Ratio, Proportion & Percentage

Snapshot

• Percentage, ratio, proportion, partnership and alligation are the engine room of the General Test: asked directly, and hidden inside profit, averages, data interpretation and work problems.
• Three ideas carry the whole topic: every percentage sits on a base, a ratio is a count of equal units, and a mixture is a balance around the mean.
• This page teaches the why behind each idea, a LaTeX formula sheet, the shortcuts and the exact situations where they fail, 14 fully worked examples, a traps table and a rapid-fire set with answers.
• Exam reality: +5 for a correct answer, −1 for a wrong one, so a clear head about which base a percentage sits on is worth more than raw speed.

Part 1 — The core idea: everything is a fraction of a base

Written and reviewed by Devendra Dubey, Mathematics & Commerce faculty at UniDrill (M.Sc. Mathematics, Osmania University · 10 years teaching). Prepared for the CUET UG General Test.

Percent means "per hundred", so 25% is just the fraction 1/4 wearing different clothes. Once you see percentages as fractions, most of the arithmetic disappears: 25% of 880 is a quarter of 880, which is 220, with no long multiplication.

Why the base matters

A percentage is meaningless until you say "of what". A rise from 80 to 100 is a rise of 20, and 20 out of the old value 80 is 25%. If you divide by the new value you get 20%, and CUET always puts that wrong answer among the options. The rule never changes: percentage change is measured on the starting value.

This single fact also explains the "flip" that surprises students. If A is 25% more than B, then A = 1.25 B, so B is A divided by 1.25, which is 0.8 A, a 20% fall. The gap (0.25 B) is the same in both statements but the base is different (B in one, A in the other).

Ratio is a count of equal units

A ratio 2 : 3 : 4 does not tell you any amount. It tells you the whole is cut into 2 + 3 + 4 = 9 equal units. Find what one unit is worth and every share is a single multiplication. This one move solves sharing, ages, mixtures and partnerships.

Proportion is a rule about what stays fixed

In direct proportion the ratio of the two quantities stays fixed (twice the workers, twice the output). In inverse proportion the product stays fixed (twice the workers, half the days). Decide which one before you write anything.

A mixture is a balance

When two things at different values are blended, the result lies between them, closer to the one you used more of. Alligation is simply the balance rule: the distances from the mean to each ingredient are in the inverse ratio of the quantities.

Pause and predict: a shopkeeper raises a price by 20% and then lowers the new price by 20%. Is the final price higher, lower, or the same as the original?

Check your answer: lower. The 20% fall is taken on the larger, raised price, so it removes more rupees than the rise added. Net change is $20 - 20 - \dfrac{20 \times 20}{100} = -4$, a 4% fall.

Fraction and percentage pairs to memorise

Part 2 — Formula sheet

Percentage and change:

$$\text{Percentage change} = \dfrac{\text{New} - \text{Old}}{\text{Old}} \times 100$$

$$\text{Two successive changes of } a\% \text{ and } b\% = \left(a + b + \dfrac{ab}{100}\right)\% \quad (\text{use a minus sign for a fall})$$

$$\text{Final value} = \text{Original} \times \left(1 \pm \dfrac{r_1}{100}\right)\left(1 \pm \dfrac{r_2}{100}\right)\cdots$$

Ratio and proportion:

$$a : b = c : d \iff ad = bc \qquad \text{Mean proportional of } a, c = \sqrt{ac}$$

Partnership and alligation:

$$\text{Profit share} \propto \text{Capital} \times \text{Time} \qquad \dfrac{\text{Quantity of cheaper}}{\text{Quantity of dearer}} = \dfrac{\text{Dearer} - \text{Mean}}{\text{Mean} - \text{Cheaper}}$$

$$\text{Pure quantity left after } n \text{ replacements} = V\left(1 - \dfrac{x}{V}\right)^n$$

Part 3 — Shortcut methods and when they fail

The multiplier habit. Turn each change into a multiplier: +12% is 1.12, −15% is 0.85, and then multiply. It never fails, and it handles three or four changes without a special formula. Use the $a + b + ab/100$ shortcut only when there are exactly two.

Direction check. After every answer ask whether it moved the right way. A price after a discount must be smaller; the original before a discount must be larger. This 3-second check catches the most common sign slips.

Part 4 — Worked examples: percentage

Worked example 1: successive change. A number is increased by 25% and then decreased by 20%. What is the net change?

Common wrong step: adding 25 and subtracting 20 to get +5%. Each change has a different base, so the rise and fall do not simply add.

Worked example 2: reverse percentage. After a 12% hike a salary is ₹28,000. What was it before?

The final value is 112% of the original, so original = $\dfrac{28000}{1.12} = 25000$, that is ₹25,000. Check: 12% of 25,000 is 3,000 and 25,000 + 3,000 = 28,000.

Common wrong step: subtracting 12% of 28,000 (₹3,360) to get ₹24,640. The 12% was taken on the old salary, not the new one.

Worked example 3: two years of growth. A town has 40,000 people. Its population grows 10% in the first year and 20% in the second. What is it after two years?

Use multipliers: $40000 \times 1.1 \times 1.2 = 40000 \times 1.32 = 52{,}800$. The overall rise is 32%, which is $10 + 20 + \dfrac{10 \times 20}{100}$.

Worked example 4: the flip. A's salary is 20% less than B's. By what percentage is B's salary more than A's?

