Question three-fifths
Convert 3/5 to a percentage.
Steps only
Multiply the fraction by 100%.
Simplify the result.
Answer
3/5=60%.
CLASS 8 · GANITA PRAKASH PART II · CHAPTER 1
Fractions in Disguise
Understand percentages as fractions, compare proportions, calculate changes, and solve profit, discount, growth and mixture problems.
108 worked answersAll 5 exercise setsAll 15 end-of-chapter questions
What you will learn
- Convert fractions, decimals and percentages.
- Find a percentage of a quantity and compare proportions.
- Calculate percentage change, profit and loss, tax, discounts and compounding.
108 worked answers · short steps first, explanations when you need them
1.1 Fractions as percentages
Textbook pages 49–54
Question nine-twentieths
Convert 9/20 to a percentage.
Steps only
Multiply the fraction by 100%.
Simplify the result.
Answer
9/20=45%.
Question one-third
Convert 1/3 to a percentage.
Steps only
(1/3)×100%=100/3%.
Answer
1/3=33 1/3% (about 33.33%).
Question seventy-two-one-fifty
Convert 72/150 to a percentage.
Steps only
Multiply the fraction by 100%.
Simplify the result.
Answer
72/150=48%.
Question seven-fourteenths
Convert 7/14 to a percentage.
Steps only
Multiply the fraction by 100%.
Simplify the result.
Answer
7/14=50%.
Question five-elevenths
Convert 5/11 to a percentage.
Steps only
(5/11)×100%=500/11%.
Answer
5/11=500/11%=45 5/11% (about 45.45%).
Question white-marbles
Nandini has 25 marbles; 15 are white. What percentage are white?
Steps only
White fraction=15/25=3/5.
3/5×100%=60%.
Answer
60%.
Question walking-students
15 of 80 students walk to school. What percentage is this?
Steps only
15/80×100%=1500/80%.
Simplify.
Answer
18.75%.
Question race-progress-visual
Four runners A, B, C and D are shown from left to right on a track. Match them to the approximate percentages completed: 55%, 20%, 38%, 72%, 84%, 93%.
Steps only
From left to right the runners are A, B, C, D.
Choose increasing percentages that best match their approximate positions.
Answer
A≈20%, B≈38%, C≈72%, D≈93%. The unused choices are 55% and 84%.
Question compare-percent-fractions
Fill the signs: (i) 50% __ 5%, (ii) 5/10 __ 50%, (iii) 3/11 __ 61%, (iv) 30% __ 1/3.
Steps only
50% is greater than 5%; 5/10=50%.
3/11≈27.3%, less than 61%; 1/3≈33.3%, more than 30%.
Answer
- (i) >,
- (ii) =,
- (iii) <,
- (iv) <.
Question yellow-paint-share
Red paint makes up 3/4 of a deep-orange mixture made from red and yellow paint. What percentage is yellow?
Steps only
Question method-three-number-line
Complete the matching values for the equal steps from 0 to 1: 0, 1/5, 2/5, 3/5, 4/5, 5/5. Write the percentages too.
Steps only
Each fifth is 20%.
The matching percentages are 0%, 20%, 40%, 60%, 80%, 100%.
Answer
0=0%, 1/5=20%, 2/5=40%, 3/5=60%, 4/5=80%, 5/5=100%.
Question why-use-percentages
If percentages are fractions, why do people use percentages instead of writing every comparison as a fraction?
Steps only
Percentages use a common denominator of 100.
That makes many comparisons easier to read and compare.
Answer
Percentages give a familiar common scale out of 100, so quantities with different denominators can be compared quickly. Fractions can still be more exact or simpler in some cases.
Question why-denominator-100
Why do we choose 100 as the denominator when writing a percentage? Why not use 10, 50, 1,000 or another number?
Steps only
“Per cent” means “per hundred”.
Using 100 gives a shared scale for comparisons.
Answer
A percentage is defined as an amount per 100. Other denominators can also be useful, but they are written as fractions or rates rather than percentages.
1.2 Percentage of some quantity
Textbook pages 55–63
Question sugar-biscuits
Madhu eats 120 g biscuits with 25% sugar; Madhav eats 95 g with 35% sugar. Who eats more sugar?
Steps only
Madhu: 0.25×120=30 g.
Madhav: 0.35×95=33.25 g.
Answer
Madhav eats more sugar: 33.25 g compared with 30 g.
Question table-25-100
Find 25% of 100.
Steps only
25% is one quarter.
