Ratio and Proportion
Ratio and Proportion
Ratio is the quiet backbone of the arithmetic block. It gives 1 to 2 questions on its own and sits inside partnership, mixture, percentage and speed problems as well. In Part-III the questions go past simple sharing into compound ratio and alligation, so prepare it properly.
Basic Rules
A ratio compares two quantities of the same kind and same unit. It has no unit of its own. Multiplying or dividing both terms by the same non-zero number leaves it unchanged.
| Term | Rule | Example |
|---|---|---|
| Equivalent ratio | Multiply or divide both terms | 2:3 = 6:9 |
| Proportion | a:b = c:d means ad = bc | 2:3 = 6:9 |
| Mean proportional | √(a × b) | Between 4 and 9 → 6 |
| Third proportional | b² ÷ a | Of 4 and 6 → 9 |
| Fourth proportional | (b × c) ÷ a | Of 2, 3, 4 → 6 |
| Compound ratio | Multiply term by term | 2:3 and 4:5 → 8:15 |
The k-substitution is the working method: write a ratio of 3 : 4 : 5 as 3k, 4k and 5k, then form one equation. It turns a multi-quantity problem into a single unknown.
Dividing in a Ratio
Divide Rs 5,400 among three people in the ratio 2 : 3 : 4. Total parts = 9, so one part is 600. The shares are 1200, 1800 and 2400.
✗ Dividing 5,400 in the ratio 2:3:4 gives 2700, 1800, 900 | ✓ Add the parts first: 9 parts, one part = 600, so 1200, 1800, 2400
Partnership
Profit in a partnership is shared in the ratio of capital multiplied by time, not capital alone.
Profit share ratio = (C₁ × T₁) : (C₂ × T₂)
If A invests Rs 20,000 for 12 months and B invests Rs 30,000 for 6 months, the ratio is 240000 : 180000, which is 4 : 3. Candidates who compare only the capitals get 2 : 3, and that wrong value will be among the options.
Mixture and Alligation
Alligation finds the ratio in which two ingredients of different values must be mixed to reach a given mean value.
Cheaper quantity : Dearer quantity = (Dearer − Mean) : (Mean − Cheaper)
To get rice worth Rs 30 per kg by mixing rice at Rs 24 and Rs 36: the ratio is (36 − 30) : (30 − 24) = 6 : 6 = 1 : 1.
The rule to remember is that the answer ratio comes out crossed — the cheaper quantity is paired with the dearer difference. Reversing it is the standard error and produces a plausible-looking wrong answer.
अनुपात का कोई मात्रक नहीं होता; दोनों राशियों की इकाई समान होनी चाहिए।
TRE pointer: Part-III arithmetic is examined five times as heavily as in Part-II, so expect the harder variants here. The capital times time rule in partnership and the crossed ratio in alligation are the two items that separate prepared candidates. Both have a ready-made wrong option built from the obvious mistake. With five choices and −1/3 for each wrong answer, write the k-substitution down rather than working it mentally.
60-Second Recap
- A ratio compares like quantities in the same unit and has no unit itself.
- Mean proportional is √(ab); third proportional is b²/a; fourth is bc/a.
- Use k-substitution: 3 : 4 : 5 becomes 3k, 4k, 5k.
- To divide an amount, add the parts first and find the value of one part.
- Partnership profit divides as capital × time, not capital alone.
- Alligation: cheaper to dearer = (dearer − mean) : (mean − cheaper), crossed.
Frequently Asked Questions
How is profit shared in a partnership?
In the ratio of each partner's capital multiplied by the time for which it was invested. Comparing capitals alone gives the wrong ratio whenever the periods differ, and that wrong value is normally offered as one of the options in the question.
What is the mean proportional between two numbers?
It is the square root of their product. Between four and nine it is six. Do not confuse it with the third proportional, which is b squared divided by a, or the fourth proportional, which is b times c divided by a. All three are asked.
How does the alligation rule work?
The ratio of the cheaper to the dearer quantity equals the dearer price minus the mean price, to the mean price minus the cheaper price. The differences come out crossed, which is the part candidates reverse. It is used for mixtures of two ingredients at different rates.
Why turn a ratio into k?
Because it replaces several unknowns with one. A ratio of three to four to five becomes 3k, 4k and 5k, so any additional condition gives a single linear equation in k. Solve for k and multiply back to get each quantity.