Aptitude · Ratio and Proportion · Model 2

Find the value of one part and the rest is multiplication

Every question in this model reduces to a single number: what one part is worth. Add the ratio terms, divide the amount by that, and the eight or nine sub-questions a paper can ask about the same split all become one multiplication each.

Split an amount and answer every sub-question →

₹10,500 in 5 : 7 : 9 is 21 parts at ₹500 a part. That single number answers every sub-question about this split.

01 The idea

The master key

₹10,500 is divided among A, B and C in the ratio 5 : 7 : 9. The ratio does not say how much money there is — it says how many equal parts the money is cut into. Add the terms: 5 + 7 + 9 = 21 parts. Divide: ₹10,500 ÷ 21 = ₹500 per part. That number is the master key, and everything else in the question is one multiplication away from it.

The three shares are 5 × 500 = ₹2,500, 7 × 500 = ₹3,500 and 9 × 500 = ₹4,500. Always add them back: 2500 + 3500 + 4500 = 10,500. That check costs three seconds and catches a wrong master key, which is the only error in the model that propagates into every part of the answer.

A paper rarely stops at the three shares. It asks for the difference between two shares, the sum of two, one share as a percentage of another, one as a fraction of another, the ratio of two groups, or what happens if someone drops out. Every one of those is a question about part counts. C − A is 9 − 5 = 4 parts, so ₹2,000, and you never compute two shares and subtract.

Two of those sub-questions do not need the money at all. C exceeds A by 4 parts out of A’s 5, so C is 80% more than A — and that is true whatever the total was. A as a fraction of C is 5/9, again independent of the total. When a sub-question compares two shares, the amount cancels, and reaching for the master key first is wasted work.

Add the terms to get the parts, divide the amount by the parts to get the master key, then multiply. Comparisons between shares skip the master key entirely.

Total partsThe sum of the ratio terms. 5 : 7 : 9 is 21 parts, which means the amount has to be divisible by 21 for the shares to be whole rupees — a fact you can use on the answer options.

Master keyThe value of one part: amount ÷ total parts. ₹10,500 ÷ 21 = ₹500. Every share, sum and difference in the question is a multiple of it.

RedistributionWhen someone leaves or a fixed amount is transferred, recompute the parts. Dropping B from 5 : 7 : 9 leaves 14 parts, so the master key rises from ₹500 to ₹750.

02 Worked example

₹10,500 in 5 : 7 : 9, and everything a paper can ask about it

One split runs the whole lesson. ₹10,500 is divided among A, B and C in the ratio 5 : 7 : 9. Find the three shares, the difference between C’s and A’s, and by what percentage C’s share exceeds A’s.

Add the termsTwenty-one equal parts, of which A takes five, B takes seven and C takes nine. Note that 10,500 is divisible by 21, which is not an accident — setters choose totals that are.5 + 7 + 9 = 21 parts

Divide to get the master keyThe value of one part. This is the only division in the entire question, however many sub-parts it has.₹10,500 ÷ 21 = ₹500 per part

Multiply out the shares and check the sumThree multiplications, then one addition as a check. If the shares do not sum to the original amount, the master key is wrong.A = ₹2,500, B = ₹3,500, C = ₹4,500 ⇒ total ₹10,500 ✓

Do the difference in parts, not in rupeesSubtract the part counts first and multiply once. This is faster and it also tells you the answer must be a multiple of 4 × 500.C − A = (9 − 5) = 4 parts = 4 × 500 = ₹2,000

The percentage question ignores the moneyC exceeds A by 4 parts, and A’s own share is 5 parts. The total never enters, so this answer would be the same for any amount at all.4/5 = 80% ⇒ C’s share is 80% more than A’s

Notice how little arithmetic the last two steps needed. Working in parts turns a five-part question into one division and a handful of small multiplications, and the comparison sub-questions do not even need the division. That is the whole advantage of this model: once the master key is on the page, the rest of the question is bookkeeping.

03 The method

The nine sub-questions, all on one split

This is the taxonomy worth learning. Every entry below is answered on the same ₹10,500 in 5 : 7 : 9, so you can see exactly how little changes between them.

Share = (its parts ÷ total parts) × amount. Equivalently, master key = amount ÷ total parts, then multiply. For a comparison between two shares, use the part counts alone: x is (x−y)/y more than y, and y is (x−y)/x less than x.

Use the part count against the options. If the answer is 4 parts, it must be a multiple of 4, which usually kills two options on sight. Two more tricks worth having: a ratio given as fractions should be multiplied by the LCM of the denominators first, so 1/2 : 1/3 : 1/4 becomes 6 : 4 : 3 — note that this reverses the order of the shares. And a fixed transfer between two people is best converted into parts: transferring ₹3,000 out of a ₹45,000 split in 4 : 5 : 6 is transferring exactly one part, so 4 : 5 : 6 becomes 5 : 5 : 5.

Sub-question	In parts	On ₹10,500 in 5 : 7 : 9
The three shares	`5, 7, 9 × 500`	₹2,500 · ₹3,500 · ₹4,500
Sum of two shares	`5 + 7 = 12 parts`	A + B = ₹6,000
Difference of two	`9 − 5 = 4 parts`	C − A = ₹2,000
Percent more	`4/5`	C is 80% more than A
Percent less	`4/9`	A is 44.44% less than C
One as a fraction of another	`5/9`	A is 5/9 of C
Ratio of two groups	`(5+7) : (7+9)`	12 : 16 = 3 : 4
One share given, find the total	`total = share × 21/its parts`	B = ₹3,500 → total ₹10,500
Someone drops out	`parts fall to 14`	master key rises to ₹750, A gets ₹3,750

05 Cheat sheet

Model 2 on one page

The method, the two shortcuts that save real time, and the traps that make this easy model cost marks anyway.