Divide the quantity by 4.
Answer
25.
Question table-25-200
Find 25% of 200.
Steps only
25% is one quarter.
Divide the quantity by 4.
Answer
50.
Question table-25-50
Find 25% of 50.
Steps only
25% is one quarter.
Divide the quantity by 4.
Answer
12.5.
Question table-25-80
Find 25% of 80.
Steps only
25% is one quarter.
Divide the quantity by 4.
Answer
20.
Question table-25-10
Find 25% of 10.
Steps only
25% is one quarter.
Divide the quantity by 4.
Answer
2.5.
Question table-25-35
Find 25% of 35.
Steps only
25% is one quarter.
Divide the quantity by 4.
Answer
8.75.
Question table-25-287
Find 25% of 287.
Steps only
25% is one quarter.
Divide the quantity by 4.
Answer
71.75.
Question percent-table-entries
Complete fraction and decimal forms for 50%, 100%, 25%, 75%, 10%, 1%, 5% and 43%.
Steps only
Write each x% as x/100.
Simplify the fraction and write the decimal.
Answer
50%=1/2=0.5; 100%=1=1; 25%=1/4=0.25; 75%=3/4=0.75; 10%=1/10=0.1; 1%=1/100=0.01; 5%=1/20=0.05; 43%=43/100=0.43.
Question 80-percent-marks
A test has 75 possible marks. Zubin needs at least 80% for an A. How many marks?
Steps only
80%=0.8.
0.8×75=60.
Answer
At least 60 marks.
Question millet-mixture
Millet and water are mixed in the ratio 2:7. What percentage of a 500 mL mixture is millet, and how much millet is that?
Steps only
Total parts=2+7=9; millet fraction=2/9.
2/9×100%=22 2/9%≈22.22%.
2/9×500=1000/9≈111.11 mL.
Answer
Millet is about 22.22%; the mixture contains about 111.1 mL millet.
Question cyclist-remaining-distance
A cyclist has travelled 92 km, which is 40% of the Delhi–Agra journey. How much farther remains?
Steps only
Total journey=92÷0.4=230 km.
Remaining=60% of 230=138 km.
Answer
138 km remain.
Question sales-target-percent
A shop’s daily target is ₹5,000. What percentages are sales of ₹2,000, ₹3,500, ₹5,000, ₹6,000, ₹7,800 and ₹9,550?
Steps only
Divide each sale by 5,000 and multiply by 100%.
Answer
40%, 70%, 100%, 120%, 156% and 191%, respectively.
Question target-150-210
If the sales target is ₹5,000, find sales at 150% and 210% of target.
Steps only
150%=1.5; 1.5×5000=7500.
210%=2.1; 2.1×5000=10500.
Answer
₹7,500 and ₹10,500.
Question harvest-growth
A farmer harvested 260 kg last year and 650 kg this year. This year’s harvest is what percentage of last year’s?
Steps only
650/260×100%=2.5×100%.
Answer
250% of last year’s harvest (2.5 times as much).
Question percentage-table-10-20-5
Complete the 10%, 20% and 5% rows for the values 100, 200, 50, 80, 10, 35 and 287.
Steps only
10% is one tenth; 20% is twice 10%; 5% is half of 10%.
| Percent | 100 | 200 | 50 | 80 | 10 | 35 | 287 |
|---|---|---|---|---|---|---|---|
| 10% | 10 | 20 | 5 | 8 | 1 | 3.5 | 28.7 |
| 20% | 20 | 40 | 10 | 16 | 2 | 7 | 57.4 |
| 5% | 5 | 10 | 2.5 | 4 | 0.5 | 1.75 | 14.35 |
Answer
10%: 10, 20, 5, 8, 1, 3.5, 28.7. 20%: 20, 40, 10, 16, 2, 7, 57.4. 5%: 5, 10, 2.5, 4, 0.5, 1.75, 14.35.
Question percent-table-observations
How can you find the values in a percentage table mentally, and what patterns do you notice?
Steps only
10% is one tenth; 5% is half of 10%; 20% is double 10%.
25% is one quarter.
Answer
Use easy benchmark percentages and combine them. For example, 15%=10%+5%, and 35%=25%+10%.
Question percent-40-and-15-table
Find 40% and 15% of 100, 200, 50, 80, 10, 35 and 287.
Steps only
40%=4×10%.
15%=10%+5%.
Answer
40%: 40, 80, 20, 32, 4, 14, 114.8. 15%: 15, 30, 7.5, 12, 1.5, 5.25, 43.05.
