Chapter 1 exercise answers: Fractions in Disguise

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(i) 3/5

1. 3/5 × 100 = 60.

Answer60%

(ii) 7/14

1. 7/14 is the same fraction as 1/2.
2. 1/2 × 100 = 50.

Answer50%

(iii) 9/20

1. 9/20 × 100 = 45.

Answer45%

(iv) 72/150

1. 72/150 simplifies to 12/25.
2. 12/25 × 100 = 48.

Answer48%

(v) 1/3

1. 1/3 × 100 = 100/3, which is 33.33…, a decimal that never ends.

Answer100/3 % ≈ 33.33% (recurring)

(vi) 5/11

1. 5/11 × 100 = 500/11, which is 45.45…, a decimal that never ends.

Answer500/11 % ≈ 45.45% (recurring)

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1. 15 of her 25 marbles are white, so the white marbles are 15/25 of all her marbles.
2. 15/25 × 100 = 60.
3. 60% is option (iv).

Answer60% — option (iv)

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1. 15 out of 80 students walk to school.
2. 15/80 × 100 = 75/4 = 18.75.

Answer18.75%

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1. The line runs from Start (0% of the race) to Finish (100%), so each runner's percentage is how far along the line they are.
2. A is a little under a third of the way along. Of the options, 38% is the nearest; 20% is too far back.
3. B is a little past halfway, so B has done about 55%.
4. C is about three quarters of the way along. 72% is nearer to three quarters (75%) than 84% is.
5. D is almost at the finish, so D has done about 93%.
6. The options 20% and 84% are left over.

AnswerA ≈ 38%, B ≈ 55%, C ≈ 72%, D ≈ 93%

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(i) 50% ____ 5%

1. 50% is ten times 5%.

Answer>

(ii) 5/10 ____ 50%

1. 5/10 is the same fraction as 1/2, which is 50%.

Answer=

(iii) 3/11 _____ 61%

1. 3/11 is a bit more than a quarter, well under 50%, while 61% is over 50%.

Answer<

(iv) 30% ____ 1/3

1. 1/3 is about 33%, a little more than 30%.

Answer<

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1. Each bar is cut into equal parts that together make 100%, so one part is worth 100% divided by the number of parts.
2. (ii) The bar has 10 equal parts, so 1 part = 100% ÷ 10 = 10%.
3. (ii) The second bar also has 10 parts and the whole bar is worth 90; the marked 6 parts are 6/10 = 60% of it, so their value is 60% of 90 = 54.
4. (iii) The bar has 4 equal parts, so 1 part = 100% ÷ 4 = 25%.
5. (iii) The second bar also has 4 parts and the whole bar is worth 140; the marked 3 parts are 3/4 = 75% of it, so their value is 75% of 140 = 105.

Answer(ii) left = 10%, (ii) right = 54; (iii) left = 25%, (iii) right = 105.

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(i) 25% of 160

1. 25% is 1/4, so 25% of 160 = 1/4 × 160 = 40.
2. Bar model: a bar for 160 cut into 4 equal parts. Each part is 25% and is worth 40.

Answer40

(ii) 16% of 250

1. 10% of 250 = 25, so 2% of 250 = 5 and 4% of 250 = 10.
2. 16% is 4 × 4%, so 16% of 250 = 4 × 10 = 40.
3. Bar model: a bar for 250 cut into 25 equal parts, each 4% and worth 10. Four parts make 16%, which is 40.

Answer40

(iii) 62% of 360

1. 1% of 360 = 3.6, so 62% of 360 = 62 × 3.6 = 223.2.
2. Bar model: a bar for 360 cut into 10 equal parts of 36 (10% each). Six parts make 60% = 216, and 2% more is 7.2, so 62% = 216 + 7.2 = 223.2.

Answer223.2

(iv) 140% of 40

1. 140% of 40 = 1.4 × 40 = 56.
2. Bar model: a bar for 40 cut into 10 equal parts of 4 (10% each). 140% is the whole bar and 4 more parts: 40 + 16 = 56.

Answer56

(v) 1% of 1 hour

1. 1 hour = 60 minutes.
2. 1% of 60 minutes = 60 ÷ 100 = 0.6 minutes, which is 36 seconds.
3. Bar model: a bar for 60 minutes cut into 100 equal parts. One part is 0.6 minutes.

