Up 20%, down 20%
Tonight's idea
A percent change is a multiplication. Up 20% means the new amount is 120% of the old: multiply by \(1.2\). Down 20% means 80%: multiply by \(0.8\). The number you multiply by is the multiplier.
Now do both, starting at 100. Up 20% is 120. The fall is then 20% of 120, not of 100: lose 24, land on 96.
With multipliers the whole trip is one product: \(1.2 \times 0.8 = 0.96\), a 4% loss. Down first, then up, also gives 0.96: the order does not matter.
Why always a loss? Write the change as a fraction \(p\) (so 20% is \(p = 0.2\)). Up then down is
\((1 + p)(1 - p) = 1 - p^2\), which is less than 1 whenever \(p \neq 0\). The loss is exactly \(p^2\): \(0.2^2 = 0.04\).
The same idea undoes a change. To undo \(\times 1.2\), divide by 1.2, which is multiplying by \(1/1.2 = 0.8333\ldots\), a fall of 16.7%, not 20%. The undo multiplier is the reciprocal, one over the original. A chain of changes is one multiplier: multiply them all. (Basics: grade 6, sheet 3.)
Watch one: a share price rises 30%, then falls 30%
Multipliers: \(1.3\) then \(0.7\). Together, \(1.3 \times 0.7 = 0.91\), so the price ends 9% below where it began. Check against the formula: \(p = 0.3\) and \(p^2 = 0.09\).
What rise would undo the 30% fall? Not 30%. The undo multiplier is \(1/0.7 = 1.4286\), a rise of about 42.9%:
\[0.7 \times \frac{1}{0.7} = 1.\]
A fall needs a bigger rise to repair it, because the rise acts on a smaller base.
Watch repeated multipliers run away from repeated additions: mathdetours.com/demos/growth.html.
Your turn
- A jacket costs 120. The price goes up 25%, then the new price is cut by 25%. (a) Find the final price. (b) What single percent change takes the original price to the final one? Check it against \(p^2\).
- (a) A rent rose 20% to 54 a night. What was it before? (b) A town's population fell 15% to 1,700. What was it before? In each case, write the multiplier first and divide.
- A shop raises every price by 40%, then advertises "40% off everything". (a) What single multiplier connects the original price to the sale price, and is the sale price above or below the original? (b) What percentage discount, after the 40% rise, would bring prices back exactly to where they were?
- (a) A stock falls 50%. What percent rise gets it back to its starting price? (b) An amount goes up \(x\%\) and then down \(x\%\), and the overall result is a loss of exactly 1%. Find \(x\).
Stretch: "50% extra free"
One bottle says "50% extra free". Another says "33% off". Which is the better deal per millilitre, or are they the same? Then: what amount of "extra free" would equal a 20% discount per millilitre? Use reciprocals, not guesswork.
Answers
- (a) \(120 \times 1.25 = 150\), then \(150 \times 0.75 = 112.5\). (b) \(1.25 \times 0.75 = 0.9375\), a fall of 6.25%. And \(0.25^2 = 0.0625\), as the formula predicts.
- (a) Multiplier 1.2, so the old rent was \(54 / 1.2 = 45\). (b) Multiplier 0.85, so the old population was \(1700 / 0.85 = 2000\).
- (a) \(1.4 \times 0.6 = 0.84\): the sale price is 16% below the original. "40% off" after a 40% rise is a real discount of 16%. (b) The undo multiplier is \(1/1.4 = 0.714\ldots\), a discount of about 28.6%.
- (a) Multiplier 0.5; the undo multiplier is \(1/0.5 = 2\), a rise of 100%. Halve a price and you must double it to get back. (b) The loss is \(p^2\), so \(p^2 = 0.01\), \(p = 0.1\): \(x = 10\).
Stretch. "50% extra free" means you get 1.5 times the volume for the same money, so the price per millilitre is multiplied by \(1/1.5 = 2/3\): a discount of \(33\tfrac13\%\). The two deals are the same (the first is a hair better if the second really is a flat 33%). For a 20% discount the price per millilitre is multiplied by 0.8, so the volume is multiplied by \(1/0.8 = 1.25\): "25% extra free".