Question mental-percent-strategies
How can you mentally find 75%, 90%, 70% and 55% of any quantity? What decimal should you multiply by to find 10%?
Steps only
75%=3/4; 90%=100%−10%; 70%=7×10%; 55%=50%+5%.
10%=0.1 times the quantity.
Answer
Use 0.75, 0.9, 0.7 and 0.55 as multipliers, or use the equivalent fraction/benchmark methods. Multiply by 0.1 to find 10%.
Question how-close-can-you-get
Choose two numbers a and b with a<b. Estimate a/b as a percentage, compare estimates, and play several rounds.
Steps only
Sample: a=3, b=8.
3/8=37.5%, so a close estimate is 38%.
Answer
Answers depend on the numbers chosen. For example, 3/8 is 37.5%, so 38% is a good estimate.
Question percentages-over-100-numberline
Write each percentage as a fraction and decimal, then order them on a number line from 0% to 400%: 90%, 110%, 200%, 250%, 15%, 173%, 358%, 28.9%, 305%.
Steps only
Divide each percentage by 100 for its decimal.
From left to right: 15%, 28.9%, 90%, 110%, 173%, 200%, 250%, 305%, 358%.
| Percent | Fraction | Decimal |
|---|---|---|
| 90% | 9/10 | 0.9 |
| 110% | 11/10 | 1.1 |
| 200% | 2 | 2 |
| 250% | 5/2 | 2.5 |
| 15% | 3/20 | 0.15 |
| 173% | 173/100 | 1.73 |
| 358% | 179/50 | 3.58 |
| 28.9% | 289/1000 | 0.289 |
| 305% | 61/20 | 3.05 |
Answer
Fractions: 9/10, 11/10, 2, 5/2, 3/20, 173/100, 179/50, 289/1000, 61/20. Decimals: 0.9, 1.1, 2, 2.5, 0.15, 1.73, 3.58, 0.289, 3.05.
Question bar-model-missing-values
In the bar models: (i) one of five equal parts is 20%; in another bar, four of five parts total 60 and the full bar totals 75. (ii) six of ten parts of a 90 total are marked. (iii) three of four parts of 140 are marked. Find the missing values.
Steps only
(i) one fifth=20%; 4/5×75=60.
(ii) one tenth=10%; 60% of 90=54.
(iii) one fourth=25%; 75% of 140=105.
Answer
- (i) The missing percentage is 20% and the marked amount is 60.
- (ii) 10% and 54.
- (iii) 25% and 105.
Question bar-model-calculations
Find: (i) 25% of 160, (ii) 16% of 250, (iii) 62% of 360, (iv) 140% of 40, (v) 1% of 1 hour, and (vi) 7% of 10 kg.
Steps only
40; 40; 223.2; 56; 0.6 minutes; 0.7 kg.
Answer
- (i) 40
- (ii) 40
- (iii) 223.2
- (iv) 56
- (v) 0.6 minutes=36 seconds
- (vi) 0.7 kg=700 g.
Question surya-red-paint
Surya makes 60 mL of deep-orange paint. Red paint is 3/4 of the mixture. How much red paint did he use?
Steps only
60÷4=15 mL per quarter.
3×15=45 mL.
Answer
Surya used 45 mL of red paint.
Question compare-percent-quantities
Choose >, < or =: (i) 50% of 510 __ 50% of 515; (ii) 37% of 148 __ 73% of 148; (iii) 29% of 43 __ 92% of 110; (iv) 30% of 40 __ 40% of 50; (v) 45% of 200 __ 10% of 490; (vi) 30% of 80 __ 24% of 64.
Steps only
(i) < (ii) < (iii) < (iv) < (v) > (vi) >.
Answer
- (i) <,
- (ii) <,
- (iii) <,
- (iv) <,
- (v) >,
- (vi) >.
Question percent-of-variable-values
Complete: (i) 30% of k=70; find 60%, 90%, 120% of k. (ii) 100% of m=215; find 10%, 1%, 6%. (iii) 90% of n=270; find 9%, 18%, 100%.
Steps only
(i) 140, 210, 280.
(ii) 21.5, 2.15, 12.9.
(iii) 27, 54, 300.
Answer
- (i) 60% of k=140; 90%=210; 120%=280.
- (ii) 10% of m=21.5; 1%=2.15; 6%=12.9.
- (iii) 9% of n=27; 18%=54; 100%=300.
