The Product Rule: Price, Consumption, Income and Area

The Product Rule in One Idea

A lot of SSC percentage questions are the same question in different clothes. The thing you care about is one quantity multiplied by another:

Once you see the product, every question is one of two kinds.

Kind 1: both factors change, find the change in the product. Multiply the multipliers. If one factor goes up 20% and the other goes down 10%, the product becomes 1.2 × 0.9 = 1.08 times, a rise of 8%. The successive-change formula $a + b + \frac{ab}{100}$ from the Percentage Change and Successive Change section gives the same: 20 − 10 − 2 = 8.

Kind 2: the product must stay the same. Then the two factors move in opposite directions. If one factor rises by x%, the other must fall by

$$\frac{x}{100 + x} \times 100\%$$

and if one factor falls by x%, the other must rise by

$$\frac{x}{100 - x} \times 100\%$$

The fraction way is quicker. A 25% rise means the price is multiplied by $\frac{5}{4}$. To keep the product fixed, consumption must be multiplied by $\frac{4}{5}$, which is a fall of $\frac{1}{5}$, or 20%. In words: a rise of $\frac{1}{n}$ is cancelled by a fall of $\frac{1}{n + 1}$, and a fall of $\frac{1}{n}$ is cancelled by a rise of $\frac{1}{n - 1}$.

When the percentage is not a neat fraction, use the formula: a 30% rise needs a fall of $\frac{30}{130} \times 100\% = 23\frac{1}{13}\%$.

[SSC Tip] The two numbers are never equal. If prices fall 20%, the same money buys 25% more, not 20% more. SSC nearly always puts the "same number" answer among the options, so that is the one to avoid.

Price, Consumption and Expenditure

Expenditure = Price × Consumption. Price questions come in four shapes.

1. Spending must not change. Use the table in The Product Rule in One Idea. If sugar goes up $12\frac{1}{2}\%$ ($\frac{1}{8}$), a family must use $\frac{1}{9}$ less, which is $11\frac{1}{9}\%$. If cooking oil gets $37\frac{1}{2}\%$ cheaper ($\frac{3}{8}$), the price becomes $\frac{5}{8}$ of what it was, so the same money buys $\frac{8}{5}$ times as much, a rise of 60%.

2. Spending may change by a given percentage. Divide the two multipliers:

$$\text{new consumption} = \frac{\text{expenditure multiplier}}{\text{price multiplier}} \times \text{old consumption}$$

Petrol goes up 25% and a family lets its petrol bill rise only 10%. Consumption becomes $\frac{1.10}{1.25} = 0.88$ times, so it must be cut by 12%. If rice gets 20% cheaper and the family decides to spend 8% less on it, consumption becomes $\frac{0.92}{0.80} = 1.15$ times, a rise of 15%.

3. Both change; find the new spending. Multiply. Price up 15% and consumption down 15% gives 1.15 × 0.85 = 0.9775, a fall of 2.25%. An equal rise and fall never cancel: the result is always a fall of $\frac{x^2}{100}\%$.

4. Real quantities. A family uses 30 kg of rice a month at ₹40 a kg, so it spends ₹1,200. The price goes up to ₹48. For the same ₹1,200 it can buy 1200 ÷ 48 = 25 kg, which is 5 kg less. That is 5 out of 30, or $16\frac{2}{3}\%$, and the price rise was 20%: the table again.

Revenue and tax work the same way. Revenue = rate × quantity sold.

• A cinema cuts its ticket price by 20% and sells 30% more tickets: 0.8 × 1.3 = 1.04, so its takings rise 4%.
• The tax on an item is cut by 25% and its sale goes up 20%: 0.75 × 1.2 = 0.9, so the tax collected falls 10%.

Purchasing power. What a salary can buy = salary ÷ price level. If the salary rises 20% and prices rise 25%, purchasing power becomes $\frac{1.2}{1.25} = 0.96$ times, a fall of 4%. Just subtracting (20 − 25 = −5%) is the usual slip.

Two price rises in a row. Up 10% and then up 20% means 1.1 × 1.2 = 1.32 times. To keep spending the same, consumption must become $\frac{1}{1.32} = \frac{25}{33}$ times, a cut of $\frac{8}{33}$, which is $24\frac{8}{33}\%$.

