Aptitude · Ratio and Proportion · Model 8

No total, one clue, and the whole split falls out

Model 2 gives you the amount and asks for the shares. This model gives you no amount at all — just a three-way ratio and one fact, such as “C gets ₹1,200 more than A”. Turn that fact into a number of parts and everything else follows.

Pick a clue and watch the whole split appear →

Count the parts in the clue, not the parts in the total. For 4 : 5 : 7, “C gets ₹1,200 more than A” is 3 parts, so one part is ₹400.

01 The idea

The ratio fixes the shape; one number fixes the scale

A prize is shared among three people in the ratio 4 : 5 : 7. You are not told what the prize is worth. You are told one thing: the third person receives ₹1,200 more than the first. That single sentence is enough to find all three shares and the total, and the route to it is short.

Look at the clue in parts rather than in rupees. The third person holds 7 parts and the first holds 4, so the gap between them is 7 − 4 = 3 parts. Those 3 parts are worth ₹1,200, so one part is ₹400. The shares are 4 × 400 = ₹1,600, 5 × 400 = ₹2,000 and 7 × 400 = ₹2,800, and the total is 16 × 400 = ₹6,400.

Nothing about that depended on the clue being a difference. “A and B together receive ₹3,600” is 9 parts, and 3600 ÷ 9 is the same ₹400. “B receives ₹2,000” is 5 parts, and again ₹400. Any single absolute number plus the ratio pins the split, and the only skill is counting how many parts the number covers.

The one kind of clue that never helps is another statement about proportions. Being told that A’s share is 4/7 of C’s adds nothing, because 4 : 5 : 7 already said that. A ratio is scale-free, so a second scale-free fact leaves you exactly where you started — and recognising a useless clue is worth as much as using a good one.

The clue is worth a number of parts. Divide the clue by that number and you have the value of one part; everything else is multiplication.

Clue in partsThe number of ratio parts the given amount covers. For 4 : 5 : 7, C − A is 3 parts, A + B is 9 parts, and B alone is 5 parts.

Scale and shapeThe ratio fixes the shape of the split and nothing else. One absolute number fixes the scale. You need exactly one of each, and a second ratio-style fact is not the missing one.

The parts divisibility checkA split into 16 parts has a total that is a multiple of 16 whenever one part is a whole number of rupees. Of ₹6,400, ₹6,500, ₹6,600 and ₹6,700, only ₹6,400 qualifies.

02 Worked example

4 : 5 : 7, and C gets ₹1,200 more than A

One split runs the whole lesson. A prize is divided among A, B and C in the ratio 4 : 5 : 7. C receives ₹1,200 more than A. Find each share and the total prize.

Count the parts, and notice what is missingSixteen parts in all. But no amount has been given, so the master key cannot be found by dividing a total — there is no total to divide.4 + 5 + 7 = 16 parts, total unknown

Turn the clue into a part countC holds 7 parts and A holds 4, so the gap between their shares is the difference of the part counts. That gap is what the ₹1,200 measures.C − A = 7 − 4 = 3 parts = ₹1,200

Divide the clue, not the totalOne division, and the master key appears exactly as it does in an ordinary split. Note that ₹1,200 is divisible by 3, as it has to be.₹1,200 ÷ 3 = ₹400 per part

Multiply out all three shares, then add for the totalHere the total is the answer rather than the given. Four hundred rupees a part, sixteen parts.A = ₹1,600, B = ₹2,000, C = ₹2,800 ⇒ total ₹6,400

Check the clue and the ratioTwo checks, both quick. The gap must come back to ₹1,200, and the three shares must still be in 4 : 5 : 7.2800 − 1600 = 1200 ✓ 1600 : 2000 : 2800 = 4 : 5 : 7 ✓

Both checks matter, and for different reasons. If you had counted the clue as 7 parts instead of 3 — a natural slip, reading “C gets ₹1,200 more” as “C gets ₹1,200” — you would have got one part = ₹171.43 and three shares still in exactly the ratio 4 : 5 : 7. The ratio check would have passed. Only re-reading the clue out of your own answer catches it.

03 The method

Every clue, and what it is worth

The whole model is the middle column of this table. Learn to read a sentence straight into a part count and there is nothing else to it.

One part = clue ÷ (parts the clue covers). Then each share is its own part count times that, and the total is the sum of the part counts times that. For A : B : C = a : b : c, a difference clue covers c − a parts, a pair clue covers a + b, and a single-share clue covers just that share’s own count.

Two divisibility checks come free, and both work on the options. First, the clue must be divisible by its part count: a clue worth 3 parts has to be a multiple of 3, so ₹1,200 is fine and ₹1,250 would mean the question is broken. Second, the total must be a multiple of the sum of the parts. With 16 parts, a total of ₹6,400 works (16 × 400) while ₹6,500, ₹6,600 and ₹6,700 do not — so a question asking for the total with those four options is answerable with no arithmetic at all.

The clue reads	Parts it covers	On 4 : 5 : 7	One part
C gets ₹1,200 more than A	`c − a`	3 parts	₹400
C gets ₹800 more than B	`c − b`	2 parts	₹400
A and B together get ₹3,600	`a + b`	9 parts	₹400
B gets ₹2,000	`b`	5 parts	₹400
B and C together get ₹4,800	`b + c`	12 parts	₹400
The total is ₹6,400	`a + b + c`	16 parts	₹400
A gets 4/7 of what C gets	`0 new parts`	already in the ratio	`no answer`

05 Cheat sheet

Model 8 on one page

One rule, two divisibility checks, and the three ways a clue can fail to be a clue.

