times 100 = 25\% \)
Thus, 25% of the plot must be left without construction.
In simple words: Subtract the building area from the total area to find the leftover space. Then find what percentage that space is of the total area.

Exam Tip: Double check that you are calculating the percentage of the plot "left without construction", and not the construction percentage.

Question 12. Mr. Sharma has a monthly salary of Rs 8,000. If he spends Rs 6,400 every month; find :
(i) his monthly expenditure as percent.
(ii) his monthly savings as percent.
Answer: Monthly salary = Rs 8000.
Monthly expenditure = Rs 6400.
First, find his monthly savings:
\( \text{Monthly savings} = \text{Rs } 8000 - \text{Rs } 6400 = \text{Rs } 1600 \)
(i) Calculate the expenditure percentage:
\( \text{Expenditure \%} = \frac{6400}{8000} \times 100 = 80\% \)
(ii) Calculate the savings percentage:
\( \text{Savings \%} = \frac{1600}{8000} \times 100 = 20\% \)
So, his monthly spending is 80% and his savings are 20%.
In simple words: Divide his spending by his salary and multiply by 100 to get the spending percent. His savings percent is the remaining amount out of 100%.

Exam Tip: You can quickly find the savings percentage by subtracting the expenditure percentage (80%) from 100%.

Question 13. The monthly salary of Rohit is Rs 24,000. If his salary increases by 12%, find his new monthly salary
Answer: Original salary of Rohit = Rs 24000.
Percentage increase = 12%.
Find the amount of increase in salary:
\( \text{Salary increase} = 12\% \text{ of } \text{Rs } 24000 = \frac{12}{100} \times 24000 = \text{Rs } 2880 \)
Add the increase to his original salary to get his new salary:
\( \text{New salary} = \text{Rs } 24000 + \text{Rs } 2880 = \text{Rs } 26880 \)
So, his new monthly salary is Rs 26,880.
In simple words: Work out 12% of Rohit's salary to find his raise, then add that raise to his old salary.

Exam Tip: You can directly multiply the original salary by 1.12 to calculate the new salary in a single calculation step.

Question 14. In a sale, the price of an article is reduced by 30%. If the original price of the article is Rs 1,800, find :
(i) the reduction in the price of the article
(ii) reduced price of the article.
Answer: Original price of the article = Rs 1800.
Percentage reduction = 30%.
(i) Find the amount reduced:
\( \text{Reduction in price} = 30\% \text{ of } \text{Rs } 1800 = \frac{30}{100} \times 1800 = \text{Rs } 540 \)
(ii) Calculate the new price after reduction:
\( \text{Reduced price} = \text{Original price} - \text{Reduction} = \text{Rs } 1800 - \text{Rs } 540 = \text{Rs } 1260 \)
So, the price reduction is Rs 540, and the new reduced price is Rs 1,260.
In simple words: Find 30% of the price to see how much money is saved, then subtract that from the old price to get the new price.

Exam Tip: Clearly label both parts of your answer with the Rs symbol to ensure you receive full credit.

Question 15. Evaluate :
(i) 30% of 200 + 20% of 450 - 25% of 600
(ii) 10% of Rs 450 - 12% of Rs 500 + 8% of Rs 500.
Answer:
(i) Calculate each part of the expression:
\( 30\% \text{ of } 200 = \frac{30}{100} \times 200 = 60 \)
\( 20\% \text{ of } 450 = \frac{20}{100} \times 450 = 90 \)
\( 25\% \text{ of } 600 = \frac{25}{100} \times 600 = 150 \)
Now substitute these back into the expression:
\( 60 + 90 - 150 = 150 - 150 = 0 \)
(ii) Calculate each part of the expression:
\( 10\% \text{ of Rs } 450 = \frac{10}{100} \times 450 = \text{Rs } 45 \)
\( 12\% \text{ of Rs } 500 = \frac{12}{100} \times 500 = \text{Rs } 60 \)
\( 8\% \text{ of Rs } 500 = \frac{8}{100} \times 500 = \text{Rs } 40 \)
Now substitute these back into the expression:
\( \text{Rs } 45 - \text{Rs } 60 + \text{Rs } 40 = \text{Rs } 85 - \text{Rs } 60 = \text{Rs } 25 \)
In simple words: Work out the value of each percentage term first, then add and subtract them from left to right.

