Chapter 1: Fractions in Disguise (Percentages)

Yellow is $1-\frac{3}{4}=\frac{1}{4}$ of the mixture, and $\frac{1}{4}=\frac{25}{100}$.

25% (100% − 75%).

Each fifth of 100% is $100÷5=20\%$, so the boxes read 20%, 40%, 60%, 80%. Saving $\frac{2}{5}$ is the same as finding $\frac{2}{5}$ of 100.

20%, 40%, 60%, 80%; so ⅖ = 40%.

Multiply each fraction by 100:

1. $\frac{3}{5}\times 100=60\%$
2. $\frac{7}{14}=\frac{1}{2}=50\%$
3. $\frac{9}{20}=\frac{45}{100}=45\%$
4. $\frac{72}{150}=\frac{48}{100}=48\%$
5. $\frac{1}{3}\times 100=33\frac{1}{3}\%\approx 33.33\%$
6. $\frac{5}{11}\times 100=45\frac{5}{11}\%\approx 45.45\%$

(i) 60% (ii) 50% (iii) 45% (iv) 48% (v) 33⅓% (vi) 45 5⁄11% ≈ 45.45%

$\frac{15}{25}=\frac{60}{100}$, so the answer is option (iv).

(iv) 60%

$\frac{15}{80}\times 100=\frac{1500}{80}=18.75$

18.75%

Measured along the track from Start to Finish, the runners are at about 32%, 53%, 75% and 95% of the way. The nearest options are:

• A → 38%
• B → 55%
• C → 72%
• D → 93%

20% and 84% are extra options that do not match anyone.

A ≈ 38%, B ≈ 55%, C ≈ 72%, D ≈ 93%.

1. 50 out of 100 is more than 5 out of 100, so >.
2. $\frac{5}{10}$ is a half, so =.
3. $\frac{3}{11}$ is less than $\frac{3}{9}=\frac{1}{3}$, which is about 33%. So <.
4. $\frac{1}{3}=33\frac{1}{3}\%$, so <.

(i) > (ii) = (iii) < (iv) <

Any denominator would work, but 100:

• fits our base-10 number system: $31\%=0.31$, so converting between fractions, decimals and percentages is quick;
• gives enough detail (per 10 is too coarse);
• is still small enough to picture easily.

A denominator like 43 would make every comparison an awkward calculation.

100 matches our decimal system and gives enough detail while staying easy to picture.

• Madhu: 25% of 120 g $=\frac{1}{4}\times 120=30$ g.
• Madhav: 35% of 95 g $=0.35\times 95=33.25$ g.

Madhav (33.25 g against 30 g).

Shortcuts:

• 10% = divide by 10
• 20% = double the 10%
• 5% = half the 10%
• 25% = a quarter
• 40% = double the 20%
• 15% = 10% + 5%

Observations:

• 20% is double 10%.
• 20% + 5% = 25%.
• When the value doubles (100 → 200), every percentage doubles.

See the table; e.g. 25% of 287 = 71.75, 15% of 35 = 5.25. Rows add up: (20% of y) + (5% of y) = 25% of y.

Build each one from easy parts (50%, 25%, 20%, 10%, 5%):

• 75% = 50% + 25% (half plus a quarter)
• 90% = 100% − 10%
• 70% = 50% + 20%
• 55% = 50% + 5%

For example, 70% of 80 = 40 + 16 = 56.

Combine easy parts: 75% = 50% + 25%, 90% = 100% − 10%, 70% = 50% + 20%, 55% = 50% + 5%.

To find 10% of a quantity, multiply it by 0.1.

Multiply by 0.1 for 10%; the table is filled above.

• Day 5: $\frac{7800}{5000}\times 100=156\%$ (56% above target)
• Day 6: $\frac{9550}{5000}\times 100=191\%$
• Day 7: 150% of 5000 $=1.5\times 5000=$ ₹7,500
• Day 8: 210% of 5000 $=2.1\times 5000=$ ₹10,500

Day 5: 156%; Day 6: 191%; Day 7: ₹7,500; Day 8: ₹10,500.

Where each one goes on the bar:

• Before the 1 mark: 15% (a little before ⅙ of the way to 1), 28.9% (just over a quarter of the way) and 90% (just before 1).
• Between 1 and 2: 110% (just after 1) and 173% (about ¾ of the way to 2).
• At or after 2: 200% (at 2) and 250% (halfway between 2 and 3).
• Beyond 3: 305% (just after 3) and 358% (a little before 3.6).