Question find-number-from-percent
Fill in: (i) 3 is what percent of 300? (ii) What number is 40% of 4? (iii) 40 is 80% of what number?
Steps only
(i) 1%. (ii) 1.6. (iii) 50.
Answer
- (i) 1%
- (ii) 1.6
- (iii) 50.
Question compare-day-week-time
Is 10% of a day longer than 1% of a week?
Steps only
10% of 24 h=2.4 h.
1% of 168 h=1.68 h.
Answer
Yes. 10% of a day is 2.4 hours, which is 0.72 hours (43.2 minutes) longer than 1% of a week.
Question mariam-bull-fodder
Each day the bull eats all but one unit: it gets 2 and eats 1 on day 1, 3 and eats 2 on day 2, continuing to 100 and 99 on day 99. Express the eaten amount as a percentage each day and describe the pattern.
Steps only
Day 1: 1/2=50%. Day 2: 2/3≈66.67%. Day 99: 99/100=99%.
Answer
The eaten percentages are 50%, 66 2/3%, 75%, …, 99%. On day n it eats (n−1)/n of the fodder; this percentage gets closer to 100% as n increases.
Question coffee-plantation-days
Workers take 18 days to pick coffee berries from 20% of a plantation. At the same rate, how long for the full plantation, and what assumption is needed?
Steps only
100% is 5×20%.
18×5=90 days.
Answer
About 90 days, assuming the picking rate stays constant and the remaining areas take comparable effort.
Question badminton-session-time
A 90-minute badminton session uses 10% for warm-up, 80% for play and 10% for cool-down. Find each time.
Steps only
10% of 90=9 minutes.
80% of 90=72 minutes.
Answer
Warm-up 9 minutes; play 72 minutes; cool-down 9 minutes.
Question northern-hemisphere-population
About 90% of the world’s population lives in the Northern Hemisphere. Using the textbook’s 2025 estimate of about 8.2 billion people worldwide, estimate the number there.
Steps only
90% of 8.2 billion=0.9×8.2 billion.
Answer
About 7.38 billion people (roughly 7.4 billion), using an 8.2-billion world-population estimate.
Question halwa-ingredient-proportions
Halwa ingredients are 40% rava, 40% sugar and 20% ghee. Do the proportions change when serving 8 instead of 4? If the total ingredients weigh 2 kg, find each weight.
Steps only
The proportions stay 40%, 40%, 20%.
40% of 2 kg=0.8 kg; 20%=0.4 kg.
Answer
The proportions do not change. In 2 kg: rava 0.8 kg, sugar 0.8 kg, ghee 0.4 kg.
Question badam-drink-table
DEF contains 99 g sugar, 30 g milk solids, 12 g badam powder and 9 g food chemicals (150 g total). Zacni contains 272 g, 64 g, 40 g and 24 g respectively (400 g total). Which has the larger badam share? Which has a smaller chemical share?
Steps only
DEF badam=12/150=8%; chemicals=6%.
Zacni badam=40/400=10%; chemicals=6%.
| Product | Sugar | Milk solids | Badam powder | Food chemicals | Total |
|---|---|---|---|---|---|
| DEF | 66% | 20% | 8% | 6% | 100% |
| Zacni | 68% | 16% | 10% | 6% | 100% |
Answer
Zacni has the larger badam share (10% vs 8%). Neither has a smaller chemical share: both are 6%.
Question check-badam-percentages
Do the ingredient percentages for DEF and Zacni add to 100%?
Steps only
DEF: 66+20+8+6=100.
Zacni: 68+16+10+6=100.
Answer
Yes. Each product’s ingredient percentages total 100%.
1.3 Compare proportions and percentage change
Textbook pages 64–69
Question test-scores
Eesha scores 42/50 in English and 70/80 in Science. Which score is a higher proportion?
Steps only
English: 42/50×100%=84%.
Science: 70/80×100%=87.5%.
Answer
Science is higher: 87.5% versus 84%.
Question tomato-price-increase
Tomato price rises from ₹30/kg to ₹42/kg. Find the percentage increase.
Steps only
Increase=42−30=₹12.
12/30×100%=40%.
Answer
40%.
Question theatre-footfall-decrease
Theatre attendance falls from 160 to 100. Find the percentage decrease.
Steps only
Decrease=160−100=60.
60/160×100%=37.5%.
Answer
37.5%.
Question population-growth-equivalence
Are “new population is 165% of the old” and “population increased by 65%” equivalent?
Steps only
165% of p=1.65p.
An increase by 65% gives p+0.65p=1.65p.