Case	Rule	On ₹10,500 in 5 : 7 : 9
Any single share	`(parts/total) × amount`	B = 7/21 × 10,500 = 3,500
Master key	`amount ÷ total parts`	10,500 ÷ 21 = 500
Difference of two shares	`(difference in parts) × key`	4 × 500 = 2,000
Percent comparison	`part counts only`	C is 80% more than A
Fractional ratio	`multiply by the LCM first`	1/2 : 1/3 : 1/4 → 6 : 4 : 3
A fixed transfer	`convert it into parts`	₹3,000 of a 4:5:6 split of 45,000 is 1 part
Someone leaves	`recompute the key`	the shares change even though the parts do not

The total must be divisible by the partsFor 5 : 7 : 9 the amount has to be a multiple of 21 for whole-rupee shares. That also works on the options: an answer that is 4 parts must be a multiple of 4, which usually eliminates half the choices without any arithmetic.

Fractional ratios reverse the order1/2 : 1/3 : 1/4 becomes 6 : 4 : 3, so the person with the largest denominator ends up with the smallest share. Splitting the same money as 2 : 3 : 4 instead gives almost exactly the opposite distribution, which is why that swap is a favourite trap.

Comparisons do not need the amount“By what percentage is C’s share more than A’s?” is 4/5 = 80% for any total. If you found yourself dividing by 21 to answer it, you did more work than the question asked for.

06 Where & why

Where this shows up

This is the most-set model in the chapter, and the one most likely to appear as a multi-part question with a shared stem.

Bank Prelims · SSC CGL

One split, four sub-questions

The stem gives the amount and the ratio, then four questions hang off it. Finding the master key once and reusing it is the whole strategy.

TCS NQT · Infosys

Fractional ratios

“₹11,700 in 1/2 : 1/3 : 1/4”. Clear the denominators with the LCM to get 6 : 4 : 3 and it becomes an ordinary split. Working with the fractions directly is possible and much slower.

Partnership questions

The same arithmetic, different words

Profit shared in the ratio of capital contributions is exactly this model. The partnerships module builds the ratio; this model divides the money.

Anything with a transfer

“C gives ₹3,000 to A”

Convert the transferred amount into parts using the master key. If it comes out to a whole number of parts, the new ratio is immediate and no further arithmetic is needed.

The habit that matters here is writing the master key down and circling it. Every sub-question after the first is then one multiplication, and the comparison ones are free.

07 Interview questions

What gets asked

Ten, all on the same split, because that is exactly how a paper asks them.

₹10,500 is divided among A, B and C in the ratio 5 : 7 : 9. Find the shares.

₹2,500, ₹3,500 and ₹4,500. The terms add to 21 parts, so one part is 10,500 ÷ 21 = ₹500, and each share is its part count times that. The three shares add back to ₹10,500, which is the check.

What is the fastest way to find the difference between C’s and A’s share?

Subtract the part counts, then multiply once: 9 − 5 = 4 parts, so 4 × 500 = ₹2,000. Computing both shares and subtracting gets the same answer with three operations instead of two.

By what percentage is C’s share more than A’s?

80%. C exceeds A by 4 parts and A’s own share is 5 parts, so the increase is 4/5. Note the total amount never enters — this answer holds for any sum divided in 5 : 7 : 9.

And by what percentage is A’s share less than C’s?

44.44%, not 80%. The gap is the same 4 parts but the base is now C’s 9 parts, so it is 4/9. Two different percentages for one gap, because “more than A” and “less than C” use different bases.

How do you divide an amount in the ratio 1/2 : 1/3 : 1/4?

Multiply all three terms by the LCM of the denominators, 12, to get 6 : 4 : 3, then split normally. For ₹11,700 that is 13 parts at ₹900, so the shares are ₹5,400, ₹3,600 and ₹2,700. Note the order reverses: the largest denominator gets the smallest share.

B’s share is ₹3,500 and the ratio is 5 : 7 : 9. Find the total.

₹10,500. B holds 7 of 21 parts, so one part is 3,500 ÷ 7 = ₹500 and the total is 21 × 500. Any single share plus the ratio determines everything, which is why setters often give you the middle share rather than the total.

A total bonus of ₹45,000 is split 4 : 5 : 6, then C gives ₹3,000 to A. Find the new ratio.

1 : 1 : 1. Fifteen parts of ₹45,000 makes one part ₹3,000, so the transfer is exactly one part. A rises from 4 to 5 and C falls from 6 to 5, giving 5 : 5 : 5. Converting the cash into parts avoids all the large numbers.

If C drops out of a 5 : 7 : 9 split of ₹10,500, what does A get?

₹4,375. The money does not change but the parts fall to 5 + 7 = 12, so the master key rises to 10,500 ÷ 12 = ₹875 and A gets 5 × 875. A’s part count never changed — only the value of a part did.

A gets 2/5 of what B and C get together, and the total is ₹10,500. Find A’s share.

Hard

₹1,170 should have been divided among P, Q and R in the ratio 1/2 : 1/3 : 1/4, but it was divided in the ratio 2 : 3 : 4 by mistake. (a) Find each person’s correct share and each person’s actual share. (b) State who gained the most and who lost the most, with the amounts. (c) Show that the gains and losses cancel exactly, and explain in one sentence why they must.

Follow-up

The two ratios are near-reverses of each other, so the error is large and lands on different people in different directions. Part (c) is the real content: the money is conserved, so the total of all gains and losses is necessarily zero, which is also the check that catches an arithmetic slip anywhere in part (a).

Show the hint

The correct ratio clears to 6 : 4 : 3 over 13 parts; the mistaken one is 9 parts. The two master keys are different numbers.