Let B = 100, so A = 80. B exceeds A by 20, and the base is now A = 80, so the rise is $\dfrac{20}{80} \times 100 = 25\%$. The shortcut: "20% less" means a fall of $\dfrac{1}{5}$, so the reverse rise is $\dfrac{1}{5-1} = \dfrac{1}{4} = 25\%$.

Worked example 5: election. In an election 10% of the 20,000 votes cast are invalid. Candidate A gets 55% of the valid votes and the only other candidate B gets the rest. By how many votes does A win?

Common wrong step: taking 55% of 20,000. The percentage is of valid votes, not of all votes cast.

Worked example 6: constant expenditure. The price of sugar rises by 25%. By what percentage must a family cut consumption so that its spending stays the same?

Use $\dfrac{r}{100 + r} \times 100 = \dfrac{25}{125} \times 100 = 20\%$. Check with numbers: 10 kg at ₹40 costs ₹400. At ₹50 the same ₹400 buys 8 kg, which is a cut of 2 kg out of 10, or 20%. The cut is not 25%: after the rise the same money buys $\dfrac{1}{1.25} = 80\%$ of the old quantity, so the fall is 20%.

Part 5 — Worked examples: ratio, proportion, partnership, mixtures

Worked example 7: ratio changes after an addition. Two numbers are in the ratio 3 : 5. When 10 is added to each, the ratio becomes 5 : 7. Find the numbers.

Check: 25 : 35 reduces to 5 : 7.

Worked example 8: joining two ratios. A : B = 2 : 3 and B : C = 4 : 5. Find A : B : C, and the three shares of ₹700.

Make B the same in both: scale the first by 4 and the second by 3. Then A : B = 8 : 12 and B : C = 12 : 15, so A : B : C = 8 : 12 : 15. The total is 35 units, one unit is ₹20, so the shares are ₹160, ₹240, ₹300.

Worked example 9: inverse proportion. 15 workers finish a job in 8 days. How long will 20 workers take?

More workers means fewer days, so the product is fixed: $15 \times 8 = 20 \times t$, giving $t = 6$ days.

Common wrong step: setting $15/8 = 20/t$ as if it were direct proportion, which gives 10.67 days, an answer that is larger even though there are more workers.

Worked example 10: partnership with different times. A invests ₹12,000 for 8 months and B invests ₹9,000 for 12 months. They earn ₹5,100. How is it shared?

Capital × time: A is $12000 \times 8 = 96000$ and B is $9000 \times 12 = 108000$. The ratio is 96 : 108 = 8 : 9, which is 17 parts. One part is $5100 \div 17 = 300$, that is ₹300. So A gets ₹2,400 and B gets ₹2,700.

Worked example 11: a partner who joins late. A starts a business with ₹20,000. After 4 months B joins with ₹30,000. At the end of the year the profit is ₹7,000. What does each get?

A's capital works for 12 months: $20000 \times 12 = 240000$. B's works for 8 months: $30000 \times 8 = 240000$. The ratio is 1 : 1, so each gets ₹3,500.

Worked example 12: alligation on price. In what ratio should rice at ₹30 per kg be mixed with rice at ₹40 per kg so that the mixture can be sold at ₹36 per kg with no gain or loss?

Note the cross: the quantity of the cheaper goes with the distance of the dearer from the mean. As a sense check, 36 is closer to 40, so you need more of the dearer rice (3 parts versus 2).

Worked example 13: adding water. A 45 L mixture has milk and water in the ratio 4 : 1. How much water must be added to make the ratio 3 : 2?

Milk = $\dfrac{4}{5} \times 45 = 36$ L and water = 9 L. Milk does not change, so 36 L must be 3 parts of the new ratio, giving one part = 12 L and water = 24 L. Water to add: $24 - 9 = $ 15 L.

Worked example 14: repeated replacement. A vessel holds 40 L of pure milk. 4 L is removed and replaced with water, and the same is done once more. How much pure milk is left?

Each step keeps a fraction $1 - \dfrac{4}{40} = 0.9$ of the milk. After two steps: $40 \times 0.9 \times 0.9 = 32.4$ L.

Common wrong step: subtracting 4 L twice to get 32 L. In the second step the 4 L removed is a mixture that already contains some water, so it carries less than 4 L of milk.

Part 6 — Common traps

Part 7 — Rapid-fire practice with answers

Do these without a calculator, with the page covered, and check yourself afterwards.

Part 8 — A timed-practice routine

Run this routine for one week and the whole topic becomes mental maths.

Marking rule for the exam. With +5 and −1, a guess between four options has an expected value of 0.25 × 5 − 0.75 × 1 = +0.5, so a guess is positive in expectation, but a blind guess is a gamble. Eliminate at least one option with a direction check first. Skip a question that needs more than 90 seconds and come back to it.

Part 9 — Where this connects

Percentage is the base for Profit, Loss and Interest, where the cost price is the base and a discount works on the marked price. Weighted means and age equations on the Averages and Ages page are alligation in another form. Inverse proportion drives Time, Speed and Work, and nearly every chart question in Data Interpretation asks for a percentage change or a ratio. For the mathematics behind repeated growth and decay, see Sequences and Series, because compounding is a geometric progression.

One-line revision: turn percentages into fractions or multipliers, divide the change by the old value, cut a ratio into equal units, keep the product fixed for inverse proportion, and cross the distances in alligation.