AFCAT Numerical Ability · Percentage

Successive Changes, Price and Consumption

Each percentage change multiplies by a factor: chain the factors, undo one, and use price times quantity to keep a budget fixed or to see how much more a cheaper price buys.

All Percentage notes

Why this matters

This is where AFCAT asks most: eight of the chapter's fourteen past-year questions are on this page, from 2015, 2018 and 2019. They are a price cut that must be restored, a budget kept fixed when the price moves, a cheaper price that buys a few kilograms more, and a school roll that rises and falls. Every one is a product of factors, and the usual slip is to add percentages instead.

Concept 1 of 5: The net effect of successive changes

A 30% rise multiplies by 1.3 and a 20% fall multiplies by 0.8. One after the other they multiply by

1.3 \times 0.8 = 1.04

, a net rise of 4%, less than the plain difference of the two percentages.

Definition

Write each change as a factor: a rise of $r\%$ is $1 + \tfrac{r}{100}$ , a fall is $1 - \tfrac{r}{100}$ . Multiply all the factors.
For two changes $a\%$ and $b\%$ (a fall is negative) the net change is $a + b + \dfrac{ab}{100}$ per cent.
The order does not matter: up 30% then down 20% equals down 20% then up 30%.
Up $x\%$ then down $x\%$ is always a loss of $\dfrac{x^2}{100}\%$ , never zero.
Given the final value, divide by the product of the factors to get the start.

Two successive changes

\text{net } \% = a + b + \dfrac{ab}{100}

Worked example

A price rises by 30% and then falls by 20%. Find the net change.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (I) 2019 — Numerical Ability · 03-203Moderate

Example 1 · Percentage · Successive Change, Price and Consumption

The strength of a school increases and decreases in every alternate year by 10%. It started with an increase in 2000. Then, the strength of the school in 2003 as compared to that in 2000 was

Up and down by the same percentage is a loss

Up 30% then down 30% is

1.3 \times 0.7 = 0.91

, a 9% fall. The fall is taken on the larger number, so it removes more than the rise added.

Do not add the percentages

Up 30% then down 20% is not the difference of the two. The second change acts on the new value, which is why the formula has the extra

\tfrac{ab}{100}

term.

Drill 17 more on the net effect of successive changes

Concept 2 of 5: Undoing a change: the percentage needed to get back

Cut a price of 100 by 40% and it becomes 60. To get back to 100 it must rise by 40 on 60, which is 66⅔%. The way back is a bigger percentage because it starts from a smaller number.

Definition

After a fall of $r\%$ , the rise that restores it is $\dfrac{r}{100 - r} \times 100\%$ .
After a rise of $r\%$ , the fall that restores it is $\dfrac{r}{100 + r} \times 100\%$ .
Both are the 'swap the base' rule from the comparison page: the same gap, measured on the new value.

The change that undoes another

\text{after a fall of } r\%: \; \dfrac{100r}{100 - r}\% \text{ rise}, \qquad \text{after a rise of } r\%: \; \dfrac{100r}{100 + r}\% \text{ fall}

Worked example

A price is cut by 40%. By what percent must the new price be raised to restore the original?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2018 · AFCAT (II) 2018 — Numerical Ability · 03-204Easy

Example 2 · Percentage · Successive Change, Price and Consumption

The price of an article has been reduced by 25%. In order to restore the original price, the new price must be increased by

The same percentage does not undo it

A 40% cut followed by a 40% rise ends at

0.6 \times 1.4 = 0.84

, still 16% short. The rise has to be measured on the smaller, cut price.

Drill 5 more on undoing a change: the percentage needed to get back

Concept 3 of 5: When a quantity is a product: area, volume, takings

Area is length times breadth, so if both grow the area grows by the product of their factors. The same holds for any product: takings are price times sales, pay is rate times hours, a fraction is a numerator over a denominator.

Definition

If $Q = X \times Y$ and X, Y change by $a\%$ , $b\%$ , then Q changes by $a + b + \tfrac{ab}{100}$ per cent, the successive-change formula again.
A square or a circle: the area changes by the factor squared, so a radius 10% longer gives $1.1^2 = 1.21$ , a 21% larger area.
A cube or a sphere: the volume changes by the factor cubed. A cylinder: radius squared times height.
A perimeter or a circumference changes by the factor itself, once.
A fraction whose numerator rises by $a\%$ and denominator falls by $b\%$ is multiplied by $\dfrac{1 + a/100}{1 - b/100}$ .

Change in a product

Q = XY \;\Rightarrow\; \text{factor}_Q = \left(1 + \tfrac{a}{100}\right)\left(1 + \tfrac{b}{100}\right)

Worked example

The length of a rectangle is increased by 30% and its breadth by 10%. Find the percentage increase in its area.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (II) 2019 — Numerical Ability · 03-201Easy

Example 3 · Percentage · Successive Change, Price and Consumption

When the price of an article was reduced by 20% its sale increased by 80%. What was the net effect on the sale?

