Ratio and Proportion for CUET General Aptitude

Q: What is a ratio and how do you simplify it?

A ratio a: b means a ÷ b. To simplify it, divide both terms by their HCF, exactly as you would reduce a fraction: 84: 126 has HCF 42, so it is 2: 3.

Q: How do you compare two ratios quickly?

Cross-multiply. For a: b and c: d, a: b is larger when a × d is greater than b × c. With four or more ratios, decimals are usually faster.

Q: What is a proportion, and what are the fourth, third and mean proportionals?

Four numbers are in proportion, written a: b:: c: d, when a: b = c: d. The key fact is that the product of the extremes equals the product of the means: a × d = b × c.

Q: What is the difference between direct and inverse variation?

In direct variation the ratio is constant: y = kx, and when one doubles, so does the other. In inverse variation the product is constant: y = k/x, and when one doubles, the other halves.

About 10 min to learn · 67 questions

Ratio and proportion sit under many CUET aptitude questions: sharing money, changing a ratio, coins in a bag, map scales, workers and days, and partners splitting a profit. The ideas are simple, but the questions are built around a few predictable slips: mixed units, the wrong order, direct taken for inverse, and time ignored in a partnership. This lesson gives the ideas, the fastest methods and the traps, with worked examples.

Syllabus area: General Mental Ability, Numerical Ability

On this page · 9 sections

01The core idea
02Ratios
03Proportion
04Variation
05Partnership
06Exam shortcuts
07Common mistakes
08Quick revision
09Check your understanding

The core idea

A ratio compares two quantities of the same kind by division, and a proportion says two ratios are equal. Almost every question in this chapter is fastest in "parts": write each quantity as so many parts, find what one part is worth, and read off everything else.

Before any arithmetic, check two things: that both quantities are in the same unit, and which way round the ratio goes. "Girls to boys" and "boys to girls" are different ratios, and both are usually among the options.

Ratios

What is a ratio and how do you simplify it?

A ratio $a : b$ means $a \div b$ . To simplify it, divide both terms by their HCF, exactly as you would reduce a fraction: $84 : 126$ has HCF 42, so it is $2 : 3$ .

Put both quantities in the same unit first: 40 minutes to 2 hours is $40 : 120 = 1 : 3$ , not $40 : 2$ .
A ratio has no unit, so $3 : 5$ of rupees and $3 : 5$ of kilograms are the same ratio.
With fractions or decimals, multiply every term by the LCM of the denominators or by a power of 10: $\tfrac{1}{3} : \tfrac{1}{4} : \tfrac{1}{6}$ times 12 is $4 : 3 : 2$ , and $0.4 : 1.5$ times 10 is $4 : 15$ .

Watch out

Fractions inside a ratio do not keep their denominators' order: $\tfrac{1}{2} : \tfrac{1}{3} : \tfrac{1}{4}$ becomes $6 : 4 : 3$ , not $2 : 3 : 4$ .

How do you compare two ratios quickly?

Cross-multiply. For $a : b$ and $c : d$ , $a : b$ is larger when $a \times d$ is greater than $b \times c$ . With four or more ratios, decimals are usually faster.

Example 1 · Comparing ratios

Which is larger, $5 : 9$ or $4 : 7$ ?

Cross-multiply: $5 \times 7 = 35$ and $9 \times 4 = 36$ .
35 is less than 36, so $5 : 9$ is the smaller ratio.

Answer

4 : 7

As decimals: $5 \div 9 \approx 0.556$ and $4 \div 7 \approx 0.571$ .

How do you divide an amount in a given ratio?

Add the ratio terms to get the number of parts, find the value of one part, and multiply. If an amount $T$ is shared in $a : b : c$ , one part is $\tfrac{T}{a + b + c}$ .

Example 2 · Sharing in a ratio

₹4,500 is shared among three cousins in the ratio $3 : 5 : 7$ . What does the middle cousin get?

Parts: $3 + 5 + 7 = 15$ .
One part: $4{,}500 \div 15 = 300$ .
Middle share: $5 \times 300 = 1{,}500$ .

Answer

₹1,500

Check: $900 + 1{,}500 + 2{,}100 = 4{,}500$ .

Exam shortcut

Parts work backwards too. If boys to girls is $7 : 4$ and there are 28 girls, 4 parts are 28, one part is 7, and there are 49 boys.

How do you combine ratios like a : b and b : c?

Make the common term equal, then read off the combined ratio. If $a : b = 2 : 3$ and $b : c = 5 : 4$ , change $b$ to 15, the LCM of 3 and 5: $a : b = 10 : 15$ and $b : c = 15 : 12$ , so $a : b : c = 10 : 15 : 12$ .