Exam Tip: Be careful with the signs (addition and subtraction) between the terms while carrying out the operations.

Exercise 16(C)

Question 1. The price of rice rises from Rs. 30 per kg to Rs. 36 per kg. Find the percentage rise in the price of rice.
Answer: Original price of rice = Rs 30 per kg.
Increased price of rice = Rs 36 per kg.
Find the increase in price:
\( \text{Price rise} = \text{Rs } 36 - \text{Rs } 30 = \text{Rs } 6 \)
Calculate the percentage rise based on the original price:
\( \text{Percentage rise} = \frac{\text{Price rise}}{\text{Original price}} \times 100 = \frac{6}{30} \times 100 = 20\% \)
So, the percentage rise in the price of rice is 20%.
In simple words: Find the rise in price by subtracting the old price from the new price. Then, divide the rise by the old price and multiply by 100.

Exam Tip: Always make sure to use the initial (original) price as the base in your denominator when calculating percentage change.

Question 2. The population of a small locality was 4000 in 1979 and 4500 in 1981, By what percent had the population increase ?
Answer:
Population in the year 1979 = 4,000
Population in the year 1981 = 4,500
Total growth in population = \( 4,500 - 4,000 = 500 \)
Percentage of increase in population = \( \frac{500}{4,000} \times 100 = 12.5\% \)
In simple words: To find the percentage growth, calculate how many more people there are now. Divide that increase by the original number of people and multiply by 100.

Exam Tip: Always divide the actual increase by the starting population (from the earlier year) to find the correct percentage growth.

Question 3. The price of a scooter was Rs. 8000 in 1975. It came down to Rs. 6000 in 1980. By what percent had the price of the scooter came down ?
Answer:
Starting cost of the scooter = Rs. 8,000
Reduced cost of the scooter = Rs. 6,000
Decrease in price = \( \text{Rs. } 8,000 - \text{Rs. } 6,000 = \text{Rs. } 2,000 \)
Percentage of price reduction = \( \frac{2,000}{8,000} \times 100 = 25\% \)
In simple words: Subtract the new price from the old price to see how much money was saved. Then, divide that amount by the starting price and multiply by 100.

Exam Tip: Be careful not to use the lower price as your denominator. Percentage calculations are always based on the starting value.

Question 4. Find the resulting quantity when :
(i) Rs. 400 is decreased by 8%.
(ii) 25 km is increased by 5%.
(iii) a speed of 600 km/h is increased by \( 12\frac{1}{2} \% \)
(iv) there is 2.5% increase in a salary of Rs. 62, 500.
Answer:
(i) Reducing Rs. 400 by 8%:
\( 8\% \text{ of Rs. } 400 = \frac{8}{100} \times 400 = \text{Rs. } 32 \)
Resulting amount = \( \text{Rs. } 400 - \text{Rs. } 32 = \text{Rs. } 368 \)

(ii) Increasing 25 km by 5%:
\( 5\% \text{ of } 25 \text{ km} = \frac{5}{100} \times 25 = 1.25 \text{ km} \)
Resulting distance = \( 25 \text{ km} + 1.25 \text{ km} = 26.25 \text{ km} \)

(iii) Increasing a speed of 600 km/h by \( 12\frac{1}{2}\% \):
\( 12\frac{1}{2}\% = \frac{25}{2}\% \)
\( \frac{25}{2}\% \text{ of } 600 \text{ km/h} = \frac{25}{2 \times 100} \times 600 = 75 \text{ km/h} \)
Resulting speed = \( 600 \text{ km/h} + 75 \text{ km/h} = 675 \text{ km/h} \)

(iv) Increasing a salary of Rs. 62,500 by 2.5%:
Amount of increase = \( 2.5\% \text{ of Rs. } 62,500 = \frac{2.5}{100} \times 62,500 = \frac{25}{1000} \times 62,500 = \text{Rs. } 1562.50 \)
Resulting salary = \( \text{Rs. } 62,500 + \text{Rs. } 1562.50 = \text{Rs. } 64062.50 \)
In simple words: To change a number by a percent, find that percent of the number first. Then add it if the question says increase, or subtract it if it says decrease.