Answer0.6 minutes (36 seconds)

(vi) 7% of 10 kg

1. 7% of 10 kg = 7/100 × 10 = 0.7 kg, which is 700 g.
2. Bar model: a bar for 10 kg cut into 100 equal parts of 0.1 kg. Seven parts make 0.7 kg.

Answer0.7 kg (700 g)

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1. Red paint is 3/4 of the 60 ml of deep orange paint.
2. 3/4 × 60 = 45.

Answer45 ml

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(i) 50% of 510 ▢ 50% of 515

1. Same percentage (50%) of the larger number (515) is bigger.

Answer<

(ii) 37% of 148 ▢ 73% of 148

1. Same base (148); 73% of it is more than 37% of it.

Answer<

(iii) 29% of 43 ▢ 92% of 110

1. 29% of 43 = 12.47; 92% of 110 = 101.2 — the second is far bigger both because the percentage and the base are bigger.

Answer<

(iv) 30% of 40 ▢ 40% of 50

1. 30% of 40 = 12; 40% of 50 = 20.

Answer<

(v) 45% of 200 ▢ 10% of 490

1. 45% of 200 = 90; 10% of 490 = 49.

Answer>

(vi) 30% of 80 ▢ 24% of 64

1. 30% of 80 = 24; 24% of 64 = 15.36.

Answer>

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(i) 30% of k is 70

1. 60% is double 30%, so 60% of k = 2 × 70 = 140.
2. 90% is three times 30%, so 90% of k = 3 × 70 = 210.
3. 120% is four times 30%, so 120% of k = 4 × 70 = 280.

Answer60% of k = 140, 90% of k = 210, 120% of k = 280

(ii) 100% of m is 215

1. 100% of m is 215, so m = 215.
2. 10% of m = 215 ÷ 10 = 21.5.
3. 1% of m = 215 ÷ 100 = 2.15.
4. 6% of m = 6 × 2.15 = 12.9.

Answer10% of m = 21.5, 1% of m = 2.15, 6% of m = 12.9

(iii) 90% of n is 270

1. 9% is a tenth of 90%, so 9% of n = 270 ÷ 10 = 27.
2. 18% is double 9%, so 18% of n = 54.
3. 100% of n is the whole of n; since 90% of n is 270, n = 270 ÷ 0.9 = 300.

Answer9% of n = 27, 18% of n = 54, 100% of n = 300

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(i) 3 is ____ % of 300

1. 3 out of 300 as a percentage: 3/300 × 100 = 1.

Answer1%

(ii) ____ is 40% of 4

1. 40% of 4 = 0.4 × 4 = 1.6.

Answer1.6

(iii) 40 is 80% of ____

1. 40 is 80% of the number, so the number = 40 ÷ 0.8 = 50.

Answer50

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1. A day has 24 hours, so 10% of a day = 10/100 × 24 = 2.4 hours.
2. A week has 7 × 24 = 168 hours, so 1% of a week = 1/100 × 168 = 1.68 hours.
3. 2.4 hours is longer than 1.68 hours.

AnswerYes — 10% of a day (2.4 hours) is longer than 1% of a week (1.68 hours).

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Open NCERT p. 13One way to think about it

1. On day n, the bull is given (n + 1) units and eats n units, so the fraction eaten is n/(n + 1).
2. As a percentage, that is n/(n + 1) × 100.
3. Day 1: 1/2 × 100 = 50%.
4. Day 2: 2/3 × 100 ≈ 66.67%.
5. Day 3: 3/4 × 100 = 75%.
6. Day 99: 99/100 × 100 = 99%.
7. As the days go on, n/(n + 1) gets closer and closer to 1, so the percentage eaten keeps climbing towards 100% — but it never actually reaches 100%, because 1 unit is always left uneaten.

In shortThe percentage eaten on day n is n/(n + 1) × 100 — 50%, 66.67%, 75%, …, 99% on day 99 — always rising towards 100% but never reaching it.

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1. 20% of the plantation takes 18 days, at a constant rate of picking.
2. 100% is 5 times as much as 20% (since 100 ÷ 20 = 5), so it takes 5 times as many days.
3. 5 × 18 = 90 days.

Answer90 days. This relies on the workers picking at the same steady rate throughout — in a real plantation some patches may be denser, harder to reach, or the workers may tire, so the actual time could differ.

