Ratio and proportion: the one skill behind partnership, mixtures and ages

Ratio questions look like a dozen different topics — sharing money, mixing solutions, comparing ages, splitting profit — but they are one habit repeated: read the ratio as a count of equal parts, find what one part is worth, and scale. Do that and the algebra you were about to write disappears.

Read a ratio as parts, not fractions

A ratio 3 : 4 does not mean the fractions 3 and 4. It means the whole is cut into 7 equal parts, one quantity holds 3 of them and the other holds 4. So the first step in almost every ratio question is to add the parts and ask what one part is worth.

Split ₹4,200 between two people in 3 : 4: total 7 parts, one part is 4200 ÷ 7 = 600, so the shares are 1,800 and 2,400. You never wrote an equation, and the arithmetic was one division and two multiplications.

The same reading handles “A is one-third more than B”. One-third more means B is 3 parts and A is 4 parts, a ratio of 4 : 3 — convert the words to parts once and the rest is counting.

Scaling: the single multiplier

When a question gives a ratio and one real value, introduce one unknown for the size of a part. If two salaries are in 5 : 7 and the larger is ₹35,000, then 7 parts equal 35,000, one part is 5,000, and the smaller salary is 25,000. Writing the parts as 5x and 7x and solving 7x = 35000 is the same move; the point is that one multiplier x carries the whole question, so you are never juggling two unknowns.

This is also why a ratio can be scaled freely: 2 : 3 is the same as 4 : 6 or 10 : 15. Reduce before you compute, and keep the numbers small.

Partnership: profit splits as capital times time

Partnership is ratio with one extra factor. Profit is divided in the ratio of each partner’s capital multiplied by the months it stayed in the business.

• A invests ₹12,000 for 12 months, B invests ₹18,000 for 8 months. Their profit ratio is 12000 × 12 : 18000 × 8 = 144000 : 144000 = 1 : 1, so they split equally despite investing different amounts — the time cancels the capital gap.
• A “sleeping” partner who only puts in money is treated no differently; only their capital and time enter the ratio. A working partner who takes a salary has it removed from the profit first, and the remainder is split by capital × time.

Reduce the capital × time products to the smallest whole-number ratio before splitting, exactly as you would any ratio.

Mixtures and alligation

When two ingredients at different prices or strengths are combined, the ratio in which they are mixed to reach a target follows the rule of alligation: the ratio of quantities is the inverse of the distances of each price from the mean. Mix rice at ₹40/kg and ₹60/kg to average ₹45/kg — the distances are 5 and 15, so the cheaper to dearer ratio is 15 : 5 = 3 : 1. The full treatment, including replacement problems where a fraction is removed and topped up, sits with mensuration, mixtures and counting.

Ages: two ratios, one unknown

Ages problems hand you a ratio at one time and a different ratio at another. Write the present ages as parts of one unknown, add or subtract the years, and set the second ratio equal.

Present ages in 4 : 5, and after 6 years the ratio becomes 5 : 6. Take present ages as 4x and 5x. Then (4x + 6) : (5x + 6) = 5 : 6. Cross-multiplying gives 6(4x+6) = 5(5x+6), so 24x + 36 = 25x + 30, giving x = 6 and present ages 24 and 30. One unknown, one equation — the ratio did the setup for you.

Where the marks leak

Treating a ratio as absolute numbers. 3 : 4 is 3 and 4 parts, not ₹3 and ₹4. Always find the value of one part first.

Forgetting the time factor in partnership. Equal capital does not mean equal profit if the money stayed invested for different periods.

Inverting alligation the wrong way. The larger quantity sits with the price closer to the mean. Sanity-check: if the average is nearer the cheaper price, you used more of the cheaper thing.

Not reducing. Carrying 144000 : 144000 instead of 1 : 1 is how arithmetic slips creep in.

Practice that transfers

Ratio is the shared spine of several money topics, so practise it next to them rather than alone. Pair it with percentage, profit and averages, because a percentage is just a ratio out of a hundred, and with the data interpretation approach, where almost every pie chart and table question is a ratio in disguise. Then take a timed full-length mock test and watch how many “equation” questions were ratios all along.