Exam Tip: When dealing with fractional percentages like \( 12\frac{1}{2}\% \), write them as improper fractions like \( \frac{25}{2}\% \) before multiplying by the quantity.

Question 5. The population of a village decreased by 12%. If the original population was 25,000, find the population after decrease ?
Answer:
Starting population of the village = 25,000
Decrease rate = 12%
Number of people reduced = \( 12\% \text{ of } 25,000 = \frac{12}{100} \times 25,000 = 3,000 \)
Final population = \( 25,000 - 3,000 = 22,000 \)
In simple words: Calculate 12 percent of the 25,000 villagers, which is 3,000. Take those 3,000 people away from the start total to find the new population of 22,000.

Exam Tip: Another direct way is to multiply the original population by 88% (since 100% - 12% = 88%) to get the final answer directly.

Question 6. Out of a salary of Rs. 13,500,1 keep 1/3 as savings. Of the remaining money, I spend 50% on food and 20% on house rent. How much do I spend on food and house rent ?
Answer:
Total monthly earnings = Rs. 13,500
Savings = \( \frac{1}{3} \times \text{Rs. } 13,500 = \text{Rs. } 4,500 \)
Remaining salary = \( \text{Rs. } 13,500 - \text{Rs. } 4,500 = \text{Rs. } 9,000 \)
Money spent on food = \( 50\% \text{ of Rs. } 9,000 = \frac{50}{100} \times 9,000 = \text{Rs. } 4,500 \)
Money spent on house rent = \( 20\% \text{ of Rs. } 9,000 = \frac{20}{100} \times 9,000 = \text{Rs. } 1,800 \)
Total combined expenditure on food and rent = \( \text{Rs. } 4,500 + \text{Rs. } 1,800 = \text{Rs. } 6,300 \)
In simple words: Save one-third of the salary first and find what is left. Then, work out the food and rent costs using that leftover money, and add them up.

Exam Tip: Read carefully: the spending percentages apply to the remaining salary after savings are deducted, not the original total salary.

Question 7. A tank can hold 50 litres of water. At present, it is only 30% full. How many litres of water shall I put into the tank so that it becomes 50% full ?
Answer:
Total capacity of the tank = 50 litres
Current water volume (30% of capacity) = \( \frac{30}{100} \times 50 = 15 \text{ litres} \)
Target water volume (50% of capacity) = \( \frac{50}{100} \times 50 = 25 \text{ litres} \)
Quantity of water to be added = \( 25 \text{ litres} - 15 \text{ litres} = 10 \text{ litres} \)
In simple words: The tank has 15 litres now, and we want it to have 25 litres. This means we must add 10 more litres of water to hit the mark.

Exam Tip: You can also find the required percentage increase first (50% - 30% = 20%) and then directly calculate 20% of 50 litres.

Question 8. In an election, there are a total of 80,000 voters and two candidates, A and B. 80% of the voters go to the polls out of which 60% vote for A. How many votes does B get.
Answer:
Total registered voters = 80,000
Total votes cast = \( 80\% \text{ of } 80,000 = \frac{80}{100} \times 80,000 = 64,000 \)
Votes received by Candidate A = \( 60\% \text{ of } 64,000 = \frac{60}{100} \times 64,000 = 38,400 \)
Votes received by Candidate B = \( 64,000 - 38,400 = 25,600 \)
In simple words: First, calculate the 64,000 people who actually voted. Since 60% voted for candidate A, the remaining 40% voted for candidate B, which equals 25,600 votes.

Exam Tip: Be sure to calculate Candidate B's votes from the total votes actually polled (64,000) rather than the overall registered voters (80,000).

Question 9. 70% of our body weight is made up of water. Find the weight of water in the body of a person whose body weight is 56 kg.
Answer:
Percentage of water in the human body = 70%
Total body mass of the person = 56 kg
Weight of water present = \( 70\% \text{ of } 56 = \frac{70}{100} \times 56 = 39.2 \text{ kg} \)
In simple words: To find how much water is inside this person, just find 70% of their total body weight of 56 kg.