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1. 10% of 90 minutes = 9 minutes, for warm up.
2. 80% of 90 minutes = 72 minutes, for play.
3. 10% of 90 minutes = 9 minutes, for cool down.
4. Check: 9 + 72 + 9 = 90 minutes.

AnswerWarm up = 9 minutes, play = 72 minutes, cool down = 9 minutes.

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Open NCERT p. 14Checked by computerAnswers can differ: one example

1. This needs a current estimate of the world's population, which the chapter does not state and which changes constantly.
2. Using a commonly cited current estimate of about 8.2 billion (8,200,000,000) people worldwide.
3. 90% of 8,200,000,000 = 0.9 × 8,200,000,000 = 7,380,000,000.

AnswerAbout 7.38 billion (7,380,000,000) people, using a world population estimate of about 8.2 billion — the exact figure depends on which year's population estimate is used.

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(i) what is the proportion of each of the above ingredients?

1. Doubling the number of people scales every ingredient's amount by the same factor.
2. So the proportion each ingredient makes up of the total does not change: rava 40%, sugar 40%, ghee 20%.

AnswerRava 40%, sugar 40%, ghee 20% — the same proportions as for 4 people.

(ii) how much rava, sugar and ghee are present?

1. Rava = 40% of 2 kg = 0.8 kg.
2. Sugar = 40% of 2 kg = 0.8 kg.
3. Ghee = 20% of 2 kg = 0.4 kg.

AnswerRava 0.8 kg, sugar 0.8 kg, ghee 0.4 kg.

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1. Profit = selling price − cost price = 110 − 75 = ₹35.
2. Profit margin with respect to the cost = profit/cost × 100 = 35/75 × 100.
3. 35/75 × 100 = 140/3 ≈ 46.67%.

Answer140/3 % ≈ 46.67%

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1. A profit margin is worked out on the cost price, so the cost price ₹475 is 100%.
2. Selling price = 100% + 50% of the cost price = 150% of the cost price.
3. Selling price = 475 × 150/100 = ₹712.50.

Answer₹712.50

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Open NCERT p. 19Checked by computerReads two ways: both answers shown

1. ₹2.5 crore = ₹2,50,00,000.
2. A 25% profit margin can be measured on the sales (revenue) or on the cost, and the question does not say which. The box on pages 18–19 uses both, depending on what we want to know.
3. Reading 1 (margin on revenue). We lead with this because the question stresses the revenue, and the book takes profit on revenue when we ask how much profit was made on the overall sales. Profit = 25% of ₹2,50,00,000 = ₹62,50,000.
4. Expenditure = ₹2,50,00,000 − ₹62,50,000 = ₹1,87,50,000.
5. Reading 2 (margin on cost, as in Questions 1 and 2 of this set): the cost is 100% and the sales are 125% of the cost.
6. Cost = ₹2,50,00,000 × 100/125 = ₹2,00,00,000.

AnswerMargin read on revenue: ₹18750000 (₹1,87,50,000, that is ₹1.875 crore). Margin read on cost: ₹20000000 (₹2,00,00,000, that is ₹2 crore).

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1. A discount is taken off the marked price, so the marked price ₹300 is 100%.
2. Anwar pays 100% − 25% = 75% of ₹300.
3. 75% of 300 = ₹225.

Answer₹225

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1. A percentage increase is measured against the starting price, ₹60, not the final price.
2. The increase is 100 − 60 = ₹40.
3. Percentage increase = 40/60 × 100 = 200/3 % = 66⅔% ≈ 66.67%.
4. Option (iv) prints this as 66.66%: 66.666…% cut short after two decimal places instead of rounded. It is the same number, so the answer is option (iv).

Answer200/3 % = 66⅔% ≈ 66.67%, which is option (iv)

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1. The 15% discount came off the original price, so ₹4,40,000 is what is left after removing 15% — that is 85% of the original price.
2. Original price = 4,40,000 ÷ 0.85.
3. Original price = ₹5,17,647.06 (rounded to the nearest paisa).

Answer8800000/17 rupees ≈ ₹5,17,647.06

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1. The winner's share is 500 out of 1600, as a percentage: 500/1600 × 100 = 125/4 = 31.25%.
2. The remaining 1100 votes went to the other candidates, and none of them can have 500 or more votes (or they would also be a winner or tie).
3. So each other candidate can hold at most 499 votes; 1100 ÷ 499 needs at least 3 such candidates (2 candidates could hold at most 998, which is not enough).
4. So the minimum number of candidates in total is 3 others + the winner = 4.