Exam shortcut

If you only need $a : c$ , multiply the fractions: $\tfrac{a}{c} = \tfrac{a}{b} \times \tfrac{b}{c} = \tfrac{2}{3} \times \tfrac{5}{4} = \tfrac{5}{6}$ . The compound ratio of two ratios multiplies them term by term: $3 : 4$ and $8 : 9$ give $24 : 36 = 2 : 3$ .

How do you solve "add a number to change the ratio" questions?

Write the quantities as multiples of one unknown, apply the change, and set the result equal to the new ratio. Anything the question does not mention stays the same.

Example 3 · Changing a ratio

Two numbers are in the ratio $5 : 8$ . If 6 is added to each, the ratio becomes $2 : 3$ . Find the numbers.

Let them be $5x$ and $8x$ . Then $\tfrac{5x + 6}{8x + 6} = \tfrac{2}{3}$ .
Cross-multiply: $15x + 18 = 16x + 12$ , so $x = 6$ .

Answer

30 and 48

Check: $36 : 54 = 2 : 3$ .

Ages and mixtures

"4 years ago" subtracts 4 from both ages, because everyone ages equally.
In a mixture, the ingredient you do not add stays fixed. 30 litres of milk and water in $4 : 1$ has 24 L of milk and 6 L of water. For $3 : 1$ , the water must become $24 \div 3 = 8$ L, so add 2 L. The answer is the amount added, not the final amount.

How do you solve coins-in-a-bag questions?

Keep the ratio of the number of coins apart from the ratio of their values. If the ratio is by number, multiply each term by the coin's face value to get the value of one set. If it is by value, divide each share of the money by the face value to get the count.

Example 4 · Coins by number

A bag has ₹1, ₹2 and ₹10 coins in the ratio $4 : 3 : 1$ by number, worth ₹200 in all. How many ₹2 coins are there?

One set is worth $4 \times 1 + 3 \times 2 + 1 \times 10 = 20$ rupees.
Number of sets: $200 \div 20 = 10$ .
₹2 coins: $3 \times 10 = 30$ .

Answer

Watch out

If the values, not the numbers, are in $5 : 4 : 1$ , split ₹200 into ₹100, ₹80 and ₹20 and divide by the face values: 100 coins of ₹1, 40 of ₹2 and 2 of ₹10. And convert paise to rupees before adding: a 50-paise coin is ₹0.5.

Proportion

What is a proportion, and what are the fourth, third and mean proportionals?

Four numbers are in proportion, written $a : b :: c : d$ , when $a : b = c : d$ . The key fact is that the product of the extremes equals the product of the means: $a \times d = b \times c$ .

Term	Condition	Formula
Fourth proportional to $a, b, c$	$a : b = c : x$	$x = \tfrac{bc}{a}$
Third proportional to $a, b$	$a : b = b : x$	$x = \tfrac{b^2}{a}$
Mean proportional between $a, b$	$a : x = x : b$	$x = \sqrt{ab}$

Example 5 · Proportionals

Find the fourth proportional to 8, 12 and 18, and the mean proportional between 8 and 18.

Fourth proportional: $12 \times 18 \div 8 = 27$ .
Mean proportional: $\sqrt{8 \times 18} = \sqrt{144} = 12$ .

Answer

27 and 12

Watch out

The mean proportional is not the average. The average of 8 and 18 is 13, a common wrong option.

Exam shortcut

Three numbers are in continued proportion when $a : b = b : c$ , so $b^2 = ac$ . Then $(a + b) : (b + c) = a : b$ and $a : c = a^2 : b^2$ .

What is the unitary method?

Find the value of one unit first, then scale up. If 15 pens cost ₹255, one pen costs ₹17, so 22 pens cost ₹374. It is the safest way through any direct proportion, and it never adds where it should multiply.

On a map of scale $1 : 50{,}000$ , 1 cm stands for 50,000 cm, which is 0.5 km, so 7 cm stands for 3.5 km.
With a limiting ingredient, work out what each ingredient allows and take the smaller. A recipe with rice and dal in $3 : 1$ can use at most 1.2 kg of rice if you have only 400 g of dal, however much rice you own.

Variation

What is the difference between direct and inverse variation?

In direct variation the ratio is constant: $y = kx$ , and when one doubles, so does the other. In inverse variation the product is constant: $y = \tfrac{k}{x}$ , and when one doubles, the other halves.

Direct: ratio constant	Inverse: product constant
Cost and quantity bought	Workers and days for a fixed job
Distance and petrol used	Speed and time for a fixed distance
Wages and days worked	People and days a food stock lasts

Example 6 · Inverse variation

The time to cook rice for a function varies inversely with the number of cookers. Four cookers take 45 minutes. How long do 6 cookers take?