Exam Tip: When solving percentage problems, set up the fraction as \( \frac{\text{percent}}{100} \times \text{total value} \) to find the required amount.

Question 10. Only one-fifth of water is available in liquid form. This limited amount of water is replenished and used by man recurrently. Express this information as percent, showing :
(i) water available in liquid form.
(ii) water available in frozen form.
Answer:
Let the entire quantity of water represent 1 unit.
(i) Portion of water in liquid state = \( \frac{1}{5} \)
Percentage of water in liquid state = \( \frac{1}{5} \times 100 = 20\% \)

(ii) Portion of water in frozen state = \( 1 - \frac{1}{5} = \frac{4}{5} \)
Percentage of water in frozen state = \( \frac{4}{5} \times 100 = 80\% \)
In simple words: Turn the fractions into percentages by multiplying them by 100. This shows that 20% of the water is liquid and the remaining 80% is frozen.

Exam Tip: Since liquid and frozen water make up the whole quantity, their percentages must always add up to 100%.

Question 11. By weight, 90% of tomato and 78% of potato is water. Find :
(i) the weight of water in 25 kg of tomato.
(ii) the total quantity, by weight, of water in 90 kg of potato and 30 kg of tomato
(iii) the weight of potato which contains 39 kg of water.
Answer:
(i) Water content in tomatoes = 90%
Weight of water in 25 kg of tomatoes = \( 90\% \text{ of } 25 = \frac{90}{100} \times 25 = 22.5 \text{ kg} \)

(ii) Water content in potatoes = 78%
Weight of water in 90 kg of potatoes = \( 78\% \text{ of } 90 = \frac{78}{100} \times 90 = 70.2 \text{ kg} \)
Weight of water in 30 kg of tomatoes = \( 90\% \text{ of } 30 = \frac{90}{100} \times 30 = 27 \text{ kg} \)
Total combined weight of water = \( 70.2 \text{ kg} + 27 \text{ kg} = 97.2 \text{ kg} \)

(iii) Let the total weight of potatoes be \( x \text{ kg} \).
Since 78% of this weight is water:
\( 78\% \text{ of } x = 39 \)
\( \frac{78}{100} \times x = 39 \)
\( x = \frac{39 \times 100}{78} = 50 \text{ kg} \)
In simple words: Use the percentages to find the water in each vegetable part. For the final part, work backwards to find that it takes 50 kg of potatoes to hold 39 kg of water.

Exam Tip: Be careful to pair each vegetable with its correct percentage: use 90% for tomatoes and 78% for potatoes.

Revision Exercise

Question 1. Rohit’s age is 12 years and Geeta’s age is 15 years. Express :
(i) Rohit’s age as a percent of Geeta’s age.
(ii) Geeta’s age as a percent of Rohit’s age.
Answer:
Rohit's age = 12 years
Geeta's age = 15 years
(i) Rohit's age compared to Geeta's age = \( \frac{12}{15} \times 100 = \frac{4}{5} \times 100 = 80\% \)

(ii) Geeta's age compared to Rohit's age = \( \frac{15}{12} \times 100 = \frac{5}{4} \times 100 = 125\% \)
In simple words: To find the percent, put the first person's age on top of the fraction and the other person's age on the bottom, then multiply by 100.

Exam Tip: Pay close attention to which age is acting as the base (the denominator) for each calculation.

Question 2. A class has 30 boys and 20 girls. Find:
(i) the percentage of girls in the class
(ii) the percentage of boys in the class
(iii) percentage of number of boys as compared with number of girls.
Answer:
Number of boys = 30
Number of girls = 20
Total number of students in the class = \( 30 + 20 = 50 \)
(i) Percentage of girls in the class = \( \frac{20}{50} \times 100 = 40\% \)

(ii) Percentage of boys in the class = \( \frac{30}{50} \times 100 = 60\% \)

(iii) Percentage of boys compared to girls = \( \frac{30}{20} \times 100 = 150\% \)
In simple words: Find the total 50 students first to calculate the class percentages. For the last part, compare the boys directly to the girls.

Exam Tip: Remember to use the total class strength (50) for parts (i) and (ii), but only the number of girls (20) for part (iii).