Answer31.25%; at least 4 candidates

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1. Inflation is the percentage increase, measured against the starting (2024) price, ₹38.
2. The increase is 42 − 38 = ₹4.
3. 4 out of 38, as a percentage, is 4/38 × 100 = 200/19 ≈ 10.53%.

Answer200/19% ≈ 10.53%

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1. Increasing a number by 20% multiplies it by 1.20.
2. So the number × 1.20 = 90, which means the number = 90 ÷ 1.20.
3. 90 ÷ 1.20 = 75.

Answer75

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1. A 5% profit and a 10% loss cannot be averaged, because they are percentages of two different cost prices — each buffalo's own cost price.
2. The buffalo sold at 5% profit had cost price = 80,000 ÷ 1.05 = ₹76,190.48 (rounded).
3. The buffalo sold at 10% loss had cost price = 80,000 ÷ 0.90 = ₹88,888.89 (rounded).
4. Total cost price ≈ ₹1,65,079.37; total selling price = ₹1,60,000.
5. Overall result = 1,60,000 − 1,65,079.37 ≈ −₹5,079.37, an overall loss.

AnswerProfit = −320000/63 rupees ≈ −₹5,079.37, so there is an overall loss of about ₹5,079.37

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1. A 5% increase means the new population is 100% + 5% = 105% of the old one.
2. 105% of p is p × 1.05, which is option (iv).
3. None of the other options equal p × 1.05.

Answer(iv)

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1. A fall of 85% means the demand now is 100% − 85% = 15% of the demand a decade ago.
2. That is exactly statement (iii).
3. None of the other statements describe that same relationship.

Answer(iii)

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1. Without compounding: the interest each year is 10% of the original ₹20,000, so total amount = p(1 + rt) = 20,000 × (1 + 0.10 × 2) = ₹24,000.
2. With compounding: each year's interest is 10% of that year's start balance, so the amount = p(1+r)^t = 20,000 × 1.10² = ₹24,200.
3. Compounding gives ₹200 more over 2 years.

Answer₹24,000 without compounding; ₹24,200 with compounding

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1. Without compounding: amount = p(1+rt) = 20,000 × (1 + 0.05 × 4) = ₹24,000.
2. With compounding: amount = p(1+r)^t = 20,000 × 1.05⁴ = ₹24,310.125.

Answer₹24,000 without compounding; ₹24,310.125 with compounding

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Open NCERT p. 23One way to think about it

1. Question 1 (10% for 2 years): ₹24,000 without compounding, ₹24,200 with compounding.
2. Question 2 (5% for 4 years): ₹24,000 without compounding, ₹24,310.125 with compounding.
3. Without compounding the two deposits give exactly the same amount. The amount is p(1 + rt), and rt is the same in both: 0.10 × 2 = 0.05 × 4 = 0.20. So each earns ₹4,000 interest.
4. With compounding they are different: 1.10² = 1.21 but 1.05⁴ = 1.21550625. The smaller rate over more years gives ₹110.125 more, because its interest is added back four times instead of two, so more of it earns interest on interest.
5. This is a 'share and discuss' question, so other observations are also valid; this is the main one.

In shortWithout compounding both give ₹24,000, because rate × time is 0.20 in both; with compounding, 5% for 4 years (₹24,310.125) gives more than 10% for 2 years (₹24,200).

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1. Without compounding, the interest is the same every year: 6% of p, which is p × 0.06.
2. Over 4 years the interest is p × 0.06 × 4. The total amount is the principal plus this interest: p + (p × 0.06 × 4) = p × 1.24. That is option (vii).
3. Options (i) to (iv) use the rate wrongly (6, 0.6, 0.6/100 and 0.06/100 instead of 6/100 = 0.06), and none of them adds back the principal p.
4. Options (v) and (vi) multiply the principal by 4 as well: for example p × 1.06 × 4 = 4.24p, which counts the principal four times. Option (v) also uses 1.6 instead of 1.06.

Answer(vii)

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1. Without compounding, interest = p × r × t = 50,000 × 0.07 × 3 = ₹10,500.
2. With compounding, amount = p(1+r)^t = 50,000 × 1.07³ = ₹61,252.15, so interest = ₹11,252.15.
3. Extra interest from compounding = 11,252.15 − 10,500 = ₹752.15.