Decimal = percent ÷ 100 (e.g. 358% = 3.58 = 179/50); each mark sits at its decimal value on the bar.

(ii) Each of the 10 parts is $100\%÷10=$ 10%. The arrow covers 6 parts, which is 60% of 90 $=$ 54.

(iii) Each of the 4 parts is 25%. The arrow covers 3 parts, which is 75% of 140 $=$ 105.

(ii) 10% and 54 (iii) 25% and 105

1. $\frac{1}{4}\times 160=40$
2. $0.16\times 250=40$
3. $0.62\times 360=223.2$
4. $1.4\times 40=56$ (more than the whole)
5. 1% of 60 min $=0.6$ min $=36$ seconds
6. $0.07\times 10$ kg $=0.7$ kg $=700$ g

(i) 40 (ii) 40 (iii) 223.2 (iv) 56 (v) 36 seconds (vi) 700 g

$\frac{3}{4}\times 60=45$

45 ml

1. Same percentage of a smaller number, so <.
2. A smaller percentage of the same number, so <.
3. A smaller percentage of a smaller number, so <.
4. 12 against 20, so <.
5. 90 against 49, so >.
6. 24 against 15.36, so >.

(i) < (ii) < (iii) < (iv) < (v) > (vi) >

1. Double, triple and quadruple the 30%: 140, 210, 280.
2. 10% = 21.5, 1% = 2.15, 6% = $6\times 2.15=$ 12.9.
3. 9% is a tenth of 90%, so 27. Then 18% = 54, and 100% = $270÷0.9=$ 300.
4. For example: "40% of p is 36: find 20%, 80% and 100% of p" (18, 72, 90). Or: "5% of q is 7: find 15%, 50% and 100% of q" (21, 70, 140).

(i) 140, 210, 280 (ii) 21.5, 2.15, 12.9 (iii) 27, 54, 300

1. $\frac{3}{300}\times 100=1\%$
2. $0.4\times 4=1.6$
3. $40÷0.8=50$

(i) 1% (ii) 1.6 (iii) 50

• 10% of 24 h $=2.4$ h $=144$ min.
• 1% of a week $=1\%$ of 168 h $=1.68$ h $=100.8$ min.

Yes, 10% of a day is longer.

Similar questions:

• Is 1% of a year more than 3 days? (Yes: 3.65 days.)
• Is 50% of an hour more than 1% of a day? (Yes: 30 min against 14.4 min.)

Yes: 144 minutes against 100.8 minutes.

Each day the bull eats $\frac{n}{n+1}$ of the fodder:

• Day 1: $\frac{1}{2}=50\%$
• Day 2: $\frac{2}{3}=66.67\%$
• Day 3: $\frac{3}{4}=75\%$
• Day 4: $\frac{4}{5}=80\%$
• Day 9: $\frac{9}{10}=90\%$
• Day 19: $\frac{19}{20}=95\%$
• Day 99: $\frac{99}{100}=99\%$

The percentage eaten keeps increasing, but more and more slowly. It gets closer and closer to 100% without ever reaching it, because one unit is always left. The unit left over becomes a smaller share each day: 50%, 33%, 25%, …, 1%.

50%, 66.7%, 75%, 80%, …, 99%: the share eaten rises towards 100% but never reaches it.

100% is 5 times 20%, so the time is $5\times 18=90$ days.

The assumption is needed because the calculation treats work done as proportional to time. If the number of workers, the weather or the density of berries changed, the rate would change and the answer would be different.

90 days, assuming the same rate (same workforce and conditions) throughout.

• Warm up: 10% of 90 = 9 min
• Play: 80% of 90 = 72 min
• Cool down: 10% of 90 = 9 min

9 min warm up, 72 min play, 9 min cool down.

Taking the world population as about 8.2 billion (2025):

$$0.9\times 8.2\text{ billion}\approx 7.4\text{ billion}$$

About 7.4 billion people (with a world population of about 8.2 billion).

(i) The proportions do not change: still 40% : 40% : 20%. Only the quantities double.