Population × use per head is the same product. If a town's population goes up 10% and each person uses 10% less water, the total water used falls 1%. It does not stay the same.

Same Money, More or Fewer Kilograms

"A fall of 20% in the price of dal lets a person buy 4 kg more for ₹1,200. Find the original price per kg."

The idea. After the fall, the quantity he used to buy costs only 80% of ₹1,200. So 20% of ₹1,200 = ₹240 is left over, and that ₹240 is exactly what buys the extra 4 kg at the new price.

$$\text{new price} = \frac{x\% \text{ of } M}{k}$$

Here M is the money spent and k is the extra kilograms. So the new price is 240 ÷ 4 = ₹60 a kg, and the original price is 60 ÷ 0.8 = ₹75 a kg. Check: 1200 ÷ 75 = 16 kg before and 1200 ÷ 60 = 20 kg after.

When the price rises. "A 25% rise in the price of apples means a person gets 3 kg less for ₹600." The same formula gives the new price again. Buying the old quantity at the new price would need 25% more money, ₹150 extra, and ₹150 is what the missing 3 kg cost at the new price. New price = 150 ÷ 3 = ₹50 a kg, and original price = 50 ÷ 1.25 = ₹40 a kg. Check: 600 ÷ 40 = 15 kg and 600 ÷ 50 = 12 kg.

So in both cases (x% of M) ÷ k gives the new price. The original is the other one. Most wrong answers come from stopping a step early or going a step too far.

Per dozen and per item. A 20% cut in the price of bananas lets a man buy 12 more for ₹48. The ₹9.60 he saves buys 12 bananas, so the reduced price is ₹9.60 a dozen (80 paise each). The original price was 9.60 ÷ 0.8 = ₹12 a dozen.

When no money is given. Use the product rule on the quantity. "Because of a 10% fall in price, a person gets 3 kg more for the same money. How much did he buy before?" A fall of $\frac{1}{10}$ in price means a rise of $\frac{1}{9}$ in quantity. So $\frac{1}{9}$ of the old quantity is 3 kg, and the old quantity was 27 kg (now 30 kg). With a 20% rise in price, the quantity falls by $\frac{1}{6}$; if that is 5 kg, the old quantity was 30 kg.

[Tier 2] A fall in rupees, not in percent. "If a pen cost ₹2 less, ₹240 would buy 4 more pens. Find the price of a pen." No percentage is given, so form the equation:

$$\frac{240}{p - 2} - \frac{240}{p} = 4$$

This gives $p(p - 2) = 120$, so p = 12. Trying the options is often faster: 240 ÷ 12 = 20 pens and 240 ÷ 10 = 24 pens.

Income, Expenditure and Savings: The Basic Table

Income = Expenditure + Savings. This type came up often in CGL papers from 2023 to 2025, so learn the method well.

Method. Take income as 100 (or use the ratio given). Make a small table with a row for before and a row for after. In each row, any two numbers fix the third.

"A man saves 20% of his income. His income goes up 25% and his expenditure goes up 15%. By what percentage do his savings go up?"

Expenditure: 80 + 15% of 80 = 92. Savings: 125 − 92 = 33. They went from 20 to 33, a rise of $\frac{13}{20}$, which is 65%.

The same table answers every version:

• Income and savings changes given, find the expenditure change. Saves 25%; income up 20%, savings up 50%. Before: 100, 75, 25. After: 120, ?, 37.5. Expenditure = 82.5, up 7.5 on 75, which is 10%.
• Savings unchanged in rupees. Saves 30%; income up 21%. Before: 100, 70, 30. After: 121, 91, 30. Expenditure went up 21 on 70, which is 30%.
• Savings can fall. Saves 25%; income up 10%, expenditure up 20%. After: 110, 90, 20. Savings fell 5 on 25, which is 20%.
• New savings as a % of new income. In the first table, 33 out of 125 is 26.4%.
• A rupee figure is given. If the savings in the first table went up by ₹2,600, then 13 units = ₹2,600, so 1 unit = ₹200 and the income was 100 × 200 = ₹20,000.

