QUANT SHORTCUT | 21 OCTOBER 2026
If 4A = 5B = 6C, what is A : B : C? The answer is 15 : 12 : 10, and Amit Sir gets there in three multiplications with what he calls the cover trick: cover the letter you want, multiply the two numbers that are left, and simplify at the end. This page explains why the trick works, when to use it, and how to check it with the LCM method.
The trap: copying the coefficients
The most common slip is to write the coefficients straight across and answer 4 : 5 : 6. That is wrong. When 4A, 5B and 6C are all equal, the letter with the biggest coefficient must be the smallest quantity: six C’s balance four A’s, so C is smaller than A. The coefficients and the shares run in opposite directions, which is why the copied ratio fails.
The cover trick, step by step
- Find A: cover A (the 4). The numbers left are 5 and 6, so A’s share is 5 × 6 = 30.
- Find B: cover B (the 5). A and C are left, so B’s share is 4 × 6 = 24.
- Find C: cover C (the 6). The numbers left are 4 and 5, so C’s share is 4 × 5 = 20.
- Simplify: 30 : 24 : 20, divided by their HCF 2, gives A : B : C = 15 : 12 : 10.
Why it works
Call the common value k, so 4A = 5B = 6C = k. Then A = k/4, B = k/5 and C = k/6, and A : B : C = 1/4 : 1/5 : 1/6. Multiplying every term by the same number does not change a ratio, so multiply each by 4 × 5 × 6 = 120. The 1/4 becomes 5 × 6 = 30, the 1/5 becomes 4 × 6 = 24 and the 1/6 becomes 4 × 5 = 20. “Cover and multiply the rest” is exactly that step done in your head.
Check it with the LCM method
The textbook route sets the common value to the LCM of the coefficients. LCM(4, 5, 6) = 60, so A = 60 ÷ 4 = 15, B = 60 ÷ 5 = 12 and C = 60 ÷ 6 = 10: again 15 : 12 : 10. Both methods agree. The cover trick simply skips the LCM, and the final division by the HCF does the tidying.
More worked examples
- 3A = 4B = 6C: 4 × 6 = 24, 3 × 6 = 18, 3 × 4 = 12, so 24 : 18 : 12 = 4 : 3 : 2.
- 2A = 5B = 10C: 5 × 10 = 50, 2 × 10 = 20, 2 × 5 = 10, so 50 : 20 : 10 = 5 : 2 : 1.
- Two letters, 3A = 2B: cover A and 2 is left; cover B and 3 is left, so A : B = 2 : 3.
- Using the ratio: if 4A = 5B = 6C and A + B + C = 74, the parts 15 + 12 + 10 = 37 make 74, so one part = 2 and A = 30, B = 24, C = 20.
If pA = qB = rC, then A : B : C = qr : pr : pq (each share is the product of the other two coefficients), reduced by the HCF. Equivalently, A : B : C = 1/p : 1/q : 1/r.
- Quantitative Techniques sets in CLAT often hide a ratio inside a passage (“equal amounts spent”, “equal work done”): the cover trick turns it into A : B : C quickly.
- Partnership and wages questions use the same form: if 4 men’s pay equals 5 women’s pay equals 6 children’s pay, their daily rates are 15 : 12 : 10.
- Sanity check every answer: the biggest coefficient must get the smallest share.
- Always reduce at the end: 30 : 24 : 20 and 15 : 12 : 10 are the same ratio, and options are usually given in lowest terms.
| Fact | Detail |
|---|---|
| Question | 4A = 5B = 6C, find A : B : C |
| Cover A | 5 × 6 = 30 |
| Cover B | 4 × 6 = 24 |
| Cover C | 4 × 5 = 20 |
| Simplify | 30 : 24 : 20 ÷ 2 = 15 : 12 : 10 |
| LCM check | LCM(4, 5, 6) = 60 → 15, 12, 10 |
| Wrong answer to avoid | 4 : 5 : 6 (copying the coefficients) |
Hide the letter you are looking for; whatever is left, multiply. Big coefficient, small share.
Sources
Amit Sir’s CLAT Gurukul class recording (the 2-Second Trick series, episode 5); the algebra above is standard ratio arithmetic.
Free in the CLAT Gurukul app
These notes (as a PDF lesson) and the 10-question CBT are in the app under Current Affairs → 21 Oct 2026. Get the app on Google Play or open app.clatgurukul.com.
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