Aptitude · Percentages · Change

A 16.66% rise is a multiplication by 7/6

Q: Give the general construction.

For a change of 1/d : multiply by (d+1)/d to increase and (d−1)/d to decrease. So a 12.5% fall is 1/8 down, giving × 7/8.

Q: State the reverse-pair rule.

A rise of 1/d is undone by a fall of 1/(d+1) , and a fall of 1/d is undone by a rise of 1/(d−1) . So up 25% comes back with down 20%, and down 20% comes back with up 25%.

Percentage change done in decimals is slow and rounds badly. Done as a single fraction multiplier it is exact and often cancels to nothing. This lesson is the table worth memorising and the one construction that uses it.

Pick a fraction and a direction, and see the multiplier →

Increase by 1/d → multiply by (d+1)/d. Decrease by 1/d → multiply by (d−1)/d.

01 The idea

One multiplier instead of two steps

Increase 600 by 16.66%. The obvious route is to find 16.66% of 600 and add it on — two steps, a repeating decimal, and a rounding decision. The better route recognises 16.66% as 1/6 and multiplies by 7/6, giving 700 exactly in one step.

The construction is worth stating precisely. If the change is the unit fraction 1/d, then increasing means you now have d + 1 of those parts where you had d, so the multiplier is (d+1)/d. Decreasing leaves d − 1 parts, so the multiplier is (d−1)/d.

That is why the fraction table matters. 12.5% is 1/8, so a 12.5% decrease is × 7/8. 11.11% is 1/9, so an 11.11% increase is × 10/9. 9.09% is 1/11, giving × 12/11 or × 10/11. Competitive exams choose these percentages precisely because the fractions cancel.

One consequence catches everybody, and it is the most examined idea in the chapter: a change and its reverse are not the same percentage. Going up by 1/6 and coming back down needs a fall of 1/7, not 1/6, because the second percentage is taken on the larger number. Undoing a multiplication means dividing by it, not applying the opposite percentage.

To undo a percentage change you divide by the same multiplier. That is a different percentage from the one you applied, because the base has moved.

Change multiplierThe single fraction that performs a percentage change in one step. (d+1)/d for a rise of 1/d, (d−1)/d for a fall.

Unit fractionA fraction with numerator 1. Nearly every percentage exams use is one — 16.66% is 1/6, 12.5% is 1/8 — which is what makes the multipliers tidy.

Reverse percentageThe change needed to undo another. A rise of 1/d is undone by a fall of 1/(d+1), not by a fall of 1/d.

02 Worked example

600 up by 16.66%, and back again

One number, one change, and the reverse that surprises people. Increase 600 by 16.66%. Then find the percentage decrease needed to return to 600.

Recognise the percentage16.66% is not a decimal to be handled; it is one sixth.16.66% = 1/6

Build the increase multiplierYou had six parts and are adding one more, so you now have seven of them.(6 + 1)/6 = 7/6

Apply it in one stepSix hundred times seven sixths. The six cancels straight into the 600.600 × 7/6 = 100 × 7 = 700

Now come back downTo undo the change you divide by the same multiplier, which is multiplying by its reciprocal.700 × 6/7 = 600 ✓

Read off the reverse percentageMultiplying by 6/7 is a fall of one part in seven, not one in six.1 − 6/7 = 1/7 = 14.28%, not 16.66%

The rise was 16.66% and the fall needed to undo it is 14.28%. The reason is that the fall is taken on 700 while the rise was taken on 600 — a larger base needs a smaller percentage to move the same absolute amount. Every “by what per cent must it be reduced to restore the original” question is this one observation, and the answer is never the percentage you started with.

03 The method

The table, and the reverse pairs it generates

Learn the left column and the rest is construction. The reverse column is what the harder questions actually ask for.

Rise of 1/d: multiply by (d+1)/d. Fall of 1/d: multiply by (d−1)/d. To undo either, multiply by the reciprocal.

The reverse pair rule: a rise of 1/d is undone by a fall of 1/(d+1), and a fall of 1/d is undone by a rise of 1/(d−1). So up 1/6 comes back with down 1/7; down 1/5 comes back with up 1/4. This is also the whole of the price-and-consumption model later in the module.

Percentage	Fraction	Rise multiplier	Fall multiplier
50%	`1/2`	`3/2`	`1/2`
33.33%	`1/3`	`4/3`	`2/3`
25%	`1/4`	`5/4`	`3/4`
20%	`1/5`	`6/5`	`4/5`
16.66%	`1/6`	`7/6`	`5/6`
14.28%	`1/7`	`8/7`	`6/7`
12.5%	`1/8`	`9/8`	`7/8`

05 Cheat sheet

Multipliers on one page

The construction, the reverse pairs, and the numbers to know by sight.