Answer
Yes. Both describe a new population of 1.65 times the old population.
Question profit-wholesaler-notebook
A notebook costs a shop ₹36. What price gives a 20% profit on cost?
Steps only
Profit=0.20×36=₹7.20.
Selling price=36+7.20.
Answer
₹43.20.
Question crayon-cost
Crayon boxes sell for ₹50 each at a 25% profit on cost. What was the cost?
Steps only
Selling price is 125% of cost.
Cost=50/1.25.
Answer
₹40.
Question company-expenditure
A company earns ₹2.5 crore revenue and has a 25% profit margin on revenue. Find expenditure.
Steps only
Profit=0.25×2.5=0.625 crore.
Expenditure=revenue−profit.
Answer
₹1.875 crore.
Question shirt-discount
A ₹300 shirt has a 25% discount. What is the sale price?
Steps only
Discount=0.25×300=₹75.
300−75=225.
Answer
₹225.
Question petrol-price-increase
Petrol rises from ₹60 to ₹100. Choose the percentage increase.
Steps only
Increase=₹40.
40/60×100%=66 2/3%.
Answer
66.67% (approximately), option (iv).
Question marked-price-after-discount
A car costs ₹4,40,000 after a 15% discount. What was the original price?
Steps only
The discounted price is 85% of marked price M.
0.85M=440000, so M=440000/0.85.
Answer
Approximately ₹5,17,647.06 (about ₹5.18 lakh).
Question winner-votes
A winner gets 500 of 1,600 votes. What percentage is this? What is the fewest candidates if the winner alone has the most votes?
Steps only
500/1600×100%=31.25%.
The other 1,100 votes need at least three candidates with fewer than 500 each.
Answer
31.25%; at least 4 candidates in total.
Question rice-inflation
Rice rises from ₹38/kg to ₹42/kg. What is the inflation rate?
Steps only
Increase=₹4.
4/38×100%=10.526…%.
Answer
About 10.53%.
Question reverse-20-percent
A number increases by 20% to become 90. What was it?
Steps only
The new amount is 120% of the original x.
1.2x=90, so x=75.
Answer
75.
Question two-buffaloes
Two buffaloes are sold for ₹80,000 each. One sale gives 5% profit, the other 10% loss. Find the overall result.
Steps only
First cost=80000/1.05≈₹76,190.48.
Second cost=80000/0.9≈₹88,888.89.
Total cost≈₹165,079.37; total selling price=₹160,000.
Answer
Overall loss≈₹5,079.37, about 3.08% of total cost.
Question elephant-population-factor
A population p increases by 5%. Express the new population.
Steps only
New amount=100%+5%=105% of p.
105%=1.05.
Answer
1.05p.
Question demand-falls-85
If camera demand falls by 85%, what fraction of the old demand remains?
Steps only
Remaining percentage=100%−85%=15%.
Answer
Demand now is 15% of the demand a decade ago (option iii).
Question sweater-supply-chain-profit
A sweater moves through three businesses: manufacturer CP ₹230, SP ₹253; wholesaler CP ₹253, SP ₹300; retailer CP ₹300, SP ₹430. Find each profit percentage on cost.
Steps only
Manufacturer: 23/230×100=10%.
Wholesaler: 47/253×100≈18.58%.
Retailer: 130/300×100≈43.33%.
Answer
Manufacturer 10%; wholesaler about 18.58%; retailer about 43.33% profit on cost.
Question rice-loss-per-kg
Rice costs ₹35 per kg. A seller sells 10 kg for ₹300, a loss of ₹50. Would calculating loss per kilogram give the same loss percentage?
Steps only
Cost per kg ₹35; selling price per kg ₹30; loss ₹5 per kg.
5/35×100≈14.29%.
Answer
Yes. The loss is about 14.29% either way. Per kg: ₹5 loss on ₹35 cost. For 10 kg: ₹50 loss on ₹350 cost.
Question strawberry-cost-price
Strawberries sell for ₹80 per kg at a 12% loss. What was the cost price per kilogram?
Steps only
At 12% loss, selling price=88% of cost.
Cost=80÷0.88≈₹90.91.
Answer
The cost price was about ₹90.91 per kg.
Question cooker-discount-profit
A cooker marked ₹1,800 has a 35% discount. If its cost price was ₹900, find the selling price and profit percentage.
Steps only
Selling price=65%×₹1,800=₹1,170.
Profit=₹270; 270/900×100=30%.
Answer
Selling price ₹1,170; profit percentage 30%.
