Percentages Model 3 — The Remaining Model

Percentages · 25 min

Aptitude · Percentages · Model 3

One word decides whether the percentages add

“He spends 25%, 15% and 20% of his salary” — add them. “40% are English, 50% of the remaining are Hindi” — you cannot. Reading which base each percentage uses is the entire model.

Switch between flat and nested on the same numbers →
Percentages of the same base add. Percentages of the remaining multiply — and you track what survives, not what is taken.

01 The idea

Flat or nested, and how to tell

A man spends 25% of his salary on household expenses, 15% on transport and 20% on education, and is left with ₹6,000. Each of those is a share of the same salary, so they add: 60% spent, 40% left. Forty per cent is ₹6,000, so the salary is ₹15,000.

Now a different sentence. In a library 40% of the books are English, 50% of the remaining are Hindi, 60% of the remaining are Sanskrit, and 7,200 books are in other languages. These cannot be added, because each percentage is taken of a smaller collection than the last.

For the nested case, flip to what survives. After English, 60% remains. Of that, 50% survives Hindi. Of that, 40% survives Sanskrit. So the leftover is 0.6 × 0.5 × 0.4 = 0.12 of the total — 12%. And 12% is 7,200, so the total is 60,000.

That flip is the whole technique. Students track what is spent and get lost; tracking what is left turns the question into a single product. The tell is the word remaining — or “of the rest”, or “of the balance”. Without it, assume the same base and add.

Look for “remaining”, “the rest” or “the balance”. If it is there, multiply the surviving fractions. If not, add the percentages.
Flat caseEvery share is a percentage of the original total, so they add and the leftover is 100% − their sum.
Nested caseEach share is a percentage of what remains, so the base shrinks each time. Multiply the surviving fractions instead of adding the shares.
Surviving fraction1 − p/100 for a step that removes p%. The quantity to track in the nested case, because it is what the next step acts on.

02 Worked example

The library: 40%, then 50% of the rest, then 60% of the rest

The nested case, which is the one worth working carefully. In a library 40% of the books are in English, 50% of the remaining are in Hindi, 60% of the remaining are in Sanskrit, and the remaining 7,200 are in other languages. Find the total number of books.

1
Spot the nestingThe second and third percentages say “of the remaining”, so each acts on a smaller collection. Adding 40 + 50 + 60 would give 150%, which is nonsense.40% of T, then 50% of the rest, then 60% of the rest
2
Flip to what survives each stepEnglish removes 40%, so 60% survives. Hindi removes 50% of that, so 50% survives. Sanskrit removes 60%, so 40% survives.survives: 0.6, then 0.5, then 0.4
3
Multiply the surviving fractionsSuccessive fractions of a shrinking base multiply.0.6 × 0.5 × 0.4 = 0.12
4
The leftover is that fraction of the totalTwelve per cent of the library is in other languages, and that is 7,200 books.0.12 × T = 7,200
5
DivideOne step from the part to the whole.T = 7,200 / 0.12 = 60,000 books

Check it forwards to be sure: 60,000 books, minus 40% English leaves 36,000; minus 50% Hindi leaves 18,000; minus 60% Sanskrit leaves 7,200. That walk-forward is worth the ten seconds, because the commonest error here is losing track of which step a percentage belongs to. And note that reading it as a flat case would have given a total of ₹18,000 — not close.

03 The method

The two cases, and the variants that mix them

Deciding which case you are in is the whole model. The last two rows are where questions combine it with something else.