Case	Rule	On 4 : 5 : 7 with C − A = ₹1,200
Difference clue	`c − a parts`	3 parts → ₹400
Pair clue	`a + b parts`	9 parts → ₹3,600
Single-share clue	`that share’s own count`	B is 5 parts → ₹2,000
Recovering the total	`(a+b+c) × one part`	16 × 400 = ₹6,400
Clue divisibility	`clue ÷ its parts must be exact`	1,200 ÷ 3 = 400 ✓
Total divisibility	`total is a multiple of a+b+c`	₹6,400 yes, ₹6,500 no
A second proportional fact	`gives no scale`	“A is 4/7 of C” is already the ratio

The ratio check does not catch a misread clueCount the clue as 7 parts instead of 3 and you get ₹171.43 a part — and the three shares are still in 4 : 5 : 7. Only reading the clue back out of your own answer catches that, so do it every time.

Any one absolute number is enoughA difference, a pair total, a single share, the whole amount — all four fix the same split. What you cannot use is a second statement about proportions, because a ratio carries no scale to add.

The parts count works on the optionsA 16-part split has a total divisible by 16. That kills three of four options in a well-set question before you compute anything, and it is the fastest route through this model under time pressure.

06 Where & why

Where this shows up

This shape appears wherever the total is deliberately withheld, which in practice means anywhere the setter wants two steps instead of one.

Bank PO · SSC CGL

“C gets ₹1,200 more than A”

The standard phrasing. Two marks, and about thirty seconds once counting the clue in parts is automatic.

Partnership questions

“B’s profit exceeds A’s by …”

The capital ratio gives the shape and one stated difference gives the scale. Identical arithmetic with different vocabulary.

Ages and salaries

“Their ages are in the ratio 4 : 5 : 7 and the eldest is 12 years older than the youngest”

Same model, and the answer to “how old is the middle one” is 5 parts. Age questions lean on this constantly.

Answer-option elimination

“Find the total amount”

The total is a multiple of the sum of the parts. With 16 parts and options at ₹6,400, ₹6,500, ₹6,600 and ₹6,700, only the first survives and no arithmetic is needed at all.

The habit is one sentence long: before you divide anything, write down how many parts the clue covers. Everything in this model follows from that number, and every wrong answer in it comes from getting that number wrong.

07 Interview questions

What gets asked

Ten. The fourth is the one that separates people who understand the model from people who have memorised one version of it.

A prize is split 4 : 5 : 7 and C gets ₹1,200 more than A. Find the total.

₹6,400. The gap between C and A is 7 − 4 = 3 parts, so one part is 1200 ÷ 3 = ₹400. The split is 16 parts, so the total is 16 × 400 = ₹6,400, with shares of ₹1,600, ₹2,000 and ₹2,800.

Why can you not just divide ₹1,200 by 16?

Because ₹1,200 is not the total. It is the gap between two shares, which covers only 3 of the 16 parts. Dividing by 16 answers a question nobody asked and gives ₹75 a part, which fails the clue check immediately.

The same split, but you are told A and B together get ₹3,600. Does anything change?

No. A + B is 4 + 5 = 9 parts, and 3600 ÷ 9 is ₹400 a part — the same answer. Any single absolute number fixes the scale; which one it is only changes the part count you divide by.

What kind of extra information would not help?

Another statement about proportions. Being told that A gets 4/7 of what C gets is already contained in 4 : 5 : 7, so it adds nothing. A ratio is scale-free, and you need exactly one fact that is not.

How do you check an answer in this model?

Twice. Read the clue back out of your numbers — here 2800 − 1600 must be ₹1,200 — and confirm the three shares are still in the given ratio. The second check alone is not enough, because a misread clue produces shares that are still in the right ratio.

The options for the total are ₹6,400, ₹6,500, ₹6,600 and ₹6,700. Can you answer without solving?

Yes. The split is 16 parts, so the total must be 16 times a whole number of rupees. Only ₹6,400 divides by 16 exactly — the others give 406.25, 412.5 and 418.75. That is the whole question, answered from the ratio alone.

Ages are in the ratio 4 : 5 : 7 and the eldest is 12 years older than the youngest. How old is the middle one?

20. The gap is 3 parts = 12 years, so one part is 4 years, and the middle person holds 5 parts. The ages are 16, 20 and 28, and 28 − 16 = 12 as required. The arithmetic is identical to the money version.

What happens if the clue compares two shares that are equal in the ratio?

It tells you nothing. For 4 : 5 : 4, a clue about C − A is worth 0 parts, so it is consistent with every possible total. A clue has to cover a positive number of parts to measure anything.

What if the clue contradicts the ratio?

Hard

A prize is shared among A, B and C in the ratio 5 : 7 : 11, and is paid entirely in ₹50 notes so that each person receives a whole number of notes. C receives ₹1,800 more than A. (a) Find the three shares and the total. (b) Show that each share really is a whole number of ₹50 notes. (c) A second prize uses the same ratio, and the four options for its total are ₹9,200, ₹9,300, ₹9,400 and ₹9,500. Show that only one of them can be correct, without finding the shares.

Follow-up

Part (c) is the transferable skill. A 23-part split forces the total to be a multiple of 23, and 23 is prime, so the divisibility test is sharp rather than loose — exactly one option survives. That turns a two-minute question into a five-second one, and the same trick works on any question in this model that asks for a total.

Show the hint

For (a) the clue covers 11 − 5 parts. For (c) divide each option by the total part count and look for the one that comes out whole.