Question check-electricity-bill
A bill lists 3 bulbs at ₹150 each, subtotal ₹450, CGST 9% ₹40.50, SGST 9% ₹40.50, total ₹531. Check it.
Steps only
3×₹150=₹450.
9% of ₹450=₹40.50 for each tax.
₹450+₹40.50+₹40.50=₹531.
Answer
The bill is correct.
Question elements-on-bills
Look at household bills. What details are commonly shown, and what may differ?
Steps only
Common: shop and date, item, quantity, price, subtotal, tax, total.
Bills can differ in layout and the taxes or fees shown.
Answer
Answers depend on the bills examined. Most show seller details, date, items, quantities, prices, subtotal, taxes or fees, and final amount.
Question geometry-box-margin
A shopkeeper buys a geometry box for ₹75 and sells it for ₹110. What is the profit margin with respect to cost?
Steps only
Profit=₹110−₹75=₹35.
35/75×100=46 2/3%.
Answer
The profit is ₹35, which is 46 2/3% of the cost price.
Question chair-profit-price
Materials for a chair cost ₹475. What selling price gives a 50% profit on cost?
Steps only
50% of ₹475=₹237.50.
₹475+₹237.50=₹712.50.
Answer
Sell the chair for ₹712.50.
Growth, compounding, depreciation and discounts
Textbook pages 70–78
Question simple-compound-10-2
₹20,000 earns 10% per year for 2 years. Find amounts with and without annual compounding.
Steps only
Without compounding: 20,000(1+0.10×2)=24,000.
With compounding: 20,000×1.1²=24,200.
Answer
Without compounding: ₹24,000. With compounding: ₹24,200.
Question simple-compound-5-4
₹20,000 earns 5% per year for 4 years. Compare simple and annual compound amounts.
Steps only
Simple amount=20,000(1+0.05×4)=24,000.
Compound amount=20,000×1.05⁴=24,310.125.
Answer
₹24,000 simple; approximately ₹24,310.13 compounded.
Question simple-interest-formula
For principal p, rate r as a decimal, and t years, write the total without compounding.
Steps only
Each year earns pr interest.
Over t years, interest is prt.
Answer
Amount=p(1+rt); interest gained=prt.
Question compound-interest-formula
For principal p, annual rate r as a decimal, and t years, write the total with annual compounding.
Steps only
Each year multiplies the balance by 1+r.
Repeat this multiplier t times.
Answer
Amount=p(1+r)^t.
Question post-office-interest
Find interest on ₹50,000 for 3 years at 7% simple interest and annual compounding.
Steps only
Simple interest=50000×0.07×3=₹10,500.
Compound amount=50000×1.07³=₹61,252.15; interest=₹11,252.15.
Answer
Simple interest: ₹10,500. Compound interest: ₹11,252.15; compounding gives ₹752.15 more.
Question loan-interest-compare
Giridhar borrows ₹12,500 at 12% simple interest for 3 years. Raghava borrows the same at 10% compounded yearly. Who pays more interest?
Steps only
Giridhar interest=12500×0.12×3=₹4,500.
Raghava amount=12500×1.1³=₹16,637.50; interest=₹4,137.50.
Answer
Giridhar pays ₹362.50 more interest.
Question doubling-time
An amount of ₹1,000 grows at 10% per year. How long to double with annual compounding versus simple growth?
Steps only
Compound: solve 1.1^t=2; t=ln2/ln1.1≈7.27 years (first whole year at or above double: 8).
Simple: 1000(1+0.1t)=2000 gives t=10 years.
Answer
About 7.27 years compounded (8 full years to pass double); 10 years without compounding. Compound growth is exponential; simple growth is linear.
Question city-population-growth
A city has 1.5 crore people and grows 3% per year. Estimate its population after 3 years with annual compounding.
Steps only
Multiply by 1.03 each year: 1.5×1.03³ crore.
1.03³=1.092727.
Answer
Approximately 1.639 crore people (about 1 crore 63.9 lakh).
Question bacteria-growth
A bacteria count of 5,06,000 grows 2.5% each hour. Find the count after 2 hours.
Steps only
Multiply by 1.025 twice.
506000×1.025²=531,616.25.
Answer
Approximately 531,616 bacteria (about 5,31,616).
Question tv-depreciation
A ₹21,000 TV loses 5% of its value in one year. What is it worth after a year?
Steps only
It keeps 95% of its value.
0.95×21000=19950.
Answer
₹19,950.