(ii) For 2 kg:

• Rava: 40% of 2 kg = 0.8 kg (800 g)
• Sugar: 40% of 2 kg = 0.8 kg (800 g)
• Ghee: 20% of 2 kg = 0.4 kg (400 g)

(i) Still 40%, 40%, 20% (ii) 800 g rava, 800 g sugar, 400 g ghee.

• Badam: Zacni has the larger share, 10% against 8%.
• Food chemicals: both have 6%, so neither uses a smaller proportion.
• Totals: each product adds up to 100%.

Both mixes are about two-thirds sugar!

Zacni has more badam (10% vs 8%); food chemicals are equal (6% each); each product's percentages add to 100%.

• Manufacturer: profit $=253-230=23$, so profit $\%=\frac{23}{230}\times 100=10\%$.
• Wholesaler: profit $=300-253=47$, so profit $\%=\frac{47}{253}\times 100\approx 18.6\%$.

(The retailer made 43.3%.)

Manufacturer 10%, wholesaler ≈ 18.6%.

• Notebook: SP $=1.2\times 36=$ ₹43.20.
• Crayons: $1.25\times \text{CP}=50$, so CP $=50÷1.25=$ ₹40.

Notebook: ₹43.20; crayon box cost: ₹40.

Per kg, the selling price is ₹30, so the loss is ₹5 per kg, and $\frac{5}{35}\times 100=14.28\%$.

This is the same answer: the cost and the selling price are both multiplied by 10, so their ratio does not change.

Yes: ₹5 loss on ₹35 per kg is also 14.28%.

The SP is $100\%-12\%=88\%$ of the CP, so CP $=80÷0.88=90.91$.

About ₹90.91 per kg.

• SP $=65\%$ of 1800 $=0.65\times 1800=$ ₹1,170.
• Profit $=1170-900=270$, so profit $\%=\frac{270}{900}\times 100=30\%$.

SP = ₹1,170; profit 30%.

• $3\times 150=450$ ✓
• 9% of 450 $=40.50$ ✓ (for each of CGST and SGST)
• $450+40.50+40.50=531$ ✓

The total GST is 18%, and $1.18\times 450=531$.

The bill is correct (18% GST in total: 9% CGST + 9% SGST).

The profit is $110-75=35$, and $\frac{35}{75}\times 100=46.67\%$.

46⅔% ≈ 46.67%

$1.5\times 475=712.50$

₹712.50

A company's profit margin is measured on its revenue, so the profit is 25% of 2.5 crore = 0.625 crore.

Costs $=2.5-0.625=$ ₹1.875 crore, i.e. ₹1,87,50,000.

(If the 25% were taken on cost instead, then costs $=2.5÷1.25=$ ₹2 crore.)

₹1.875 crore (₹1,87,50,000).

$0.75\times 300=225$

₹225

The increase is ₹40 on a base of ₹60, and $\frac{40}{60}\times 100=66\frac{2}{3}\%$, which is option (iv).

(iv) 66.66%

The price paid is 85% of the original, so the original price $=440000÷0.85\approx 517647$.

About ₹5,17,647.

The winner got $\frac{500}{1600}\times 100=31.25\%$.

The other candidates share the remaining 1100 votes, and each must get fewer than 500. Two of them could get at most $2\times 499=998<1100$ votes, so there must be at least 3 other candidates.

Minimum number of candidates: 4 (for example, 500, 400, 400, 300).

31.25%; at least 4 candidates.

$\frac{4}{38}\times 100\approx 10.53\%$

About 10.5%

$1.2x=90$, so $x=75$.

75

• CP of the first $=80000÷1.05\approx 76,190.48$
• CP of the second $=80000÷0.90\approx 88,888.89$
• Total CP $\approx 1,65,079.37$, while the total SP is $1,60,000$.

Loss $\approx$ ₹5,079.37, which is $\frac{5079.37}{165079.37}\times 100\approx 3.08\%$.

An overall loss of about ₹5,079 (≈ 3.1%).

$p+0.05p=1.05p$, which is option (iv).

(iv) p × 1.05

A fall of 85% leaves $100\%-85\%=15\%$, so the demand now is 15% of the demand a decade ago. That is statement (iii).

None of the others say this. For example, (ii) and (v) take today's demand as the base.

Only (iii): the demand now is 15% of the demand a decade ago.

