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Multipliers

Ratio, rates and money · level 6, lesson 1 of 7

A question to start

A jacket costs 120 dollars. It is on sale for 20% off, and then 13% sales tax is added at the till. Your friend says: "You should ask them to add the tax first and take the 20% off after, because then the 20% comes off a bigger number and you save more."

Is your friend right? Try it both ways before you read on.

Both ways give exactly 108.48 dollars. That is not luck. In this lesson you will learn the one tool that makes this obvious, and that solves almost every percent problem in a single step: the multiplier.

The idea

From two steps to one

In level 4 you increased an amount by a percent in two steps: find the percent, then add it on. To increase 250 by 8%:

  • 8% of 250 is \(0.08 \times 250 = 20\).
  • \(250 + 20 = 270\).

Look at what happened in algebra. Starting from an amount \(x\), the new amount is

\[x + 0.08x = (1 + 0.08)x = 1.08x.\]

(This is the distributive law backwards: taking out the common factor \(x\).) So increasing by 8% is the same as multiplying by 1.08, in one step: \(1.08 \times 250 = 270\).

The number 1.08 is called the multiplier (or growth factor, or decay factor when it is less than 1). The same algebra works for any percent. With \(p\) as the percent:

\[\text{increase by } p\%: \quad \times \left(1 + \frac{p}{100}\right), \qquad \text{decrease by } p\%: \quad \times \left(1 - \frac{p}{100}\right).\]

In words: you keep the whole amount (that is the 1, or 100%) and add or take away the percent.

100% 108% = × 1.08 85% −15% 85% = × 0.85

A table of multipliers

Change Multiplier Why
up 8% 1.08 \(100\% + 8\% = 108\%\)
up 13% (Ontario HST; check the current rate) 1.13 \(100\% + 13\%\)
up 0.5% 1.005 \(100\% + 0.5\%\)
up 100% (doubled) 2 \(100\% + 100\%\)
up 150% 2.5 \(100\% + 150\%\)
no change 1 \(100\%\)
down 15% 0.85 \(100\% - 15\%\)
down 20% 0.80 \(100\% - 20\%\)
down 62% 0.38 \(100\% - 62\%\)
down 100% (all gone) 0 \(100\% - 100\%\)

Three things to notice:

  • A multiplier bigger than 1 means growth; smaller than 1 (but not negative) means shrinking; exactly 1 means no change.
  • Small percents need care with decimals. Up 5% is 1.05, not 1.5 (which is up 50%). Up 0.5% is 1.005.
  • "Percent of" is a multiplier too. 35% of an amount is \(\times 0.35\). "Down 65%" is also \(\times 0.35\): they describe the same thing, what is left.

Reading a multiplier backwards

Given a multiplier, you can read off the change: subtract 1 and turn it into a percent.

  • \(\times 1.4\): \(1.4 - 1 = 0.4\), so up 40%.
  • \(\times 0.93\): \(0.93 - 1 = -0.07\), so down 7%.
  • \(\times 3\): \(3 - 1 = 2\), so up 200% (tripling is a 200% increase, as in level 5).

And given an old and a new amount, the multiplier is just

\[\text{multiplier} = \frac{\text{new}}{\text{old}}.\]

From 64 to 80: \(80 \div 64 = 1.25\), so up 25%. This is the percent change from level 5 in a new outfit: \(\dfrac{\text{new} - \text{old}}{\text{old}} = \dfrac{\text{new}}{\text{old}} - 1\).

On Mars, gravity is about 38% of Earth's, so anything weighs \(\times 0.38\) there: \(0.38 - 1 = -0.62\), a drop of about 62%.

Why order does not matter

Now the jacket. Taking 20% off is \(\times 0.80\). Adding 13% tax is \(\times 1.13\). So

  • discount then tax: \(120 \times 0.80 \times 1.13\);
  • tax then discount: \(120 \times 1.13 \times 0.80\).