Question village-population-decline
A village has 1,250 people and declines 10% each decade. Estimate after three decades.
Steps only
Each decade retains 90%, so multiply by 0.9³.
1250×0.9³=911.25.
Answer
About 911 people (roughly 910).
Question choose-percentage-vs-fixed-discount
A store offers 20% off or ₹50 off. Which saves more on purchases of ₹180, ₹225 and ₹300?
Steps only
At ₹180: 20%=₹36, so choose ₹50.
At ₹225: 20%=₹45, so choose ₹50.
At ₹300: 20%=₹60, so choose 20%.
Answer
Choose ₹50 at ₹180 and ₹225; choose 20% at ₹300.
Question stacked-discount
For a ₹200 cake, compare 30% then 20% off with a single 50% discount.
Steps only
After 30% off: 200×0.7=₹140.
Then 20% off: 140×0.8=₹112.
A single 50% discount gives ₹100.
Answer
The single 50% discount is cheaper; successive 30% and 20% discounts equal 44% off, not 50%.
Question markup-50-discount-50
A shop marks goods 50% above cost, then gives 50% off the marked price. Is there no profit or loss?
Steps only
If cost is x, marked price=1.5x.
After 50% off, selling price=0.5×1.5x=0.75x.
Answer
There is a 25% loss.
Question markup-50-specific-loss
In the previous situation, goods sell for ₹12,000 after discount. Find original cost and loss.
Steps only
Selling price=75% of cost, so cost=12000/0.75=₹16,000.
Loss=16000−12000=₹4,000.
Loss percentage=4000/16000×100%=25%.
Answer
Cost ₹16,000; loss ₹4,000, or 25%.
Question break-even-discount
If goods are marked 50% above cost, what discount on marked price gives no profit or loss?
Steps only
Marked price=1.5x; break-even selling price=x.
Discount fraction=(1.5x−x)/(1.5x)=1/3.
Answer
33 1/3% (about 33.33%).
Question arun-marble-percent
Ariba has 120% as many marbles as Arun. What percentage of Ariba’s marbles does Arun have?
Steps only
Let Arun have 100 units; Ariba has 120.
Arun as a share of Ariba=100/120×100%=83 1/3%.
Answer
Arun has 83 1/3% of Ariba’s marbles.
Question compare-two-interest-observations
In the two examples, ₹20,000 grows at 10% for 2 years or 5% for 4 years. What pattern do you notice?
Steps only
Simple interest: both use 20% total interest, so both reach ₹24,000.
Compound amounts: ₹24,200 and ₹24,310.13.
Answer
Without compounding, both amounts are ₹24,000 because rate×time is 20%. With annual compounding, the 5%-for-4-years amount is slightly higher: ₹24,310.13 versus ₹24,200.
Question compound-interest-gained-formula
For principal p, annual rate r as a decimal and t years, write the interest gained after annual compounding.
Steps only
Amount=p(1+r)^t.
Interest gained=amount−principal.
Answer
Compound interest gained=p[(1+r)^t−1].
Question choose-prize-percentage-gain
Option A: deposit ₹100 and get back ₹300. Option B: deposit ₹1,000 and get back ₹1,500. What percentage gain does each give, and which has the higher percentage gain?
Steps only
A gain=₹200; 200/100×100=200%.
B gain=₹500; 500/1000×100=50%.
Answer
Option A gives a 200% gain; Option B gives a 50% gain. Option A has the larger percentage gain, though Option B returns more rupees in total.
Figure it Out — chapter review
Textbook pages 79–80
Question bengaluru-population
Bengaluru’s 2025 population is about 250% of its 2000 population of 50 lakh. Find the 2025 estimate.
Steps only
250%=2.5.
50×2.5=125 lakh.
Answer
125 lakh, or 1.25 crore.
Question country-population-shares
Estimate the percentage shares for Germany 83 million, India 1.46 billion, Bangladesh 175 million and USA 347 million.
Steps only
Question phone-gst
A phone costs ₹8,250 before 18% GST. Which expressions give the final price, and what is it?
Steps only
GST=0.18×8250=₹1,485.
Final=8250+1485=₹9,735=8250×1.18.
Answer
Expressions (v) and (vi) are correct; final price ₹9,735.
Question mouse-monthly-changes
Population changes by +5%, then −2%, then −3%, starting from p. Which expression is correct, and is the result above or below p?
Steps only
Use multipliers 1.05, 0.98 and 0.97.
Product=1.05×0.98×0.97=0.99813.