Flat: leftover% = 100 − (p₁ + p₂ + …), then scale up. Nested: leftover fraction = ∏(1 − pᵢ/100), then divide the known balance by it.
A ratio split inside a remainder. If 10% goes to mutual funds and the remaining 90% is split between groceries and rent in the ratio 2 : 3, then rent is 3/5 of 90% = 54% of the whole salary. Convert the ratio share into a percentage of the original total, and the question collapses to one unitary step. If rent is ₹21,600 then 54% is ₹21,600, so the salary is ₹40,000.
The sentence saysCaseHandle it by
“25% on X, 15% on Y, 20% on Z”Flatadd: 60% spent
“50% of the remaining”Nestedmultiply survivors
“of the rest” / “of the balance”Nestedmultiply survivors
“the remaining he saves”Flat100% minus the sum
Remainder split in a ratioMixedratio share of the remainder %
A fraction then a percentageNested1/3 saved → 2/3 survives
Adding nested percentagesmeaninglesscan exceed 100%

05 Cheat sheet

Model 3 on one page

Two cases, the words that distinguish them, and the mixed variant.

CaseRouteWorked
Flat, leftover wanted100 − Σp25+15+20 → 40% left
Flat, total from leftoverleft × 100/leftover%6,000/0.4 = 15,000
Nested, surviving fraction∏(1 − p/100)0.6×0.5×0.4 = 0.12
Nested, total from leftoverleft ÷ product7,200/0.12 = 60,000
Remainder split in a ratioshare × remainder%3/5 of 90% = 54%
A fraction saved firstconvert to a survivor1/3 saved → 2/3 survives
Adding nested percentagesmeaningless40+50+60 = 150%
Track what survives, not what goesIn the nested case the surviving fractions multiply cleanly, while the amounts taken do not. Flipping to survivors is the whole technique.
One word decides the case“Remaining”, “the rest” or “the balance” means nested. Without one of them, the shares are of the same base and add.
Nested percentages can exceed 10040 + 50 + 60 = 150% is perfectly possible in a nested question and is a sign you are in the nested case, not a sign the question is broken.

06 Where & why

Where Model 3 shows up

A reading model. The arithmetic is easy and the marks go to whoever parsed the sentence correctly.

Bank PO · SSC CGL
Salary and expenditure breakdowns

Usually flat, occasionally nested, and the two are set close together to check whether you read rather than pattern-matched.

TCS NQT · Infosys
“Of the remaining” chains

Books in a library, students in a college, votes among candidates. Three nested steps is the standard depth.

Mixed ratio-and-percentage
A remainder split in a ratio

“10% to mutual funds, the rest 2 : 3 between groceries and rent.” Convert the ratio share to a percentage of the original.

Repeated replacement, elsewhere
The same product of survivors

Drawing off and replacing liquid in a vessel is this model with equal steps, which is why it produces a power. Recognising the shared shape shortens both.

Before writing anything, underline every percentage and note what each is a percentage of. That single pass decides the whole question, and the arithmetic afterwards is two lines.

07 Interview questions

What gets asked

Nine, and the first two are the two cases the model is built on.

A man spends 25%, 15% and 20% of his salary and is left with ₹6,000. Find the salary.
₹15,000. All three are percentages of the same salary, so they add to 60% spent, leaving 40%. That 40% is ₹6,000, so the salary is 6,000 × 100/40 = ₹15,000.
40% of books are English, 50% of the remaining are Hindi, 60% of the remaining are Sanskrit, and 7,200 are other. Find the total.
Sixty thousand. Track what survives: 60% after English, then 50% of that, then 40% of that, giving 0.6 × 0.5 × 0.4 = 0.12. So 12% is 7,200 and the total is 60,000.
Why can’t you add 40, 50 and 60 in that question?
Because each percentage is taken of a different, smaller collection. Adding them gives 150%, which is meaningless as a share of the whole. The word “remaining” is what tells you the base has changed.
What word should you look for?
“Remaining”, or an equivalent like “of the rest” or “of the balance”. Its presence means nested and you multiply the surviving fractions; its absence means the shares are of the same base and you add them.
Why track what survives rather than what is spent?
Because the surviving fractions multiply cleanly, while the amounts taken do not. Each step acts on what the previous step left, so what is left is exactly the quantity the next step needs. It turns a chain of subtractions into one product.
A person invests 10% in mutual funds and splits the rest between groceries and rent in the ratio 2 : 3. If rent is ₹21,600, find the salary.
₹40,000. The remainder is 90% of the salary and rent is three-fifths of that, which is 54% of the salary. So 54% is ₹21,600, making 1% equal to ₹400 and the salary ₹40,000.
How do you handle a fraction rather than a percentage?
Convert it to a surviving fraction the same way. If a third is saved, two-thirds survives to the next step. Mixing fractions and percentages is fine as long as everything becomes a survivor before you multiply.
Can a nested question’s percentages sum to more than 100?
Yes, easily — 40 + 50 + 60 = 150%. That is not a broken question; it is a signal that you are in the nested case. If the percentages sum above 100 and the question mentions a leftover, it must be nested.
How would you check a nested answer?
Walk it forwards. From 60,000: remove 40% to get 36,000, remove 50% to get 18,000, remove 60% to get 7,200 — matching the stated leftover. It takes ten seconds and catches a step applied to the wrong base, which is the model’s only real failure mode.

