Formula

Consumption cut that absorbs a price rise

Arithmetic • Percentage • Price and Consumption

Formula

reduction = \frac{100 r}{100 + r} %

Expenditure is price

\times

consumption, so if the price rises by

r %

and expenditure is to stay the same, consumption must fall by

\frac{100 r}{100 + r} %

; if instead the price falls by

r %

, consumption may rise by

\frac{100 r}{100 - r} %

How to use it

1Read it as two multipliers whose product must be $1$ : $(1 + \frac{r}{100}) (1 - \frac{x}{100}) = 1$ .
2The reduction is measured on the old consumption, so it is always smaller than the rise itself.
3For a price fall the denominator becomes $100 - r$ and the change is an increase in consumption.

The price of rice rises by

25%

. By what percentage must a family cut its consumption to keep the rice bill unchanged?

Step-by-step

1
$\frac{100 \times 25}{100 + 25} = \frac{2500}{125} = 20$
Why: Apply the relation with $r = 25$ .
2
Check: $1.25 \times 0.80 = 1$ , so the bill is unchanged

Answer

20%

Watch out

A rise of

r %

is not cancelled by a cut of

r %

: the cut is taken on a larger base, so it is always the smaller percentage, and answering

r %

is the standard trap.

Builds on

Also known as

price rise consumption reduction, keep expenditure the same, 100r/(100+r), percentage reduction in consumption, expenditure unchanged