Question 3. Mrs. Sharma went to the market with Rs. 800 in her purse. When she returned to her home, Rs. 240 were still left in her purse. What percent of her money did she spend in the market ?
Answer:
Total initial amount = Rs. 800
Remaining balance = Rs. 240
Amount spent = \( \text{Rs. } 800 - \text{Rs. } 240 = \text{Rs. } 560 \)
Percentage of total spent = \( \frac{560}{800} \times 100 = 70\% \)
In simple words: Find how much money was spent by subtracting the leftover money. Then, divide the spent money by the starting amount and multiply by 100.

Exam Tip: Be sure to base the percentage calculation on the total money she started with (Rs. 800), not what she had left.

Question 4. In a mixture of two liquids A and B, 35% is liquid B. If the total quantity of the mixture is 20 kg, find the quantity of A, by weight.
Answer:
Total quantity of the mixture = 20 kg
Percentage of liquid B = 35%
Weight of liquid B in the mixture = \( 35\% \text{ of } 20 = \frac{35}{100} \times 20 = 7 \text{ kg} \)
Weight of liquid A in the mixture = \( 20 \text{ kg} - 7 \text{ kg} = 13 \text{ kg} \)
In simple words: Find the weight of liquid B first by calculating 35% of the total. Subtract that from the total mixture weight to get liquid A.

Exam Tip: You can also find the percentage of liquid A directly (100% - 35% = 65%) and then calculate 65% of 20 kg to get the same result.

Question 5. A girl got 375 marks out of 500 in the first term examination, 560 marks out of 800 in the second term examination and 840 marks out of 1200 in the third term examination. Find:
(i) her percentage score in the first term examination.
(ii) her percentage score in the second term examination.
(iii) her percentage score in the third term examination.
(iv) the total marks secured in all the three examinations.
(v) the total marks scored in all the three examinations.
(vi) her percentage score on the whole in all the three examinations.
Answer:
(i) First term percentage score = \( \frac{375}{500} \times 100 = 75\% \)

(ii) Second term percentage score = \( \frac{560}{800} \times 100 = 70\% \)

(iii) Third term percentage score = \( \frac{840}{1200} \times 100 = 70\% \)

(iv) Sum of maximum marks for all terms = \( 500 + 800 + 1200 = 2500 \)

(v) Sum of marks obtained across all terms = \( 375 + 560 + 840 = 1775 \)

(vi) Overall percentage score = \( \frac{1775}{2500} \times 100 = 71\% \)
In simple words: Find the percentage for each term individually. To find the overall percentage, divide all the marks she earned by the total possible marks and multiply by 100.

Exam Tip: To find the cumulative percentage in part (vi), do not average the percentages of each term. Calculate it using the total marks scored and total maximum marks.

Question 6. Out of his monthly income of Rs. 2,500; a man spends Rs. 1,750. What percent of his income does he save every month?
Answer:
Monthly income = Rs. 2,500
Monthly expenditure = Rs. 1,750
Monthly savings = \( \text{Rs. } 2,500 - \text{Rs. } 1,750 = \text{Rs. } 750 \)
Percentage of monthly income saved = \( \frac{750}{2500} \times 100 = 30\% \)
In simple words: Find the money saved first by subtracting spending from income. Then, divide the savings by the total monthly income and multiply by 100.

Exam Tip: Double check that you calculate the saving percentage based on the income (Rs. 2,500), not the expenses.

Question 7. Mr. Singh’s monthly salary is Rs. 15,000. This month he was promoted with an increment of Rs. 3,000 in his salary. Express his increment as a percent of his original salary.
Answer:
Original monthly salary = Rs. 15,000
Promotion increment = Rs. 3,000
Increment percentage = \( \frac{3000}{15000} \times 100 = 20\% \)
In simple words: Divide the salary increase of Rs. 3,000 by the original salary of Rs. 15,000, then multiply by 100 to get the percentage.

Exam Tip: Always use the original salary as the denominator when finding the percentage of a salary increase.