08 Practice problems

Six on the base

For each one, first write down what every percentage is a percentage of. Then the arithmetic is two lines.

Flat

Easy
A person’s income is ₹50,000. He spends 20% on rent, 30% on food and 15% on other expenses, and saves the rest. Find his savings.
Follow-up
All three shares are of the same income, so they add. Note the total is given here, so this is the forward direction rather than a reverse question.
Show the hint
Add the three percentages and subtract from 100.

Flat, reversed

Easy
A man spends 40% of his salary on food, 20% on rent, 10% on travel and 10% on entertainment, and is left with ₹1,500. Find his salary.
Follow-up
The leftover is given and the total is wanted, so it is one unitary step after the addition. Check your answer by computing the spending.
Show the hint
The four shares add to 80%, so what fraction is ₹1,500?

Nested, two steps

Medium
In a college 20% of the students study Arts, and 30% of the remaining study Commerce. If 17,500 students study neither, find the total number of students.
Follow-up
Two nested steps. Flip to survivors before multiplying, and check whether adding the percentages would have given a plausible-looking but wrong answer.
Show the hint
After Arts, 80% survives; after Commerce, 70% of that survives.

Which case is it?

Medium
Two questions. (a) A man spends 30% of his salary on rent and 40% on food, and saves ₹9,000. (b) A man spends 30% of his salary on rent and 40% of the remaining on food, and saves ₹9,000. Find the salary in each case.
Follow-up
Identical numbers, one word of difference, and two different answers. Doing both is the only reliable way to stop misreading these under time pressure.
Show the hint
In (a) the shares add; in (b) the second acts on what the first left.

Remainder split in a ratio

Medium
Out of his monthly salary, a man invests 20% in a pension fund. The remainder he divides between household expenses and school fees in the ratio 5 : 3, and saves nothing. If the school fees come to ₹18,000, find his monthly salary.
Follow-up
The ratio applies to the remainder, not to the salary. Converting the ratio share into a percentage of the original salary is what makes this one line.
Show the hint
Fees are three-eighths of the remaining 80% — what percentage of the salary is that?

Nested with a fraction, and a general result

Hard
A man saves one-third of his salary. Of the remainder he pays 60% as EMI and spends the rest on a watch. (a) Express the amount spent on the watch as a percentage of his salary. (b) If the difference between his savings and the watch amount is ₹7,000, find his salary. (c) Show that for any first-step fraction f and any second-step percentage p, the final leftover is (1 − f)(1 − p/100) of the total, and explain why this expression is unchanged if the two steps are applied in the opposite order.
Follow-up
Part (c) is the structural point: nested steps are a product of surviving fractions, and multiplication commutes, so the order of the deductions never changes what is left. That is the same fact that makes successive discounts order-independent, and seeing it twice is what makes it stick.
Show the hint
For (a), a third saved means two-thirds survives; then 40% of that survives the EMI.