Answer₹10,500 without compounding; ₹752.15 more with compounding

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1. Giridhar's interest (no compounding) = p × r × t = 12,500 × 0.12 × 3 = ₹4,500.
2. Raghava's amount (compounded) = 12,500 × 1.10³ = ₹16,637.50, so his interest = ₹4,137.50.
3. Giridhar pays more, and the difference is 4,500 − 4,137.50 = ₹362.50.

AnswerGiridhar pays more, by ₹362.50

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1. Without compounding, amount = 1000(1 + 0.10t); doubling needs 1 + 0.10t = 2, so t = 10 years exactly.
2. With compounding, amount = 1000 × 1.10^t; doubling needs 1.10^t ≥ 2. 1.10⁷ ≈ 1.949 (not yet double), 1.10⁸ ≈ 2.144 (past double), so it takes 8 full years.
3. Without compounding the amount grows in equal steps every year (a straight line) — that is linear growth. With compounding each year's growth is a fixed multiple of an ever-growing amount — that is exponential growth.

Answer10 years without compounding; 8 years with compounding; yes, compounding is exponential growth and not-compounding is linear growth

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1. Rising by 3% every year means each year's population is 1.03 times the year before − the base moves each time, so this is compounding, not plain addition.
2. 1.5 crore = 1,50,00,000.
3. After 3 years the population is 1,50,00,000 × 1.03 × 1.03 × 1.03 = 1,50,00,000 × (1.03)³.
4. (1.03)³ = 1.092727.
5. 1,50,00,000 × 1.092727 = 1,63,90,905.

AnswerAbout 1,63,90,905.

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1. A 2.5% rise every hour means the count is multiplied by 1.025 each hour, with the new, larger count as the base for the next hour.
2. After 2 hours the count is 5,06,000 × 1.025 × 1.025 = 5,06,000 × (1.025)².
3. (1.025)² = 1.050625.
4. 5,06,000 × 1.050625 = 5,31,616.25.

Answer5,31,616.25 by the formula; since bacteria are counted in whole numbers, this rounds to about 5,31,616.

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1. Here, 2000's population is the base (the yardstick), and 250% means 2.5 times that base − it is allowed to be more than 100% because it is not a share of a whole, just a comparison.
2. 250% of 50 lakhs = 2.5 × 50 lakhs = 125 lakhs.

Answer1,25,00,000, i.e., 125 lakhs.

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1. Write every population in millions so all of them are compared with the same whole: the world is 8.2 billion = 8,200 million, and India is 1.46 billion = 1,460 million.

2. Country	Population	Share of 8,200 million	Nearest tile
   Germany	83 million	83 ÷ 8,200 × 100 ≈ 1.01%	1%
   India	1,460 million	1,460 ÷ 8,200 × 100 ≈ 17.8%	18%
   Bangladesh	175 million	175 ÷ 8,200 × 100 ≈ 2.13%	2%
   USA	347 million	347 ÷ 8,200 × 100 ≈ 4.23%	about 4% (no such tile)

3. Estimating in standard form gives the same picture: India is 1.46 × 10⁹ out of 8.2 × 10⁹, a little under 0.2, so a little under 20%, which is 18%.
4. The USA's share is about 4.2%. The page has no 4% tile: it shows 2% twice, and 4.2% is about double Bangladesh's 2%, so it is not approximately 2% (or 8%). The honest match for the USA is about 4%.

AnswerGermany 1%, India 18%, Bangladesh 2%, USA about 4% (4.23%, which is not among the printed tiles).

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1. Adding 18% GST means the final price is the price plus 18% of the price.
2. 18% of 8,250 = 0.18 × 8,250 = 1,485, so the final price is 8,250 + 1,485 = ₹9,735.

3. Option	Value	Final price?
   (i) 8250 + 18	8,268 (adds ₹18, not 18%)	No
   (ii) 8250 + 1800	10,050 (1,800 is not 18% of 8,250)	No
   (iii) 8250 + 18/100	8,250.18 (adds 18 paise)	No
   (iv) 8250 × 18	1,48,500 (18 times the price)	No
   (v) 8250 × 1.18	9,735	Yes
   (vi) 8250 + 8250 × 0.18	9,735	Yes
   (vii) 1.8 × 8250	14,850 (180% of the price)	No

4. (v) and (vi) are the same calculation written two ways: 8,250 + 8,250 × 0.18 = 8,250 × (1 + 0.18) = 8,250 × 1.18.