Question 8. (i) The price of an article increased from Rs. 16 to Rs. 20; find the percentage increase.
(ii) The price of an article decreased from Rs 20 to Rs 16; find the percentage decrease.
Answer:
(i) Original price = Rs. 16
Increased price = Rs. 20
Price increase = \( 20 - 16 = \text{Rs. } 4 \)
Percentage increase = \( \frac{4}{16} \times 100 = 25\% \)

(ii) Original price = Rs. 20
Decreased price = Rs. 16
Price decrease = \( 20 - 16 = \text{Rs. } 4 \)
Percentage decrease = \( \frac{4}{20} \times 100 = 20\% \)
In simple words: Even though the price change is Rs. 4 in both parts, the percentage changes because the starting value is different in each case.

Exam Tip: Remember that the denominator is always the starting value of the price before the change happened.

Question 9. (i)) The salary of a man is Rs. 7,200 per month, which is now increased by 8%. Find his new salary per month.
(ii) The salary of Mr. Sahni is Rs. 8,400 per month, which is now decreased by 8%. Find his new salary per month.
Answer:
(i) Original monthly salary = Rs. 7,200
Increment = \( 8\% \text{ of Rs. } 7,200 = \frac{8}{100} \times 7,200 = \text{Rs. } 576 \)
New monthly salary = \( \text{Rs. } 7,200 + \text{Rs. } 576 = \text{Rs. } 7,776 \)

(ii) Original salary of Mr. Sahni = Rs. 8,400
Salary reduction = \( 8\% \text{ of Rs. } 8,400 = \frac{8}{100} \times 8,400 = \text{Rs. } 672 \)
New monthly salary = \( \text{Rs. } 8,400 - \text{Rs. } 672 = \text{Rs. } 7,728 \)
In simple words: For the first part, calculate 8% and add it to the original pay. For the second part, find 8% and subtract it from the original pay.

Exam Tip: You can quickly find an 8% increase by multiplying the base by 1.08, and an 8% decrease by multiplying the base by 0.92.

Question 10. Find the percentage change from the first quantity to the second :
(i) Rs. 80, Rs. 120
(ii) 75 kg, 60 kg
(iii) 50 cm, 45 cm
Answer:
(i) From Rs. 80 to Rs. 120:
Increase = \( 120 - 80 = \text{Rs. } 40 \)
Percentage increase = \( \frac{40}{80} \times 100 = 50\% \)

(ii) From 75 kg to 60 kg:
Decrease = \( 75 - 60 = 15 \text{ kg} \)
Percentage decrease = \( \frac{15}{75} \times 100 = 20\% \)

(iii) From 50 cm to 45 cm:
Decrease = \( 50 - 45 = 5 \text{ cm} \)
Percentage decrease = \( \frac{5}{50} \times 100 = 10\% \)
In simple words: Find the change between the two numbers, then divide that change by the very first number and multiply by 100.

Exam Tip: Always divide the difference by the first quantity given to get the correct percentage change.

Question 11. The original price of an article is Rs. 640. Find its new price when its price is :
(i) increased by 30%
(ii) decreased by 20%
Answer:
Original price of the article = Rs. 640
(i) New price with a 30% increase:
New price = \( 640 \times \frac{100 + 30}{100} = 640 \times \frac{130}{100} = \text{Rs. } 832 \)

(ii) New price with a 20% decrease:
New price = \( 640 \times \frac{100 - 20}{100} = 640 \times \frac{80}{100} = \text{Rs. } 512 \)
In simple words: An increase of 30% means the new price is 130% of the old one. A decrease of 20% means the new price is 80% of the old one.

Exam Tip: Using the formula \( \text{Original Value} \times \frac{100 \pm \text{Percentage}}{100} \) helps you solve these problems in a single step.

Question 12. Find the number that is :
(i) 50% more than 48
(ii) 30% less than 70
Answer:
(i) To find the number 50% more than 48:
Required value = \( 48 \times \frac{100 + 50}{100} = 48 \times \frac{150}{100} = 72 \)

(ii) To find the number 30% less than 70:
Required value = \( 70 \times \frac{100 - 30}{100} = 70 \times \frac{70}{100} = 49 \)
In simple words: Finding 50% more is the same as calculating 150% of the number. Finding 30% less is the same as calculating 70% of the number.

Exam Tip: You can also find the percentage amount first (e.g. 50% of 48 is 24) and then add it to the original number (48 + 24 = 72).