• Without compounding: interest $=prt$, so the amount is $p(1+rt)$.
• With compounding: the amount is $p(1+r{)}^t$, so the interest is $p(1+r{)}^t-p$.

Simple: interest = prt. Compound: interest = p(1 + r)ᵗ − p.

• Without: $20000+2\times 2000=$ ₹24,000.
• With: $20000\times 1.1\times 1.1=$ ₹24,200.

₹24,000 without compounding; ₹24,200 with compounding (₹200 more).

• Without: $20000+4\times 1000=$ ₹24,000.
• With: $20000\times 1.05^4=20000\times 1.21550625\approx$ ₹24,310.13.

₹24,000 without compounding; ≈ ₹24,310 with compounding.

Without compounding the two banks give the same ₹24,000, because $10\%\times 2=5\%\times 4=20\%$.

With compounding, the 4-year 5% deposit gives more (₹24,310 against ₹24,200). Interest is added back more times, and each time it earns interest of its own.

Equal simple interest (20% either way), but more terms of compounding give a larger amount.

The amount is the principal plus the interest: $p+p\times 0.06\times 4=1.24p$. This is option (vii).

The other options either use the wrong rate (6% is 0.06, not 6, 0.6 or 0.06/100) or multiply instead of adding the interest.

Only (vii) p + (p × 0.06 × 4).

• Without: interest $=50000\times 0.07\times 3=$ ₹10,500.
• With: amount $=50000\times 1.07^3=50000\times 1.225043=61,252.15$, so the interest is ₹11,252.15.

₹10,500 simple interest; compounding gives ₹752.15 more (₹11,252.15).

• Giridhar: $12500\times 0.12\times 3=$ ₹4,500.
• Raghava: $12500\times 1.1^3-12500=16637.50-12500=$ ₹4,137.50.

Giridhar pays ₹362.50 more.

Without compounding: ₹100 is added every year, so it takes 10 years to reach ₹2000.

With compounding:

• After 7 years: $1.1^7=1.949$, so ₹1949
• After 8 years: $1.1^8=2.144$, so ₹2144

So it doubles during the 8th year, in about 7.3 years.

Yes. Simple interest adds the same amount each year, so its graph is a straight line: linear growth. Compounding multiplies by the same factor each year, so the yearly additions keep growing: exponential growth.

10 years without compounding; about 7–8 years with compounding (it has doubled by the end of year 8). Yes: compounding is exponential, simple interest is linear.

$1.5\text{ crore}\times 1.03^3=1.5\times 1.092727\approx 1.639$ crore

About 1.64 crore (≈ 1,63,90,900).

$506000\times 1.025^2=506000\times 1.050625=531616.25$

About 5,31,616 bacteria.

• A: gain ₹200 on ₹100, which is 200%.
• B: gain ₹500 on ₹1000, which is 50%.

A has the bigger percentage, but you can use each option only once. B gives more money (₹500 against ₹200), so choose B if you can afford the deposit.

A: 200% gain (₹200); B: 50% gain (₹500). Choose B for the larger amount.

• ₹180: 20% off is ₹36, so choose ₹50 off.
• ₹225: 20% off is ₹45, so choose ₹50 off.
• ₹300: 20% off is ₹60, so choose 20% off.

The two options are equal at ₹250, where 20% = ₹50.

₹50 off for ₹180 and ₹225; 20% off for ₹300.

On ₹200, Cakely gives ₹140 and then ₹112, while Cakify gives ₹100.

"30% + 20%" is applied one after the other: $0.7\times 0.8=0.56$, which is only a 44% discount.

Cakify (50% off) is cheaper; 30% + 20% is just 44% off.

If Arun has $x$, Ariba has $1.2x$. Arun's share of Ariba's is $\frac{x}{1.2x}=\frac{5}{6}\approx 83.3\%$.

"I have about 83⅓% of Ariba's marbles", i.e. about 16⅔% fewer, not 20% fewer.

$2.5\times 50$ lakh $=125$ lakh

About 125 lakh (1.25 crore).

• Germany: $\frac{83}{8200}\approx 1.0\%$, so 1%.
• India: $\frac{1460}{8200}\approx 17.8\%$, so 18%.
• Bangladesh: $\frac{175}{8200}\approx 2.1\%$, so 2%.
• USA: $\frac{347}{8200}\approx 4.2\%$, so about 4%.