Answer(v) and (vi): both give ₹9,735.

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1. Each month's change acts on that month's own population (the base moves), so the three changes multiply rather than add.
2. Population after 3 months = p × 1.05 × 0.98 × 0.97 − this matches statement (ii).
3. 1.05 × 0.98 × 0.97 = 0.99813, which is less than 1.
4. So the population after 3 months is less than p − this matches statement (vi).
5. Statements (i) and (iii) wrongly use the raw percentages (0.05, −0.02, −0.03) instead of the multipliers, and (iv), (v) are ruled out since the net multiplier is below 1.

Answer(ii) and (vi) are true.

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1. Let the cost price be x.
2. A 35% profit margin makes the marked selling price 1.35x.
3. A 30% discount is taken off this selling price, not off x: the final price is 1.35x × 0.70 = 0.945x.
4. 0.945x is less than the cost price x, by 0.055x, which is 5.5% of the cost price.

AnswerHe makes a loss of 5.5%, not a profit − the 30% discount is bigger than it looks because it acts on the marked-up price, not on the original cost.

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1. Counting the dot grid, the whole figure is a rectangle 8 units wide and 8 units tall, so it holds 64 unit squares in all.
2. Region A (top left) is 4 × 4 = 16 unit squares.
3. Region B (top right) is 4 × 6 = 24 unit squares, and region C (bottom right) is 4 × 2 = 8 unit squares.
4. The remaining 4 × 4 = 16 unit squares at the bottom left are cut by the diagonal into two equal triangles, D and E, of 8 unit squares each.
5. E's share of the whole figure is 8 out of 64.
6. 8/64 = 1/8 = 12.5%.

Answer12.5% (region E is 1/8 of the whole figure).

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1. 5% of 40 = (5/100) × 40 = 2, and 40% of 5 = (40/100) × 5 = 2 − the same answer.
2. 25% of 12 = 3, and 12% of 25 = 3 − the same answer again.
3. 15% of 60 = 9, and 60% of 15 = 9 − the same once more.
4. In every pair, swapping which number is the percentage and which is the amount does not change the answer.
5. In algebra: x% of y is (x/100) × y, and y% of x is (y/100) × x. Multiplication does not care about order, so (x/100) × y = (y/100) × x for any x and y.

AnswerAll three pairs give equal results (2, 3 and 9), and this always happens: x% of y is always equal to y% of x, since both equal xy/100.

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(i) What percentage of the students going … are Grade 8 girls?

1. Grade 8 students are 40% of all the students, and 60% of those Grade 8 students are girls.
2. So Grade 8 girls are 60% of 40% of all the students: (60/100) × (40/100) = 24/100 = 24%.

Answer24% of the students going on the excursion are Grade 8 girls.

(ii) how many of them are Grade 8 girls?

1. 40% of 160 = (40/100) × 160 = 64 Grade 8 students.
2. 60% of 64 = (60/100) × 64 = 38.4. Part (i) gives the same: 24% of 160 = 38.4.
3. A number of girls must be a whole number, so the percentages in the question must be rounded ones. The nearest whole number is 38: 38 girls out of 64 is about 59%, while 39 out of 64 is about 61%.

AnswerThe percentages give 38.4, so there are about 38 Grade 8 girls.

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1. Let the cost price of one pencil be c. No actual rupee price is given, and none is needed.
2. Selling price of 3 pencils = cost price of 5 pencils, so 3 × (selling price of one pencil) = 5c, giving selling price of one pencil = 5c/3.
3. 5c/3 is more than c, so he makes a profit.
4. Profit per pencil = 5c/3 − c = 2c/3.
5. Profit percentage = (2c/3) ÷ c × 100 = 200/3 = 66.67% (approximately).

AnswerHe makes a profit of 200/3%, i.e. about 66.67%.

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1. Each year's increase acts on that year's fare, so the two increases multiply: the final fare is the original fare × 1.03 × 1.04.
2. 1.03 × 1.04 = 1.0712.
3. The overall change is 1.0712 − 1 = 0.0712, i.e. 7.12%.
4. This is more than 3% + 4% = 7%, because the 4% rise also acts on the 3% already added.