Question 13. Evaluate :
(i) 8% of 900 - 12% of 750 + 20% of 165.
(ii) 70% of 70 + 90% of 90 - 120% of 120.
Answer:
(i) Evaluating \( 8\% \text{ of } 900 - 12\% \text{ of } 750 + 20\% \text{ of } 165 \):
\( = \left(\frac{8 \times 900}{100}\right) - \left(\frac{12 \times 750}{100}\right) + \left(\frac{20 \times 165}{100}\right) \)
\( = 72 - 90 + 33 \)
\( = 105 - 90 = 15 \)

(ii) Evaluating \( 70\% \text{ of } 70 + 90\% \text{ of } 90 - 120\% \text{ of } 120 \):
\( = \left(\frac{70 \times 70}{100}\right) + \left(\frac{90 \times 90}{100}\right) - \left(\frac{120 \times 120}{100}\right) \)
\( = 49 + 81 - 144 \)
\( = 130 - 144 = -14 \)
In simple words: Work out each percentage block separately first. Once you have those three numbers, add and subtract them from left to right.

Exam Tip: Be careful with signs in part (ii) since subtracting a larger number (144) from a smaller one (130) results in a negative value.

Question 14. Approximately 97.3% water on the earth is not fit for drinking. Find :
(i) the percentage of water on the earth that is fit for drinking.
(ii) The total volume of water available in certain part of the earth where there is 21,600 \( \text{m}^3 \) of drinking water.
Answer:
(i) Percentage of water unfit for drinking = 97.3%
Percentage of water suitable for drinking = \( 100\% - 97.3\% = 2.7\% \)

(ii) Let the total water volume be \( V \).
Since 2.7% of this total volume is safe drinking water:
\( 2.7\% \text{ of } V = 21,600 \text{ m}^3 \)
\( \frac{2.7}{100} \times V = 21,600 \)
\( V = 21,600 \times \frac{100}{2.7} \)
\( V = 21,600 \times \frac{1000}{27} \)
\( V = 800 \times 1,000 = 800,000 \text{ m}^3 \)
In simple words: Subtract 97.3% from 100% to find that only 2.7% of water is fit to drink. If 2.7% equals 21,600 cubic meters, dividing by 0.027 gives a total of 800,000 cubic meters of water.

Exam Tip: To simplify dividing by a decimal, multiply both parts of the fraction by 10 to turn \( \frac{21600}{2.7} \) into \( \frac{216000}{27} \).

Question 15. Air is an important inexhaustible natural resource. It is essential for the survival of human beings, microbes, plants and animals. The following table shows the percentage of various gases in air.

(i) In 800 \( \text{m}^3 \) of air, calculate the approximate quantities of nitrogen, oxygen and other gases.
(ii) If a certain quantity (by volume) of air contains 4,200 litres of oxygen, find the total quantity of air taken and the amount of nitrogen in it.
Answer:
(i) Given total air quantity = 800 \( \text{m}^3 \)
Approximate quantity of Nitrogen = \( 78\% \text{ of } 800 = \frac{78}{100} \times 800 = 624 \text{ m}^3 \)
Approximate quantity of Oxygen = \( 21\% \text{ of } 800 = \frac{21}{100} \times 800 = 168 \text{ m}^3 \)
Approximate quantity of other gases = \( 1\% \text{ of } 800 = \frac{1}{100} \times 800 = 8 \text{ m}^3 \)

(ii) Volume of oxygen present = 4,200 litres
Since oxygen makes up 21% of the air:
\( 21\% \text{ of total air volume} = 4,200 \text{ litres} \)
Total quantity of air = \( \frac{4,200 \times 100}{21} = 20,000 \text{ litres} \)
Amount of nitrogen in this air = \( 78\% \text{ of } 20,000 = \frac{78}{100} \times 20,000 = 15,600 \text{ litres} \)
In simple words: Use the percentages from the table to find each gas amount. For the second part, since 21% of the air is 4,200 litres of oxygen, the total air is 20,000 litres, containing 15,600 litres of nitrogen.
Exam Tip: Be sure to calculate the volume of other gases (1%) in part (i) as well, as requested by the question.