Weighted-average shortcut. Income is expenditure plus savings, so the % change in income is the average of the other two % changes, weighted by their sizes:

$$I \times i = E \times e + S \times s$$

where i, e and s are the three percentage changes. For the first table: 100 × 25 = 80 × 15 + 20 × s, so 20s = 1300 and s = 65.

It is fastest when the question asks what % of income was saved. "Income goes up 16%, expenditure up 10% and savings up 40%. What % of his income did he save?" Here 16 lies between 10 and 40, and by alligation E : S = (40 − 16) : (16 − 10) = 24 : 6 = 4 : 1. So he saved $\frac{1}{5}$ of his income, 20%.

[SSC Tip] The three percentage changes are on three different bases. Expenditure going up 15% does not make savings go down 15%. Always go through the actual numbers.

Income and Savings: Two Years, Ratios and Falls

"Total expenditure in two years is double the first year's." This line simply means the expenditure did not change from year 1 to year 2. So all the extra income went into savings.

"Ravi's income rises 20% in the second year and his savings rise 80%. His total expenditure over the two years is double his expenditure in the first year. What % of his income did he save in the first year?"

Take income 100 and savings S. Expenditure is the same in both years, so

$$100 - S = 120 - 1.8S$$

which gives 0.8S = 20 and S = 25. He saved 25%.

Shortcut: extra income = extra savings, so i% of I = s% of S, and $\frac{S}{I} = \frac{i}{s}$. Here $\frac{20}{80} = \frac{1}{4}$, which is 25%.

Income and expenditure as a ratio. Income : expenditure = 5 : 3. Income rises 20% and expenditure rises 10%.

Savings rise 70 on 200, which is 35%. Multiplying the ratio by 100 keeps the decimals away.

Savings as a part of expenditure. "He saves $\frac{1}{4}$ of what he spends" means S : E = 1 : 4, so savings are $\frac{1}{5}$ of income, 20%. In general, if savings are $\frac{1}{n}$ of expenditure, they are $\frac{1}{n + 1}$ of income.

When income falls. Income falls 20%, expenditure falls 25% and savings fall 5%. What % of income was saved? With income 100 and savings S:

$$80 - 0.75(100 - S) = 0.95S$$

so 5 + 0.75S = 0.95S, which gives S = 25, or 25%.

Monthly figures in rupees. "Suresh saves ₹4,500 a month, which is 18% of his salary. His salary goes up 12% and his expenditure goes up 10%. What does he save now?" Salary = 4500 ÷ 0.18 = ₹25,000 and expenditure = ₹20,500. Now the salary is ₹28,000 and the expenditure ₹22,550, so he saves ₹5,450.

The other two-year line. "Total savings are double the first year's" means the savings did not change, so all the extra income went into expenditure. Read that last line twice before you set up the table.

Two People: Incomes, Spending and Savings

When a question has two earners, write each person's savings as (100 − spending %) of income. Then build one equation from the line that links them.

Equal savings, different spending. "A and B together earn ₹72,000 a month. A spends 80% of his salary and B spends 70% of hers. Their savings are equal. Find their salaries."

A saves 20% and B saves 30%. 20% of A = 30% of B, so A : B = 30 : 20 = 3 : 2. A gets $\frac{3}{5}$ of ₹72,000 = ₹43,200 and B gets ₹28,800. Check: 20% of 43,200 = 30% of 28,800 = ₹8,640.

A ratio of incomes and a ratio of expenditures. "The incomes of A and B are in the ratio 3 : 2 and their expenditures in the ratio 5 : 3. Each saves ₹2,000. Find their incomes."

Let the incomes be 3x and 2x and the expenditures 5y and 3y. The savings are equal, so 3x − 5y = 2x − 3y, which gives x = 2y. Then A's savings are 3(2y) − 5y = y = 2000. So y = 2000 and x = 4000. The incomes are ₹12,000 and ₹8,000, and the expenditures ₹10,000 and ₹6,000.

Checking options also works well here: an option is right only if both people end up with the stated savings.

One person earns x% more. "Asha earns 25% more than Bina. Asha saves 20% of her salary and Bina saves 30% of hers. Together they save ₹22,000. Find Bina's salary."