Answer7.12%, slightly more than simply adding 3% and 4%.

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1. Let the original length be L and breadth be B, so the area is L × B.
2. The new length is 1.1L. For the area to stay the same, the new breadth must be (L × B) ÷ 1.1L = B/1.1 = 10B/11.
3. The decrease in breadth is B − 10B/11 = B/11.
4. As a percentage of the original breadth: (B/11) ÷ B × 100 = 100/11 %, which is about 9.09%.

Answer100/11%, i.e. about 9.09% − not 10%.

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1. Ingredient	Share	Weight in 65 g
   Potato	70%	45.5 g
   Vegetable oil	24%	15.6 g
   Salt	3%	1.95 g
   Spices	3%	1.95 g

2. The four shares are 70% + 24% + 3% + 3% = 100%, so they make up the whole packet with nothing left over.
3. Multiply each share by the total weight, 65 g: e.g. potato = 70% of 65 = 45.5 g.

AnswerPotato 45.5 g, vegetable oil 15.6 g, salt 1.95 g, spices 1.95 g.

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(i) what is the effective price per item in each shop?

1. Shop A: pay for 1 item (₹100) and take home 2, so each item costs 100 ÷ 2 = ₹50.
2. Shop B: pay for 2 items (₹200) and take home 3, so each item costs 200 ÷ 3 = ₹200/3, about ₹66.67.
3. Shop C: pay for 3 items (₹300) and take home 4, so each item costs 300 ÷ 4 = ₹75.

AnswerShop A ₹50, Shop B ₹200/3 (about ₹66.67), Shop C ₹75 per item. From cheapest to costliest: A, B, C.

(ii) calculate the percentage discount on the items

1. Compare the free items with all the items you take home.
2. Shop A: 1 free out of 2 = 1/2 = 50%.
3. Shop B: 1 free out of 3 = 1/3 = 100/3%, about 33.33%.
4. Shop C: 1 free out of 4 = 1/4 = 25%.
5. This agrees with part (i): at Shop A each ₹100 item costs ₹50, a saving of 50%.

AnswerShop A 50%, Shop B 100/3% (about 33.33%), Shop C 25%.

(iii) Suppose you need 4 items. Which shop would you choose?

1. Shop A: buy 2 and get 2 free, so 4 items cost ₹200.
2. Shop B: buy 2 and get 1 free gives 3 items for ₹200; the 4th item costs ₹100 more, so ₹300 in all.
3. Shop C: buy 3 and get 1 free gives exactly 4 items for ₹300.

AnswerShop A, because 4 items cost ₹200 there, against ₹300 at Shop B and at Shop C.

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1. 99% of 100 people are left-handed, so exactly 1 person is right-handed − and that person never leaves.
2. After some left-handed people leave, the right-handed person must be 2% of the new, smaller room.
3. If the new room size is x, then 1 = 2% of x, so x = 1 ÷ 0.02 = 50.
4. The room must shrink from 100 to 50 people, so 100 − 50 = 50 people must leave, and all of them are left-handed.

Answer50 left-handed people must leave.

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1. Age group	Women	Men
   Children	4%	≈ 4.8% (drawn, not printed)
   Teenage	24%	29%
   Twenties	26%	37%
   Thirties	14%	25%
   Forties	7%	14%
   Fifties	4%	9%
   Seniors	2%	4%

2. Each bar is the percentage of that age group (women or men) who can use a computer.
3. (i) Valid. The twenties have the longest bar for women (26%) and for men (37%), so they are the highest group.
4. (ii) Valid. In every group the women's bar is shorter than the men's. For Children the men's bar has no number, but it is drawn longer than the women's 4% bar and ends just short of the 5% line.
5. (iii) Not valid. The bars are percentages of each group, not numbers of people, so the graph cannot say which group has more people.
6. (iv) Not valid. In the thirties 14% of women and 25% of men can use computers. The figure for the whole group lies between these, so it is less than 25%, not more than a quarter.
7. (v) Valid. For those aged 60 and above (Seniors), 2% of women and 4% of men can use computers. Both are below 10%, so fewer than 1 in 10 of the group can.
8. (vi) Not valid. For the twenties the figures are 26% and 37%; half would be 50%.

Answer(i), (ii) and (v) are valid.