Multiplication can be done in any order (it is commutative: \(a \times b = b \times a\)). So both give the same answer, \(120 \times 0.904 = 108.48\) dollars. Your friend's idea, "the 20% comes off a bigger number", is true, but the tax was also added to a bigger number in the other order, and the two effects cancel exactly.

The combined multiplier \(0.80 \times 1.13 = 0.904\) says the whole deal is a 9.6% saving on the original price. Changes in a row are the subject of the next lesson.

Why multipliers are worth it

  • One step instead of two.
  • Any direction: multiply to go forwards, divide to go backwards (the next lesson).
  • Chains: several changes are one multiplication.
  • Repeats: the same change many times is a power, like \(1.05^{10}\). That is compound interest (lesson 4).

Worked examples

1. Increase. Increase 250 by 8%.

  • Multiplier 1.08. \(1.08 \times 250 =\) 270.

2. Decrease. Decrease 60 by 15%.

  • Multiplier 0.85. \(0.85 \times 60 =\) 51.

3. From two amounts. A ticket price goes from 64 to 80 dollars. Find the multiplier and the percent change. Then do the same for 80 back to 64.

  • \(80 \div 64 = 1.25\): up 25%.
  • \(64 \div 80 = 0.80\): down 20%. (The level 5 effect again: the two multipliers are reciprocals, \(1.25 \times 0.80 = 1\).)

4. Discount and tax. The 120 dollar jacket, 20% off, plus 13% HST.

  • \(120 \times 0.80 \times 1.13 =\) 108.48 dollars, in either order.

5. A raise. A salary of 52 000 dollars a year goes up 3.5%.

  • Multiplier \(1 + 0.035 = 1.035\). \(52\,000 \times 1.035 =\) 53 820 dollars.

Try it

  1. Write the multiplier for: (a) up 6%; (b) down 6%; (c) up 150%; (d) down 0.5%; (e) up 0.5%.
  2. What percent change does each multiplier mean? (a) 1.07; (b) 0.93; (c) 0.6; (d) 3; (e) 1.005.
  3. Increase 340 by 15% in one step.
  4. Decrease 85 by 40% in one step.
  5. A price goes from 48 to 60 dollars. Find the multiplier and the percent change.
  6. A coat costs 150 dollars. It is 30% off, then 13% HST is added. What do you pay?
  7. A town of 12 500 people grows by 2% one year and shrinks by 2% the next. How many people now? Why is it not 12 500?
  8. Use algebra to show that taking \(p\%\) off an amount \(x\) leaves \(\left(1 - \dfrac{p}{100}\right)x\).
Answers
  1. (a) 1.06; (b) 0.94; (c) 2.5; (d) 0.995; (e) 1.005.
  2. (a) up 7%; (b) down 7%; (c) down 40%; (d) up 200%; (e) up 0.5%.
  3. \(1.15 \times 340 = 391\).
  4. \(0.60 \times 85 = 51\).
  5. \(60 \div 48 = 1.25\), so up 25%.
  6. \(150 \times 0.70 \times 1.13 = 118.65\) dollars.
  7. \(12\,500 \times 1.02 \times 0.98 = 12\,495\) people. The overall multiplier is \(1.02 \times 0.98 = 0.9996\), slightly less than 1, because the 2% drop is taken from a bigger number (12 750) than the 2% rise was (12 500). More on this in the next lesson.
  8. \(p\%\) of \(x\) is \(\dfrac{p}{100}x\). Taking it away leaves \(x - \dfrac{p}{100}x\). Both terms have the factor \(x\), so this is \(\left(1 - \dfrac{p}{100}\right)x\) (the distributive law).

Common mistake

Small percents with the wrong decimal. Up 5% is \(\times 1.05\), not \(\times 1.5\). Up 0.5% is \(\times 1.005\), not \(\times 1.05\). Say it as "100% plus 5% is 105%, which is 1.05".

Using the percent as the multiplier. For a 15% decrease, multiplying by 0.15 gives the amount taken off, not the amount left. The amount left is \(\times 0.85\).

Where it leads