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Successive changes and reverse percent

Ratio, rates and money · level 6, lesson 2 of 7

A question to start

A share in a company is worth 100 dollars. On Monday its price jumps up 50%. On Tuesday it falls 50%. Up 50, down 50: back to 100 dollars?

Work it out. Monday: 50% of 100 is 50, so it is worth 150. Tuesday: 50% of 150 is 75, so it falls to \(150 - 75 = 75\) dollars. A quarter of the money is gone, after an "equal" rise and fall.

And it is not just big percents. Up 10% then down 10% also loses money, just less. Today you will see why it always loses, how to undo a change properly, and how to find a price before tax was added.

The idea

Changes in a row multiply

From lesson 1, each percent change is a multiplication by its multiplier. So two changes, one after the other, are two multiplications:

\[\text{new} = \text{old} \times m_1 \times m_2\]

These are called successive changes (changes that follow one another). The overall multiplier is \(m_1 \times m_2\), and you read the overall percent change from it.

Up 10% then down 10%:

100 110 99 × 1.10 × 0.90 overall × 0.99: down 1%

The overall multiplier is \(1.10 \times 0.90 = 0.99\), so the result is down 1%. The 10% rise was 10% of 100 (that is 10), but the 10% fall was 10% of 110 (that is 11). The fall is bigger because it is a percent of a bigger number.

Why "up then down by the same percent" always loses

Write the percent as a decimal \(x\) (so 10% is \(x = 0.1\)). Up then down is

\[(1 + x)(1 - x) = 1 - x + x - x^2 = 1 - x^2.\]

Since \(x^2\) is positive for any \(x\) that is not 0, the overall multiplier \(1 - x^2\) is always less than 1. You always end up below where you started, and the loss is exactly \(x^2\):

Up and down by Overall multiplier \(1 - x^2\) Overall change
10% \(1 - 0.01 = 0.99\) down 1%
20% \(1 - 0.04 = 0.96\) down 4%
50% \(1 - 0.25 = 0.75\) down 25%

And since \((1 - x)(1 + x)\) is the same product in the other order, down then up loses exactly the same.

Adding percents is wrong; multiplying multipliers is right

Two discounts of 30% and then 20% are not a 50% discount:

\[0.70 \times 0.80 = 0.56,\]

so you pay 56%: a 44% discount. Three yearly rises of 5% are not 15%:

\[1.05 \times 1.05 \times 1.05 = 1.05^3 \approx 1.1576,\]

a rise of about 15.76%, a bit more than 15%, because each rise is on a bigger amount. That extra is the seed of compound interest in lesson 4.

Undoing a change

To get back where you started, you must multiply by the reciprocal (1 divided by the multiplier), because \(m \times \dfrac{1}{m} = 1\).

After a 20% drop (\(\times 0.80\)), the recovery needs \(\times \dfrac{1}{0.80} = \times 1.25\), a 25% rise. The deeper the fall, the bigger the climb back:

Loss Multiplier Rise needed to recover
10% 0.90 \(\dfrac{1}{0.9} \approx 1.111\): about 11.1%
20% 0.80 \(\dfrac{1}{0.8} = 1.25\): 25%
25% 0.75 \(\dfrac{1}{0.75} \approx 1.333\): about 33.3%
50% 0.50 \(\dfrac{1}{0.5} = 2\): 100%
75% 0.25 \(\dfrac{1}{0.25} = 4\): 300%
90% 0.10 \(\dfrac{1}{0.1} = 10\): 900%

This is why investors fear big losses: lose half your money and you need to double what is left just to get back.

Reverse percent: finding the "before"

Often you know the amount after a change and want the amount before. Since

\[\text{before} \times m = \text{after}, \qquad \text{before} = \text{after} \div m.\]

Divide by the multiplier. A receipt shows a total of 50.85 dollars including 13% HST (Ontario's rate since 2010; check the current rate). The price before tax was \(50.85 \div 1.13 = 45\) dollars, and the tax was \(50.85 - 45 = 5.85\) dollars.

