Math 7 · 7.RP.A.3

The multiplier for a percent change

A herd grows by twenty percent. Is there one number you can multiply by to get its new size, and if you only know the new size, can you get back to the old one?

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The multiplier for a percent change

Two percent changes one after the other

Why this exists

A herd of 50 goats grows by 20\%. To find its new size the long way, you find 20\% of 50, which is 10, and add it on: 50 + 10 = 60 goats. That is two steps, and it works.

Now turn the question round. A museum had 6{,}250 visitors this month, which is 25\% more than last month. How many came last month? The long way breaks down here. The 25\% was 25\% of last month’s number, which is the very number you do not know, so you cannot work out the increase and take it off.

Both questions become one step once you notice that a percent change is a multiplication. Growing by 20\% leaves you with 120\% of what you had, and 120\% of an amount is 1.2 times it. So the herd’s new size is 50 \times 1.2 = 60, and the museum’s two months are tied by “last month \times 1.25 = this month”. Going forward you multiply by 1.25; going back you divide by it. That single number is the multiplier for the change.

This lesson builds on percent increase and decrease, which measures a change against the original amount, and on finding the whole from a part and a percent, which is the backward question in disguise. It is taught over two days. Part 1 is the multiplier itself and the new amount it gives. Part 2 runs it backwards to find the original amount.

Definition

Write the percent of the change as a decimal and call it r: a change of 20\% has r = 0.2, and a change of 4.5\% has r = 0.045.

The multiplier for a percent change is the single number that the original amount is multiplied by to give the new amount:

for an increase of r, the multiplier is 1 + r;
for a decrease of r, the multiplier is 1 - r.

So a 20\% increase has multiplier 1.2, a 20\% decrease has multiplier 0.8, and a 150\% increase has multiplier 2.5. The multiplier is the new amount written as a fraction of the original: 1.2 is 120\%, 0.8 is 80\%.

Other books call it the growth factor for an increase and the decay factor for a decrease.

Intuition

Draw the original amount as a bar and call the whole bar 100\%. In the top bar below, a herd of 50 is cut into five pieces of 10, each 20\% of the herd. A 20\% increase sticks one more piece of the same size on the end. The new bar is six pieces, 120\% of the old one, so it is the old one times 1.2.

Figure 1A rise of 20\% leaves 120\% of the original, so it multiplies by 1.2. A fall of 25\% leaves 75\% of the original, so it multiplies by 0.75.

The bottom bar is a count of 80 fish in a stream, cut into four pieces of 20, each 25\%. A 25\% decrease takes one piece away. Three pieces are left, 75\% of the original, so the new count is the old one times 0.75.

In both bars the change is a piece measured against the original bar, which is why the 1 in 1 + r and 1 - r stands for the original amount itself. An increase adds pieces to that 1; a decrease takes pieces off it.

Once the change is a multiplication, undoing it is division. Multiplying by 1.25 carries the museum’s last month to this month; dividing by 1.25 carries it back.

Figure 2Multiplying by the multiplier goes forward from the original to the new amount. Dividing by the same number comes back.

Results

For an original amount and a change of r written as a decimal:

\begin{aligned} \text{increase by } r: \quad \text{new} &= (1 + r) \times \text{original} \\ \text{decrease by } r: \quad \text{new} &= (1 - r) \times \text{original} \end{aligned}

Calling the multiplier m, so m = 1 + r or m = 1 - r, both directions are one step:

\text{new} = m \times \text{original} \qquad \text{original} = \frac{\text{new}}{m}

The multiplier also says which way the amount moved and by how much:

m > 1 is an increase of (m - 1) \times 100\%;
m < 1 is a decrease of (1 - m) \times 100\%;
m = 1 is no change at all.

When both amounts are known, the multiplier is their quotient, \displaystyle m = \frac{\text{new}}{\text{original}}.