Take Bina's salary as B. Asha earns 1.25B and saves 0.2 × 1.25B = 0.25B. Together 0.25B + 0.3B = 0.55B = 22,000, so B = ₹40,000. Asha earns ₹50,000.

Who saves more? In that example Asha earns more but saves ₹10,000, while Bina saves ₹12,000. Bina's savings are 20% more than Asha's, and Asha's are $16\frac{2}{3}\%$ less than Bina's. A higher income does not mean higher savings.

Ratios after a raise. Incomes in the ratio 3 : 2 get raises of 20% and 25%. The new ratio is 3.6 : 2.5 = 36 : 25.

[SSC Tip] "Their savings are equal", with spending percentages given, is a percentage-with-ratio question in disguise: a% of A = b% of B gives A : B = b : a. Put the savings percentages into that ratio, not the spending ones.

Area and Perimeter When Sides Change

Area is a product, so every area question is the product rule.

Rectangle. Area = length × breadth. If the length changes by a% and the breadth by b%, the area changes by $a + b + \frac{ab}{100}$ percent.

• Length up 20%, breadth down 10%: 1.2 × 0.9 = 1.08, area up 8%.
• Length up 10%, breadth down 10%: 1.1 × 0.9 = 0.99, area down 1%.

Keeping the area the same. Exactly like price and consumption: if the length goes up 25%, the breadth must go down 20%; if the length goes down 20%, the breadth must go up 25%.

Square. Area = side × side, so both factors change by the same x%, and the area changes by $2x + \frac{x^2}{100}$ percent.

• Side up 10%: area up 21%. Side up 20%: area up 44%. Side up 50%: area up 125%.
• Side down 10%: area down 19%. Side down 20%: area down 36%.

Circle. Area = $\pi r^2$, so it behaves like a square in the radius. If the radius, the diameter or the circumference changes by x%, the area changes by $2x + \frac{x^2}{100}$ percent, because all three grow in step with r.

Triangle. Area = $\frac{1}{2}$ × base × height, so treat base and height like length and breadth. The $\frac{1}{2}$ changes nothing. An equilateral triangle's area depends on side × side, so it behaves like a square.

Going backwards: from area to side. Take the square root of the area multiplier.

Perimeter is a sum, not a product. If every side changes by x%, the perimeter changes by exactly x%. So when a square's area goes up 21%, its perimeter goes up only 10%. If only the length of a rectangle changes, you need the actual sides: a 20 m × 10 m plot with its length up 10% has a perimeter of 2(22 + 10) = 64 m instead of 60 m, a rise of $6\frac{2}{3}\%$.

The diagonal of a square changes in step with its side. "The diagonal of a square goes up 20%" means the area goes up 44%.

Volume and Surface Area When Dimensions Change

Solids add a third factor. Volume multiplies three lengths; surface area multiplies two.

Cube. Volume = $a^3$ and surface area = $6a^2$. Cube the edge multiplier for the volume and square it for the surface area.

The same table works for a sphere: the volume goes with $r^3$ and the surface area with $r^2$.

Cuboid. Volume = l × b × h, so it is three successive changes. Length up 10%, breadth up 20% and height down 25%: 1.1 × 1.2 × 0.75 = 0.99, so the volume falls 1%.

Cylinder and cone. Volume = $\pi r^2 h$ (a cone has an extra $\frac{1}{3}$, which changes nothing). Square the radius multiplier and multiply by the height multiplier. The curved surface area of a cylinder is $2\pi rh$, so there you just multiply the two.

• Radius up 10%, height down 20%: volume 1.21 × 0.8 = 0.968, a fall of 3.2%; curved surface 1.1 × 0.8 = 0.88, a fall of 12%.
• Radius down 20%, height up 20%: volume 0.64 × 1.2 = 0.768, a fall of 23.2%.

Keeping the volume the same. If a cylinder's radius goes up 25%, $r^2$ becomes $\frac{25}{16}$ times, so the height must become $\frac{16}{25}$ times, a fall of 36%. If the radius goes down 20%, $r^2$ becomes $\frac{16}{25}$ times and the height must go up by $\frac{9}{16}$, which is 56.25%.