None of the printed options (13, 8, 18, 10, 1, 35, 2, 2, 0.1) is close to 4%. The nearest are 2% and 8%; the correct share is about 4%.

Germany ≈ 1%, India ≈ 18%, Bangladesh ≈ 2%, USA ≈ 4% (not among the options).

Price + 18% of the price $=8250\times 1.18=$ ₹9,735. Options (v) and (vi) give this.

(v) and (vi): ₹9,735.

Each month multiplies the population by its factor: $p\times 1.05\times 0.98\times 0.97=0.99813p$.

That is slightly less than $p$, so (ii) and (vi) are true.

(ii) and (vi) are true.

$1.35\times 0.70=0.945$ of the cost, so he sells at 94.5% of the cost.

A loss of 5.5%.

The large square is 8 × 8 = 64 unit squares. The bottom-left 4 × 4 square (16 units) is cut by its diagonal into D and E, so E = 8 units.

$\frac{8}{64}\times 100=12.5\%$

(For reference: A = 25%, B = 37.5%, C = 12.5%, D = 12.5%.)

12.5%

• 5% of 40 = 2 = 40% of 5
• 25% of 12 = 3 = 12% of 25
• 15% of 60 = 9 = 60% of 15

In general: $x\%$ of $y=\frac{x}{100}\times y=\frac{xy}{100}=\frac{y}{100}\times x=y\%$ of $x$.

This is a useful trick: 8% of 25 is the same as 25% of 8, which is 2.

They are always equal: x% of y = xy/100 = y% of x.

(i) 60% of 40% $=0.6\times 0.4=0.24$, so 24%.

(ii) 24% of 160 = 38.4. A number of students must be a whole number, so the data given are not exact. Grade 8 has 64 students, and 60% of 64 is about 38 girls.

(i) 24% (ii) 38.4, i.e. about 38 girls (the figures given do not give a whole number).

Let one pencil cost ₹1. Then 3 pencils sell for ₹5, so each sells for ₹$\frac{5}{3}$.

Profit per pencil $=\frac{2}{3}$ on a cost of 1, which is $66\frac{2}{3}\%$.

A profit of 66⅔%.

$1.03\times 1.04=1.0712$

7.12% (not 7%).

New breadth $=\frac{b}{1.1}=\frac{10}{11}b$, so the decrease is $\frac{1}{11}=9\frac{1}{11}\%\approx 9.09\%$.

9 1⁄11% ≈ 9.09%

• Potato: $0.70\times 65=45.5$ g
• Oil: $0.24\times 65=15.6$ g
• Salt: $0.03\times 65=1.95$ g
• Spices: 1.95 g

The total is 65 g ✓

Potato 45.5 g, oil 15.6 g, salt 1.95 g, spices 1.95 g.

(i) From cheapest to costliest: A, B, C.

(iii) For exactly 4 items:

• A: pay for 2, so ₹200.
• B: pay ₹200 for 3 and ₹100 more for the 4th, so ₹300.
• C: pay ₹300.

Choose Shop A.

(i) A ₹50, B ₹66.67, C ₹75 (A cheapest) (ii) 50%, 33⅓%, 25% (iii) Shop A: 4 items for ₹200.

There is 1 right-handed person. For left-handers to be 98%, that one person must be 2% of the room, so the room must have $1÷0.02=50$ people.

Hence $100-50=50$ left-handed people must leave.

50 people (a surprisingly large number).

1. Valid: the twenties have the highest bars for both females (26%) and males (37%).
2. Valid: in every age group, the female bar is shorter than the male bar.
3. Not valid: the graph shows the percentages who can use computers, not how many people are in each group.
4. Not valid: 14% of females and 25% of males, so together less than a quarter.
5. Valid: seniors are at 2% and 4%, well under 10%.
6. Not valid: the share is 26% to 37%, not 50%.

Valid: (i), (ii) and (v).

A knight always jumps 2 squares one way and 1 square the other. So a knight never attacks a square in its own row, because that would need a jump of 0 rows.

So all 8 knights in one row are safe:

Another way: a knight always lands on a square of the opposite colour. Knights placed only on squares of one colour can never attack each other. For example, put them on the 8 dark squares of the first two rows.

In fact, up to 32 knights (all of one colour) can be placed peacefully.

Put all 8 knights in one row (or all on squares of the same colour): no knight attacks another.