Why it is true

Forward. An increase of r adds r times the original to the original:

\text{new} = \text{original} + r \times \text{original}

The original is 1 \times \text{original}, so both terms have the factor “original”, and the The distributive property gathers them into one product:

1 \times \text{original} + r \times \text{original} = (1 + r) \times \text{original}

A decrease takes r times the original away, and the distributive property gathers 1 \times \text{original} - r \times \text{original} into (1 - r) \times \text{original} in the same way.

Backward. The forward rule is an equation, \text{new} = m \times \text{original}. When the new amount is known and the original is not, divide both sides by m, which is allowed because m is not 0:

\text{original} = \frac{\text{new}}{m}

Division undoes multiplication by the same number, so this is the only original that the change could have started from.

The same fact read as a percent sentence: after a 25\% rise, the new amount is 125\% of the original. That is “a part is 125\% of the whole, find the whole”, the question of Finding the whole from a part and a percent, with a part that happens to be bigger than the whole. As percents above 100 allows, the part may be the larger number. Its answer is the part divided by 1.25, the same division.

Worked examples

Examples 1 to 4 go with Part 1, the multiplier and the new amount. Examples 5 to 7 go with Part 2, finding the original amount.

Example 1

An ingredient cost going up

A bakery pays 18.40 dollars for a sack of flour. The next month the price is 15\% higher. What does a sack cost now?

Solution

An increase of 15\% has r = 0.15, so the multiplier is 1 + 0.15 = 1.15. The new price is the old price times the multiplier:

18.40 \times 1.15 = 21.16

A sack now costs 21.16 dollars.

Check the long way: 15\% of 18.40 is 0.15 \times 18.40 = 2.76, and 18.40 + 2.76 = 21.16. The two routes agree because, by the distributive property, they are the same calculation.

Example 2

Fish counted in a stream

Volunteers counted 260 trout in a stream. A year later the count was 35\% lower. How many trout did they count the second year?

Solution

A decrease of 35\% leaves 100\% - 35\% = 65\% of the fish, so the multiplier is 1 - 0.35 = 0.65:

260 \times 0.65 = 169

They counted 169 trout. The multiplier is less than 1, so the answer had to be smaller than 260, and it is.

Check: 35\% of 260 is 91, and 260 - 91 = 169.

Example 3

Writing and reading multipliers

(a) Write the multiplier for a rise of 7\%, a fall of 7\%, a rise of 150\% and a fall of 0.5\%.

(b) Say what change each multiplier stands for: 1.6, 1.06, 0.6 and 0.06.

Solution

(a) Write each percent as a decimal, then add it to 1 for a rise or take it from 1 for a fall.

Change	r	Multiplier
rise of 7\%	0.07	1.07
fall of 7\%	0.07	0.93
rise of 150\%	1.5	2.5
fall of 0.5\%	0.005	0.995

A rise of 150\% adds one and a half times the original to the original, leaving two and a half times it, so the multiplier is more than 2.

(b) A multiplier above 1 is an increase of m - 1; one below 1 is a decrease of 1 - m.

1.6: an increase of 0.6 = 60\%.
1.06: an increase of 0.06 = 6\%.
0.6: a decrease of 1 - 0.6 = 0.4 = 40\%.
0.06: a decrease of 1 - 0.06 = 0.94 = 94\%. Only 6\% of the amount is left.

The digits 6 appear in all four, but their place matters: 1.6 and 1.06 differ by a factor of ten in the size of the increase.

Example 4

The multiplier from two counts

A farmer’s flock had 150 sheep in the spring and 192 a year later. A second flock went from 150 sheep to 117. Find the multiplier for each flock, and the percent change it stands for.

Solution

The rule \text{new} = m \times \text{original} means the multiplier is the new count divided by the original count.

First flock:

m = \frac{192}{150} = 1.28

That is more than 1, an increase of 0.28 = 28\%.

Second flock:

m = \frac{117}{150} = 0.78

That is less than 1, a decrease of 1 - 0.78 = 0.22 = 22\%.