Going backwards. If the volume of a cube goes up 33.1%, the edge multiplier is $\sqrt[3]{1.331} = 1.1$, so the edge went up 10%. If a sphere's volume becomes 8 times, its radius doubles and its surface area becomes 4 times, a rise of 300%.

[Tier 2] One dimension balancing the other two. "The length of a box goes up 50% and its breadth goes down 20%. By what % must the height change to keep the volume the same?" Length × breadth becomes 1.5 × 0.8 = 1.2 = $\frac{6}{5}$ times, so the height must become $\frac{5}{6}$ times, a fall of $16\frac{2}{3}\%$.

Speed and Time, Workers and Days, and Fractions

Speed and time. For a fixed distance, speed × time stays the same, so the product rule gives the new time.

• Speed up 25% ($\frac{1}{4}$) means time down $\frac{1}{5}$, which is 20%.
• Speed down to 75% of the usual (a fall of $\frac{1}{4}$) means time up $\frac{1}{3}$.

The usual twist gives you the minutes saved or lost. "A car goes 25% faster than usual and arrives 10 minutes early. Find its usual time." The time falls by $\frac{1}{5}$ of the usual time, and that is 10 minutes, so the usual time is 50 minutes. "A man walks at $\frac{3}{4}$ of his usual speed and is 20 minutes late." His time goes up by $\frac{1}{3}$ of the usual, so the usual time is 60 minutes and today he took 80.

Workers and days. For a fixed job, workers × days stays the same. If the number of workers goes up 25%, the job takes 20% fewer days. If 20% of the workers leave, the job takes 25% more days.

A fraction with its top and bottom changed. A fraction is top ÷ bottom, so divide the multipliers.

"The numerator of a fraction is increased by 20% and the denominator is decreased by 25%. The new fraction is $\frac{8}{15}$. Find the original."

The fraction gets multiplied by $\frac{1.2}{0.75} = \frac{6}{5} \times \frac{4}{3} = \frac{8}{5}$. So the original is $\frac{8}{15} \times \frac{5}{8} = \frac{1}{3}$.

Check it forwards: $\frac{1 \times 1.2}{3 \times 0.75} = \frac{1.2}{2.25} = \frac{8}{15}$.

Products and quotients of two numbers.

• One number goes up 30% and another goes down 30%: their product becomes 1.3 × 0.7 = 0.91 times, a fall of 9%.
• x goes up 20% and y goes down 20%: $\frac{x}{y}$ becomes $\frac{1.2}{0.8} = 1.5$ times, a rise of 50%.
• If $\frac{x}{y}$ must stay the same and x goes up 15%, y must go up 15% too.

Price per item is a quotient too. If a packet's price goes up 10% and its weight goes up 25%, the price per gram becomes $\frac{1.1}{1.25} = 0.88$ times, which is 12% cheaper per gram.

Solved Examples

Question 1

The price of onions goes up by 60%. By what percentage should a family cut its onion consumption so that its spending on onions does not change?

Answer. 60% is $\frac{3}{5}$, so the price becomes $\frac{8}{5}$ times. To keep the spending the same, consumption must become $\frac{5}{8}$ times, which is a fall of $\frac{3}{8}$. As a percentage, $\frac{3}{8}$ = 37.5%. The formula gives the same: $\frac{60}{160} \times 100 = 37.5$.

Question 2

The price of petrol goes up by 30%, but a family lets its petrol bill go up by only 17%. By what percentage must it reduce its petrol use?

Answer. Spending = price × consumption, so consumption = spending ÷ price. The new consumption is $\frac{1.17}{1.30} = 0.9$ times the old one, so the family must cut its use by 10%. Just subtracting 30 − 17 = 13% doesn't work, because consumption is spending divided by price.

Question 3

Kiran's salary goes up by 26%, and the prices of everything she buys go up by 20%. What happens to her purchasing power? (a) rises 6% (b) rises 5% (c) falls 5% (d) rises 4.8%

Answer. What she can buy is salary ÷ prices: $\frac{1.26}{1.20} = 1.05$. So she can buy 5% more than before. Simply subtracting 20 from 26 would give option (a). The answer is (b) rises 5%.