Why you cannot just take 13% off the total. The 13% was 13% of the price before tax, not 13% of 50.85. Taking 13% off 50.85 gives about 44.24 dollars, which is too small: check it, \(44.24 \times 1.13 \approx 49.99\), not 50.85. Taking a percent off the "after" amount always uses the wrong base.

Several changes, backwards

If a price went through several changes, divide by the overall multiplier. A price after 25% off and then 13% tax is 67.80 dollars. The overall multiplier is \(0.75 \times 1.13 = 0.8475\), and \(67.80 \div 0.8475 = 80\) dollars before.

Worked examples

1. Up 10%, down 10%. Overall: \(1.10 \times 0.90 = 0.99\). Down 1%.

2. The share price. Up 50% then down 50%: \(1.5 \times 0.5 = 0.75\). Down 25%: 100 dollars becomes 75.

3. Three changes. A price rises 10%, falls 10%, rises 10%.

  • \(1.10 \times 0.90 \times 1.10 = 1.089\). Overall up 8.9%, not 10%.

4. Recovering. A bike's value dropped 20% in a sale. By what percent must the sale price rise to return to the original?

  • \(\dfrac{1}{0.80} = 1.25\): a 25% rise.
  • Check with numbers: 100 down 20% is 80; 80 up 25% is 100. ✓

5. Before tax. A total of 50.85 dollars includes 13% HST.

  • \(50.85 \div 1.13 =\) 45 dollars before tax. Tax: \(50.85 - 45 = 5.85\) dollars.

6. Before a discount. A sale price of 68 dollars is after 15% off. Original price?

  • \(68 \div 0.85 =\) 80 dollars. Check: \(80 \times 0.85 = 68\). ✓

7. Two steps backwards. After 25% off and then 13% tax, you pay 67.80 dollars. Original price?

  • Overall multiplier \(0.75 \times 1.13 = 0.8475\).
  • \(67.80 \div 0.8475 =\) 80 dollars.

Try it

  1. A price goes up 20% and then down 20%. What is the overall percent change?
  2. A shop takes 30% off and then a further 20% off the sale price. What is the overall discount? Is it 50%?
  3. A rent rises 5% a year for three years. What is the total percent rise? Is it 15%?
  4. A restaurant bill including 13% HST is 101.70 dollars. What was it before tax?
  5. After a 25% raise, a salary is 50 000 dollars. What was it before?
  6. An investment loses 40% of its value. What percent gain does it need to get back to where it started?
  7. After a 20% discount and then 13% tax, a coat costs 90.40 dollars. What was the original price?
  8. Show that for any \(x\) between 0 and 1, going up by \(x\) (as a decimal) and then down by \(x\) leaves less than you started with. How much less, as a fraction of the start?
Answers
  1. \(1.20 \times 0.80 = 0.96\): down 4%.
  2. \(0.70 \times 0.80 = 0.56\): you pay 56%, an overall discount of 44%, not 50%. The second 20% is taken off the already reduced price.
  3. \(1.05^3 = 1.157625\): up about 15.76%, a little more than 15%, because each rise is 5% of a bigger rent.
  4. \(101.70 \div 1.13 = 90\) dollars.
  5. \(50\,000 \div 1.25 = 40\,000\) dollars. (Not \(50\,000 \times 0.75 = 37\,500\).)
  6. Multiplier after the loss: 0.60. To recover: \(\dfrac{1}{0.60} \approx 1.667\), a gain of about 66.7%.
  7. Overall multiplier \(0.80 \times 1.13 = 0.904\). \(90.40 \div 0.904 = 100\) dollars.
  8. The overall multiplier is \((1 + x)(1 - x) = 1 - x^2\). For \(x\) between 0 and 1, \(x^2\) is positive, so \(1 - x^2 < 1\): you end below the start, by exactly \(x^2\) of it. (For 10%, \(x^2 = 0.01\), which is 1%.)

Common mistake

Adding percents in a row. Up 10% and down 10% is not "no change", and 30% off then 20% off is not 50% off. Multiply the multipliers.

Reversing with the wrong base. To remove a 13% tax from a total, divide by 1.13. Taking 13% of the total and subtracting it uses the after-tax amount as the base, and gives too little.

Where it leads