This is the same answer Percent increase and decrease gives by dividing the change by the original: \displaystyle \frac{192 - 150}{150} = \frac{42}{150} = 0.28. The new count is the original plus the change, so \displaystyle \frac{\text{new}}{\text{original}} = 1 + \frac{\text{change}}{\text{original}}, and the two routes always differ by exactly that 1.

Example 5

Museum visitors before a rise

A museum had 6{,}250 visitors this month, which is 25\% more than last month. How many visitors came last month?

Solution

Last month is the original, so

\text{last month} \times 1.25 = 6{,}250

Divide both sides by the multiplier:

\text{last month} = \frac{6{,}250}{1.25} = 5{,}000

5{,}000 visitors came last month.

Check: 5{,}000 \times 1.25 = 6{,}250. It also shows why taking 25\% off this month’s number does not work: 25\% of 6{,}250 is 1{,}562.5, and 6{,}250 - 1{,}562.5 = 4{,}687.5, which is not even a whole number of people. The rise was 25\% of 5{,}000, which is 1{,}250, not 25\% of 6{,}250.

Example 6

A herd before a fall

After a dry summer, a herd of red deer on an estate is 20\% smaller than it was, with 176 animals. How many deer were there before?

Solution

A 20\% decrease has multiplier 1 - 0.2 = 0.8, so

\text{before} \times 0.8 = 176

\text{before} = \frac{176}{0.8} = 220

There were 220 deer. Dividing by a number less than 1 gives a bigger answer, which is right: the herd shrank, so it used to be bigger.

Check: 20\% of 220 is 44, and 220 - 44 = 176.

Example 7

Same number now, different numbers before

This year a zoo had 52{,}800 visitors, 10\% more than last year. The museum next door also had 52{,}800 visitors this year, but that was 12\% fewer than last year. Which had more visitors last year, and by how many?

Solution

Each place’s last year is the original for its own change, so divide this year’s number by that place’s multiplier.

The zoo rose by 10\%, multiplier 1.1:

\frac{52{,}800}{1.1} = 48{,}000

The museum fell by 12\%, multiplier 1 - 0.12 = 0.88:

\frac{52{,}800}{0.88} = 60{,}000

Last year the museum had more visitors, by 60{,}000 - 48{,}000 = 12{,}000. The two places met at the same number this year from opposite sides: the zoo climbing up to it, the museum falling down to it. Check: 48{,}000 \times 1.1 = 52{,}800 and 60{,}000 \times 0.88 = 52{,}800.

Boundaries

The percent is a percent of the original. The multiplier carries the original to the new amount. Going back means dividing by the same multiplier; taking the same percent off the new amount is a different calculation and gives a different answer, as Example 5 shows.
A decrease of 100\% has multiplier 0. Every original goes to 0, so a new amount of 0 says nothing about where it started, and \displaystyle \frac{0}{0} is not defined. An amount of things cannot fall by more than 100\%, so a multiplier for a count is never negative.
Counts and rounded percents. When the percent has been rounded, the multiplier can give a number of animals or visitors that is not whole. Then the true count is the nearest sensible whole number, and the answer is given as “about”.
One change at a time. The multiplier describes one change measured against one original. Two changes in a row multiply their multipliers, and the percents do not simply add; that is Two percent changes one after the other.

Where this goes next

Multipliers make a run of changes easy to follow, because two changes in a row are two multiplications, and that is the subject of two percent changes one after the other.

When the same multiplier is applied again and again, once a year or once a week, the amount after n steps is the original times m^n. That repeated multiplier is the growth or decay factor, and it drives Exponential growth and decay.

Many everyday percent questions are this one with a new name. A price after a discount, with tax added, or after a markup is the old price times one multiplier, and finding a price before tax or before a sale is a division by it; see discounts, sales tax, tips and commission.

Exercises

Part 1: writing a multiplier

Write the multiplier for each change.

An increase of 30\%.
A decrease of 30\%.
An increase of 4.5\%.
A decrease of 2\%.
An increase of 200\%.