Question 4

A fall of 15% in the price of sugar lets a shopkeeper buy 6 kg more for ₹1,020. Find the original price of sugar per kg.

Answer. The 15% he saves on ₹1,020 is 0.15 × 1020 = ₹153, and that buys the extra 6 kg at the new price. So the new price is 153 ÷ 6 = ₹25.50 a kg. That is 85% of the original price, so the original was 25.50 ÷ 0.85 = ₹30 a kg. Check: 1020 ÷ 30 = 34 kg before and 1020 ÷ 25.50 = 40 kg now, which is 6 kg more.

Question 5

After a 20% rise in the price of oranges, a man gets 8 fewer oranges for ₹240. How many oranges did he get for ₹240 before the rise?

Answer. To buy his old number of oranges at the new price he would need 20% more money, ₹48 extra, and ₹48 is the cost of the 8 oranges he now misses. So the new price is 48 ÷ 8 = ₹6 each and the old price was 6 ÷ 1.2 = ₹5. Before the rise, ₹240 bought 240 ÷ 5 = 48 oranges (now it buys 40).

A second way: a rise of $\frac{1}{5}$ in price means $\frac{1}{6}$ fewer oranges. $\frac{1}{6}$ of the old number is 8, so the old number was 48.

Question 6

The tax on a commodity is cut by 20%, and as a result its consumption goes up by 15%. Find the change in the tax collected from it.

Answer. Tax collected = tax per unit × units sold. The new amount is 0.8 × 1.15 = 0.92 times the old one, so the tax collected falls by 8%. With the formula: −20 + 15 − $\frac{20 \times 15}{100}$ = −20 + 15 − 3 = −8.

Question 7

Anil saves 30% of his income. His income goes up by 20% and his expenditure by 10%. By what percentage do his savings go up?

Answer. I take his income as 100, so his expenditure is 70 and his savings 30. After the change: income 120 and expenditure 70 × 1.1 = 77, so savings are 120 − 77 = 43. Savings went from 30 to 43, up 13 on 30, which is $\frac{13}{30}$ = $43\frac{1}{3}\%$.

Question 8

Priya saves 20% of her income. Her income rises by 15% and her savings rise by 35%. By what percentage does her expenditure rise?

Answer. Income 100 means expenditure 80 and savings 20. Next year: income 115 and savings 20 × 1.35 = 27, so expenditure = 115 − 27 = 88. It went from 80 to 88, up 8 on 80, which is 10%.

Question 9

Raju saves 25% of his monthly income. His income goes up by 20% and his expenditure by 10%, and his monthly savings go up by ₹2,500. Find his original monthly income.

Answer. With income 100: expenditure 75 and savings 25. After: income 120, expenditure 82.5 and savings 37.5. Savings went up by 12.5 units, and that is ₹2,500. So 1 unit = ₹200 and his income was 100 × 200 = ₹20,000.

Question 10

A man's income goes up by 30% in the second year and his savings go up by 75%. His total expenditure over the two years is double his expenditure in the first year. What percentage of his income did he save in the first year?

Answer. "Total expenditure is double the first year's" just means he spent the same in both years. So the whole extra income went into savings: 30% of income = 75% of savings. That gives savings ÷ income = 30 ÷ 75 = $\frac{2}{5}$, so he saved 40%. Check with income 100: savings 40 and expenditure 60. Next year the income is 130 and savings 70, so the expenditure is still 60.

Question 11

The ratio of Sunil's income to his expenditure is 7 : 5. His income goes up by 15% and his expenditure by 9%. By what percentage do his savings go up?

Answer. I take income 700 and expenditure 500, so savings are 200. New income = 805 and new expenditure = 545, so new savings = 260. Savings went up 60 on 200, which is 30%.

Question 12

A woman's income falls by 10% and her expenditure falls by 20%. Her savings go up by 30%. What percentage of her income was she saving at first?

Answer. Say her income is 100 and her savings S, so her expenditure is 100 − S. After the change, income is 90 and expenditure is 0.8(100 − S) = 80 − 0.8S. Savings = 90 − 80 + 0.8S = 10 + 0.8S, and this must equal 1.3S. So 0.5S = 10 and S = 20. She was saving 20% of her income.

