Aptitude · Ratio and Proportion · Model 1
When two multiples are equal, the numbers swap
The whole of Model 1 is one move. Given 3A = 5B, the ratio A : B is 5 : 3 — each letter takes the other one’s coefficient. Extend it to three equal multiples and the same idea gives you a three-term ratio with no algebra at all.
Set the coefficients and watch the ratio fall out →01 The idea
One equation, read backwards
“Three times A’s load is exactly equal to five times B’s load.” That is 3A = 5B, and the ratio it hides is A : B = 5 : 3. Divide both sides by 3 and by B and you get A/B = 5/3, which is the ratio. In an exam you skip the algebra: the number next to B moves to A, the number next to A moves to B.
The direction feels wrong the first time. A has the smaller coefficient and the larger share. That is exactly right, because a smaller multiplier reaching the same total needs a bigger quantity to work on. Use it as your check — whichever letter carries the smaller coefficient is the bigger quantity.
The same reasoning handles percentages and fractions without any new rule. “40% of S equals 60% of P” is 40S = 60P once the per-cent signs cancel, so S : P = 60 : 40 = 3 : 2. And “K’s payment is 2/7 of J’s” is even more direct: the first name named takes the top number and the second takes the bottom, so K : J = 2 : 7.
Three equal multiples are the hard version, and they are worth learning properly because they are set often. For 2A = 3B = 4C, call the common value k. Then A = k/2, B = k/3 and C = k/4. Choosing k as the LCM of 2, 3 and 4 clears every fraction at once: k = 12 gives 6, 4 and 3. There is a shortcut that skips even that, and section 02 builds it.
02 Worked example
2A = 3B = 4C, worked two ways
One equation runs the whole lesson. Two times A’s hidden logs equals three times B’s equals four times C’s. Find A : B : C, and confirm that the pair A : B inside your answer agrees with the two-term rule.
The two routes to 6 : 4 : 3 are worth doing once each. The LCM route explains why the answer is what it is; the hide-and-multiply route is what you use under time pressure. And the final substitution is not optional — a swap performed in the wrong direction produces 4 : 6 : 12-style answers that look plausible and fail the check instantly.
03 The method
Every sentence shape this model uses
Four phrasings, one underlying equation each. The right-hand column is the answer you should be able to write without a pen.
| Sentence | As an equation | Ratio |
|---|---|---|
| 3 times A equals 5 times B | 3A = 5B | A : B = 5 : 3 |
| K’s pay is 2/7 of J’s pay | K = (2/7)J | K : J = 2 : 7 |
| 40% of S equals 60% of P | 40S = 60P | S : P = 3 : 2 |
| Half of A equals a third of B | A/2 = B/3 | A : B = 2 : 3 |
| 2A = 3B = 4C | three equal products | 6 : 4 : 3 |
| 3x = 4y = 5z | LCM 60 | 20 : 15 : 12 |
| Reading pA = qB as A : B = p : q | the wrong direction | gives 3 : 5 instead of 5 : 3 |
05 Cheat sheet
Model 1 on one page
One rule, one shortcut, one check, and the four sentence shapes that trigger them.
| Case | Rule | Example |
|---|---|---|
| Two equal multiples | pA = qB ⇒ q : p | 3A = 5B → 5 : 3 |
| Percentage equality | cancel the % and swap | 40%S = 60%P → 3 : 2 |
| Fractional share | numerator : denominator | K = (2/7)J → 2 : 7 |
| Equal fractions | denominators, in place | A/2 = B/3 → 2 : 3 |
| Three equal multiples | qr : pr : pq | 2A=3B=4C → 6 : 4 : 3 |
| Direction check | small coefficient, big share | A has the 2 and gets 6 parts |
| Swapping the wrong way | the only real error here | 3A = 5B is not 3 : 5 |
06 Where & why
Where this shows up
Model 1 is line one of hundreds of questions. Its value is that it is almost free — once the swap is automatic, the whole model costs no time at all.
Asked directly, worth a mark, done in three seconds. The only failure mode is swapping the wrong way, which the direction check eliminates.
“If 3x = 4y = 5z, find x : y : z” gives 20 : 15 : 12. Hide-and-multiply beats the LCM route once the numbers stop being small.
Once you have the ratio you may substitute x = 8 and y = 9 literally, because the expression is itself a ratio and the scale cancels. That gives 4 : 42 = 2 : 21.
Models 2 to 9 all begin with a clean integer ratio. Model 1 is how you get one out of whatever sentence the paper actually gave you.
07 Interview questions
What gets asked
Ten. The first is the whole model; the rest are the variations a setter reaches for once the first one is too easy.
If 3A = 5B, what is A : B?
Why do the coefficients swap?
“40% of S equals 60% of P.” Find S : P.
“K’s payment is 2/7 of J’s.” Find K : J.
A/2 = B/3 gives what ratio, and why is it not 3 : 2?
If 2A = 3B = 4C, find A : B : C.
Explain the hide-and-multiply shortcut.
If 3x = 4y = 5z, find x : y : z.
If A : B = 8 : 9, find (5A − 4B) : (3A + 2B).
When can you substitute the ratio terms as if they were the actual values?
08 Practice problems
Six on finding the ratio
The first two should take seconds. The middle three are the substitution questions that make this model worth marks, and the last one is the three-term case pushed further than a paper usually goes.