The product is fixed: $4 \times 45 = 180$ cooker-minutes.
Six cookers: $180 \div 6 = 30$ minutes.

Answer

30 minutes

Exam shortcut

Ask one question: if this goes up, does the other go up or down? Up means direct, down means inverse. Variation can also be with a square: a plate with 1.5 times the diameter has $1.5^2 = 2.25$ times the area. For joint variation such as $y \propto \tfrac{x}{z^2}$ , write $y = \tfrac{kx}{z^2}$ , find $k$ , then substitute.

How do you solve men-days-food and work questions?

Treat the job, or the food stock, as a fixed total of person-days, then divide by the new number of people. Men times days times hours, divided by the work, stays constant. If something changes partway through, subtract the part already used first.

Work and people

\frac{M_1 \times D_1 \times H_1}{W_1} = \frac{M_2 \times D_2 \times H_2}{W_2}

Example 7 · Rations after people leave

A relief camp has rations for 300 people for 40 days. After 10 days, 60 people leave. How long will the rest of the rations last?

Rations left: $300 \times 30 = 9{,}000$ person-days.
People now: $300 - 60 = 240$ .
Days: $9{,}000 \div 240 = 37.5$ .

Answer

37.5 days

Key idea

With both size and time, use the whole formula. Ten workers who build 50 m of drain in 6 days need $10 \times 6 \div 50 = 1.2$ worker-days a metre, so 90 m in 9 days needs $90 \times 1.2 \div 9 = 12$ workers.

Partnership

How is profit shared in a partnership?

Profit is shared in the ratio of capital times time for each partner. If everyone stays for the same time, use the capital ratio; if the capital is the same, use the time ratio.

A partner who joins after $m$ months has money in for $12 - m$ months.
A partner who leaves or adds money partway through has two blocks: capital times months for each.
A working partner's salary is paid out first, and only the rest is shared. Their total is the salary plus their share.

Example 8 · A late partner

Rekha starts a shop with ₹48,000. After 3 months Dinesh joins with ₹64,000. The year's profit is ₹42,000. How is it shared?

Rekha: $48{,}000 \times 12 = 5{,}76{,}000$ .
Dinesh: $64{,}000 \times 9 = 5{,}76{,}000$ .
The ratio is $1 : 1$ , so each gets half.

Answer

₹21,000 each

Exam shortcut

To work backwards, divide: capital is proportional to profit divided by time. If profits are $3 : 4$ and times $3 : 2$ , the capitals are $\tfrac{3}{3} : \tfrac{4}{2} = 1 : 2$ .

Exam shortcuts

Work in parts. Find one part, then read off every share.
Check with the total. The shares must add back to the amount.
Equal products, equal shares. In a partnership, compute each capital times time early.
Person-days. One multiplication and one division solve most men-days-food questions.
Direction first. Decide direct or inverse before you calculate.

Common mistakes

Mixed units. Minutes against hours, paise against rupees, map centimetres against kilometres.
The wrong order. "A to B" is not "B to A".
Answering the wrong quantity. Final water instead of water added, present age instead of age in 4 years.
Adding instead of multiplying. Proportion scales by a factor.
Direct for inverse. More workers means fewer days.
Forgetting the part already done in work and food questions.
Ignoring time in a partnership.
Forgetting the salary of a working partner.
The average as the mean proportional.
Fractions in a ratio. $\tfrac{1}{2} : \tfrac{1}{3} : \tfrac{1}{4}$ is $6 : 4 : 3$ .

Quick revision

Ratios

Simplify by the HCF, same units first
Compare by cross-multiplying
One part $= \tfrac{T}{a + b + c}$
Combine: make the common term equal

Proportion

$a : b :: c : d$ means $ad = bc$
Fourth $\tfrac{bc}{a}$ , third $\tfrac{b^2}{a}$ , mean $\sqrt{ab}$
Continued: $b^2 = ac$

Variation and work

Direct: ratio constant. Inverse: product constant
$\tfrac{M_1 D_1 H_1}{W_1} = \tfrac{M_2 D_2 H_2}{W_2}$
Subtract what was used before a change

Partnership

Share in capital × time
Joins after $m$ months: $12 - m$ months
Salary first, then share the rest

Mixing liquids at different strengths is worked in mixtures and alligation, rates of work in time and work, and age ratios in ages and number problems.

Next · 08Quick revision

Check your understanding

67 questions, grouped by type, easy to hard. Pick an answer and check it. The worked solution opens when you ask for it.

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Ratios and sharing

25 questions

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Ratios and sharing

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Proportion

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Direct and inverse variation

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