Question 13

The incomes of A and B are in the ratio 5 : 4 and their expenditures are in the ratio 3 : 2. If each of them saves ₹8,000 a month, find A's monthly income.

Answer. Incomes 5x and 4x, expenditures 3y and 2y. Both save the same, so 5x − 3y = 4x − 2y, which gives x = y. Then A's savings are 5x − 3x = 2x = 8000, so x = 4000. A's income is 5 × 4000 = ₹20,000. (B earns ₹16,000; their expenditures are ₹12,000 and ₹8,000.)

Question 14

X and Y together earn ₹60,000 a month. X spends 85% of his salary and Y spends 75% of hers. If their savings are equal, find X's salary.

Answer. X saves 15% and Y saves 25%. Equal savings: 15% of X = 25% of Y, so X : Y = 25 : 15 = 5 : 3. X's salary is $\frac{5}{8}$ × 60,000 = ₹37,500. Check: 15% of 37,500 = ₹5,625 and 25% of 22,500 = ₹5,625.

Question 15

Meena's salary is 20% more than Reena's. Meena saves 30% of her salary and Reena saves 25% of hers. Together they save ₹14,640 a month. Find Reena's salary.

Answer. If Reena earns R, she saves 0.25R. Meena earns 1.2R and saves 0.3 × 1.2R = 0.36R. Together 0.61R = 14,640, so R = ₹24,000. (Meena earns ₹28,800 and saves ₹8,640; Reena saves ₹6,000.)

Question 16

The length of a rectangle is increased by 40%. (i) By what percentage must its breadth be reduced to keep the area the same? (ii) If the breadth is reduced by 25% instead, what happens to the area?

Answer. (i) 40% is $\frac{2}{5}$, so the length becomes $\frac{7}{5}$ times and the breadth must become $\frac{5}{7}$ times. That is a fall of $\frac{2}{7}$, which is $28\frac{4}{7}\%$. (ii) The area becomes 1.4 × 0.75 = 1.05 times, so it goes up by 5%.

Question 17

The area of a circle goes up by 69%. By what percentage did its radius go up, and what happened to its circumference?

Answer. Area goes with $r^2$, so the radius multiplier is $\sqrt{1.69} = 1.3$. The radius went up by 30%. The circumference $2\pi r$ grows in step with the radius, so it also went up by 30%, not 69%.

Question 18

The radius of a cylinder is increased by 20% and its height is decreased by 25%. Find the percentage change in (i) its volume and (ii) its curved surface area.

Answer. (i) Volume = $\pi r^2 h$: 1.2 × 1.2 × 0.75 = 1.08, so the volume goes up by 8%. (ii) Curved surface = $2\pi rh$: 1.2 × 0.75 = 0.9, so it goes down by 10%. The same change in the radius pushes the two in different directions because the volume uses the radius twice.

Question 19

Each edge of a cube is increased by 30%. Find the percentage increase in its total surface area and in its volume.

Answer. The edge becomes 1.3 times. Surface area goes with $a^2$: 1.3 × 1.3 = 1.69 times, an increase of 69%. Volume goes with $a^3$: $1.3^3$ = 2.197 times, an increase of 119.7%.

Question 20

A train runs 20% slower than usual and reaches 30 minutes late. Find its usual travel time.

Answer. The distance is fixed, so speed × time stays the same. Speed × $\frac{4}{5}$ means time × $\frac{5}{4}$, so the trip takes $\frac{1}{4}$ of the usual time extra. That $\frac{1}{4}$ is 30 minutes, so the usual time is 30 × 4 = 120 minutes, or 2 hours.

Question 21

The numerator of a fraction is increased by 25% and its denominator is decreased by 10%. The new fraction is $\frac{25}{27}$. Find the original fraction.

Answer. The fraction gets multiplied by $\frac{1.25}{0.9} = \frac{5}{4} \times \frac{10}{9} = \frac{25}{18}$. So the original fraction is $\frac{25}{27} \times \frac{18}{25} = \frac{18}{27}$ = $\frac{2}{3}$. Check: $\frac{2 \times 1.25}{3 \times 0.9} = \frac{2.5}{2.7} = \frac{25}{27}$.