Answer
p×1.05×0.98×0.97 (statement ii); it is slightly less than p (statement vi).
Question profit-discount-35-30
A seller sets a price 35% above cost and then discounts that selling price by 30%. Profit or loss?
Steps only
Marked selling price=1.35×cost.
After discount: 1.35×0.70=0.945×cost.
Answer
A 5.5% loss.
Question x-percent-y
Evaluate the pairs and prove the general relationship.
Steps only
x% of y=(x/100)y=xy/100.
y% of x=(y/100)x=xy/100.
Answer
They are equal. For example, 5% of 40=40% of 5=2; 25% of 12=12% of 25=3; 15% of 60=60% of 15=9.
Question grade8-girls
40% of excursion students are in Grade 8; 60% of those are girls. What percentage of all students are Grade 8 girls? How many if there are 160 students?
Steps only
0.40×0.60=0.24, so 24% of all students.
0.24×160=38.4.
Answer
24%; the proportional estimate is 38.4 students (about 38, noting exact counts must be whole people).
Question pencils-profit
Selling 3 pencils brings the same revenue as the cost of 5 pencils. Find profit or loss percentage.
Steps only
Let one pencil’s cost be c. Revenue for 3 is 5c, so SP per pencil=5c/3.
Profit per pencil=5c/3−c=2c/3.
Profit percentage=(2c/3)/c×100%.
Answer
A profit of 66 2/3%.
Question two-year-bus-increase
Bus fares rise 3% one year and 4% the next. Find the total percentage increase.
Steps only
Combined multiplier=1.03×1.04=1.0712.
Increase factor=0.0712.
Answer
7.12%.
Question fixed-area-breadth-decrease
A rectangle’s length increases by 10%, but its area stays constant. By what exact percentage must breadth decrease?
Steps only
New length=1.1L.
To keep area, new breadth=B/1.1=10B/11.
Decrease=(1−10/11)B=B/11.
Answer
Breadth decreases by 1/11 of its old value, or 9 1/11% (about 9.09%).
Question free-item-deals
Items cost ₹100 each. Shop A: buy 1 get 1 free; B: buy 2 get 1 free; C: buy 3 get 1 free. Compare effective prices and discounts; if you need 4 items, which is best?
Steps only
A: pay ₹100 for 2, so ₹50 each and 50% off.
B: pay ₹200 for 3, so ₹66.67 each and 33 1/3% off.
C: pay ₹300 for 4, so ₹75 each and 25% off.
Answer
Cheapest per item: A, then B, then C. For exactly 4 items, Shop A can supply 4 for ₹200 (two deals), the cheapest.
Question left-handed-room
In a room of 100 people, 99 are left-handed. How many left-handed people must leave so that 98% of those remaining are left-handed?
Steps only
Let x left-handed people leave: (99−x)/(100−x)=0.98.
99−x=98−0.98x, so 1=0.02x and x=50.
Answer
50 left-handed people must leave; then 49 of 50 people (98%) are left-handed.
Question computer-graph-statements
Which claims are supported by the chapter’s age-and-gender chart: twenties highest; women lower in every age group; twenties have more people than teens; over a quarter of thirties can use computers; under 10% of seniors can; half of twenties can?
Steps only
The chart shows ability percentages by age and gender, not population counts.
The twenties have the chart’s highest rates; female rates are below male rates; seniors are below 10%.
It does not show age-group population sizes; the thirties entry is not above 25%, and the twenties rate is not 50%.
Answer
Valid: twenties have the highest computer-use rates; women’s rate is lower in each age group; fewer than 1 in 10 seniors can use computers (i, ii and v).
Question region-e-area-percent
A square grid has 64 equal unit squares, and region E covers 8 of them. What percentage of the grid is region E?
Steps only
8/64=1/8.
1/8×100%=12.5%.
Answer
Region E covers 12.5% of the grid.
Question chips-packet-ingredient-weights
A 65 g chips packet is 70% potato, 24% vegetable oil, 3% salt and 3% spices. Find each ingredient’s weight.
Steps only
Potato 45.5 g; oil 15.6 g; salt 1.95 g; spices 1.95 g.
| Ingredient | Share | Weight |
|---|---|---|
| Potato | 70% | 45.5 g |
| Vegetable oil | 24% | 15.6 g |
| Salt | 3% | 1.95 g |
| Spices | 3% | 1.95 g |
Answer
Potato 45.5 g, vegetable oil 15.6 g, salt 1.95 g, spices 1.95 g.