Area squares the change

A side 10% longer gives an area 21% larger, not 10%. A perimeter, though, grows by the same 10%.

Drill 11 more on when a quantity is a product: area, volume, takings

Concept 4 of 5: Keeping expenditure fixed when the price changes

Expenditure is price times consumption. If the price rises by 60%, consumption must fall so that the two factors multiply back to 1. That is the undo-a-change rule from earlier on this page.

Definition

Expenditure = price × consumption. For the bill to stay the same, the consumption factor must cancel the price factor.
Price up $r\%$ : cut consumption by $\dfrac{r}{100 + r} \times 100\%$ .
Price down $r\%$ : consumption can rise by $\dfrac{r}{100 - r} \times 100\%$ .
If both move by given amounts, the change in spending is the product of the two factors: price up 12% and consumption down 10% gives $1.12 \times 0.9 = 1.008$ , 0.8% more spent.

Consumption cut for a fixed budget

\text{cut} = \dfrac{r}{100 + r} \times 100\%

Worked example

The price of oil rises by 60%. By what percent must a family cut its use so that it spends the same?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2018 · AFCAT (II) 2018 — Numerical Ability · 03-207Easy

Example 4 · Percentage · Successive Change, Price and Consumption

If there is a 25% increase in the cost of sugar, by what % should consumption be decreased in order to maintain the expenditure?

The cut is smaller than the rise

A 60% price rise needs only a 37.5% cut in use, because the cut is taken on the old consumption while the rise was on the old price.

Drill 7 more on keeping expenditure fixed when the price changes

Concept 5 of 5: A cheaper price buys more for the same money

When the price of rice falls by 25%, the same ₹540 saves 25% of ₹540, which is ₹135, compared with the old price. That ₹135 is what pays for the extra kilograms, at the new price.

Definition

The saving method: a price cut of $r\%$ on a fixed sum $M$ saves $\tfrac{r}{100} M$ . That buys the extra quantity at the reduced price, so reduced price $= \dfrac{rM/100}{\text{extra quantity}}$ .
The original price is then the reduced price divided by $1 - \tfrac{r}{100}$ .
A price rise means fewer kilograms. With the price up a quarter, the money buys $\tfrac{1}{1.25} = \tfrac 45$ of the old quantity, so the shortfall is a fifth of the old quantity.

Reduced price from the extra quantity

\text{reduced price} = \dfrac{\tfrac{r}{100} \times M}{\text{extra quantity}}

Worked example

A 25% fall in the price of rice lets a man buy 6 kg more for ₹540. Find the reduced price per kg.

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (II) 2019 — Numerical Ability · 03-202Moderate

Example 5 · Percentage · Successive Change, Price and Consumption

A reduction of 20% in the price of apples enables a man to buy 16 kg more apples for ₹4000. What is the reduced price per kg of apples?

The saving method gives the reduced price

The money saved buys the extra kilograms at the new price. If the question asks for the original price, divide by

1 - \tfrac{r}{100}

at the end.

Per dozen, per kg, per item

Read the unit asked. A reduced price of ₹1.10 an apple is ₹13.20 a dozen, and both are usually among the options.

Drill 4 more on a cheaper price buys more for the same money

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (5)

The net effect of successive changes
Two successive changes
$\text{net } \% = a + b + \dfrac{ab}{100}$
Undoing a change: the percentage needed to get back
The change that undoes another
$\text{after a fall of } r\%: \; \dfrac{100r}{100 - r}\% \text{ rise}, \qquad \text{after a rise of } r\%: \; \dfrac{100r}{100 + r}\% \text{ fall}$
When a quantity is a product: area, volume, takings
Change in a product
$Q = XY \;\Rightarrow\; \text{factor}_Q = \left(1 + \tfrac{a}{100}\right)\left(1 + \tfrac{b}{100}\right)$
Keeping expenditure fixed when the price changes
Consumption cut for a fixed budget
$\text{cut} = \dfrac{r}{100 + r} \times 100\%$
A cheaper price buys more for the same money
Reduced price from the extra quantity
$\text{reduced price} = \dfrac{\tfrac{r}{100} \times M}{\text{extra quantity}}$

Watch out for (7)

Up and down by the same percentage is a loss
→ The net effect of successive changes
Do not add the percentages
→ The net effect of successive changes
The same percentage does not undo it
→ Undoing a change: the percentage needed to get back
Area squares the change
→ When a quantity is a product: area, volume, takings
The cut is smaller than the rise
→ Keeping expenditure fixed when the price changes
The saving method gives the reduced price
→ A cheaper price buys more for the same money
Per dozen, per kg, per item
→ A cheaper price buys more for the same money

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