Case	Route	Example
Rise of 1/d	`× (d+1)/d`	up 1/6 → × 7/6
Fall of 1/d	`× (d−1)/d`	down 1/8 → × 7/8
Undo a change	`× the reciprocal`	× 7/6 undone by × 6/7
Reverse of a rise 1/d	`fall of 1/(d+1)`	up 1/6, back down 1/7
Reverse of a fall 1/d	`rise of 1/(d−1)`	down 1/5, back up 1/4
Applying the same % back	`wrong`	16.66% up needs 14.28% down
Working in decimals	`rounds and is slower`	0.1666 × 600

The reverse is always the smaller percentageUndoing a rise needs a smaller percentage than the rise, because the fall is taken on a bigger base. Undoing a fall needs a bigger one, for the mirror reason.

Exam percentages are unit fractions16.66%, 12.5%, 11.11%, 9.09%, 14.28%. Setters choose them so the multipliers cancel, so an awkward-looking percentage is a hint rather than a problem.

One multiplication, not two stepsNever compute the change and add it. Build the multiplier and apply it once — fewer operations, no rounding, and it chains for successive changes.

06 Where & why

Where multipliers show up

The multiplier is the single most transferable tool in the syllabus, because every chapter that applies a percentage uses one.

Bank PO · SSC CGL

“By what per cent must it be reduced…”

The reverse-pair question, set constantly. The trap answer is always the original percentage.

Price and consumption

Keeping expenditure unchanged

A 10% price fall allows an 11.11% consumption rise. That is exactly the reverse pair, and it is a model later in this module.

Profit, discount and interest

Every forward and reverse calculation

A 25% profit is × 5/4 forwards and ÷ 5/4 backwards. The whole reason those chapters warn against subtracting a percentage is this asymmetry.

Successive changes

Multipliers chain

Two changes are two multipliers multiplied. That is why successive percentages never add, in this chapter or in discounts.

Memorise the fraction table properly — it is perhaps two hours of work and it pays back in every quantitative paper you ever sit.

07 Interview questions

What gets asked

Ten, and the reverse-percentage question is the one you are most likely to meet.

How do you increase a number by 16.66% in one step?

Multiply by 7/6. Sixteen point six six per cent is 1/6, so an increase turns your six parts into seven. On 600 that is 600 × 7/6 = 700, with the six cancelling straight into the 600.

Give the general construction.

For a change of 1/d: multiply by (d+1)/d to increase and (d−1)/d to decrease. So a 12.5% fall is 1/8 down, giving × 7/8.

600 increased by 16.66% gives 700. What percentage decrease returns it to 600?

14.28%, not 16.66%. Undoing a multiplication by 7/6 means multiplying by 6/7, which is a fall of 1/7. The fall is taken on 700 while the rise was taken on 600, so a bigger base needs a smaller percentage to move the same amount.

State the reverse-pair rule.

A rise of 1/d is undone by a fall of 1/(d+1), and a fall of 1/d is undone by a rise of 1/(d−1). So up 25% comes back with down 20%, and down 20% comes back with up 25%.

Why is the reverse of a rise always a smaller percentage?

Because it acts on a larger base. To undo a ₹100 rise on ₹600 you must remove ₹100 from ₹700, and ₹100 is a smaller share of 700 than of 600. The absolute change is the same; the base is not.

The price of petrol falls 10%. By how much may consumption rise with no change in expenditure?

11.11%. A 10% fall is × 9/10, so to hold the product constant consumption must be × 10/9, which is a rise of 1/9 = 11.11%. This is the reverse-pair rule doing the work, and it is a standard model in its own right.

Why work in fractions rather than decimals here?

Because the multipliers cancel. 600 × 7/6 is exact and immediate, whereas 600 × 1.1666 needs real multiplication and a rounding decision. Exams choose these percentages precisely so the fractions work out.

What are the most useful entries in the fraction table?

The unit fractions from a half to a twelfth: 50%, 33.33%, 25%, 20%, 16.66%, 14.28%, 12.5%, 11.11%, 10%, 9.09% and 8.33%. Those cover the overwhelming majority of exam percentages, and recognising one on sight is usually the whole question.

A number is increased by 20% and then decreased by 20%. Where does it end up?

Hard

A price is increased by 25%, then by a further 20%, and then decreased by 25%. (a) Find the single equivalent percentage change. (b) Find the percentage change now needed to return the price to its original value, exactly. (c) Show that performing the three original changes in any other order gives the same final price, and state the property of multiplication that guarantees it.

Follow-up

Part (c) is the structural fact: the changes are multipliers and multiplication is commutative, so the order cannot matter. That is the same reason successive discounts can be applied in any order — worth seeing once here so it is obvious there.

Show the hint

Build all three multipliers as fractions and multiply them; the arithmetic stays exact if you never convert to decimals.