Part 1: reading a multiplier

Say whether each multiplier is an increase or a decrease, and by what percent.

1.18
0.85
1.03
2.6

Part 1: the new amount

A herd of 45 goats grows by 40\%. How many goats are in the herd now?
A museum had 12{,}600 visitors in March. In April it had 5\% fewer. How many visitors came in April?
A bakery pays 2.40 dollars for a tray of eggs. The price rises by 35\%. What does a tray cost now?
Volunteers counted 360 fish in a stream one year and 15\% fewer the next. How many did they count the next year?
A zoo had 2{,}800 visitors on a rainy Sunday. The next Sunday was sunny and attendance was 125\% higher. How many visitors came on the sunny Sunday?
A stream’s fish count goes from 400 to 332. Find the multiplier, and the percent change it stands for.

Part 2: the original amount

After an increase of 20\%, an amount is 72. What was it before?
After a decrease of 40\%, an amount is 54. What was it before?
A flock of hens grew by 25\% and now has 95 birds. How many hens were in the flock before?
After a 6\% rise, a bakery pays 31.80 dollars for a sack of sugar. What did a sack cost before the rise?
The number of fish counted in a stream fell by 12\% to 264. How many were counted the year before?
A zoo’s attendance for a month rose by 8\% to 19{,}440. What was it the month before?
A museum reports 4{,}000 visitors this month, about 30\% fewer than last month. Estimate last month’s visitors to the nearest whole number.

Reasoning

Over a dry year a herd of goats shrank by 25\%, to 120 goats. Coralie finds the size before by adding 25\% of 120 back on to 120, and gets 150. Find the size before, and explain why her method gives a different number.
Explain why the multiplier for a decrease of less than 100\% is always between 0 and 1. What change does a multiplier of 1 describe, and what does a multiplier of 0 describe?
Esperanza notices that dividing by 1.25 gives the same answer as multiplying by 0.8. Show that this is true. Then explain what it says: after a 25\% rise, by what percent is the original smaller than the new amount?

Answers

1.3, since 1 + 0.3 = 1.3.
0.7, since 1 - 0.3 = 0.7.
1.045, since 4.5\% = 0.045 and 1 + 0.045 = 1.045.
0.98, since 1 - 0.02 = 0.98.
3, since 200\% = 2 and 1 + 2 = 3. Growing by 200\% triples the amount.
An increase of 18\%, since 1.18 - 1 = 0.18.
A decrease of 15\%, since 1 - 0.85 = 0.15.
An increase of 3\%, since 1.03 - 1 = 0.03.
An increase of 160\%, since 2.6 - 1 = 1.6.
63 goats. The multiplier is 1.4, and 45 \times 1.4 = 63.
11{,}970 visitors. The multiplier is 1 - 0.05 = 0.95, and 12{,}600 \times 0.95 = 11{,}970.
3.24 dollars. The multiplier is 1.35, and 2.40 \times 1.35 = 3.24.
306 fish. The multiplier is 0.85, and 360 \times 0.85 = 306.
6{,}300 visitors. The multiplier is 1 + 1.25 = 2.25, and 2{,}800 \times 2.25 = 6{,}300.
The multiplier is \displaystyle \frac{332}{400} = 0.83, a decrease of 1 - 0.83 = 0.17 = 17\%.
60, since \displaystyle \frac{72}{1.2} = 60. Check: 60 \times 1.2 = 72.
90, since the multiplier is 0.6 and \displaystyle \frac{54}{0.6} = 90. Check: 90 \times 0.6 = 54.
76 hens, since \displaystyle \frac{95}{1.25} = 76. Check: 76 \times 1.25 = 95.
30 dollars, since \displaystyle \frac{31.80}{1.06} = 30. Check: 30 \times 1.06 = 31.80.
300 fish, since the multiplier is 0.88 and \displaystyle \frac{264}{0.88} = 300. Check: 300 \times 0.88 = 264.
18{,}000, since \displaystyle \frac{19{,}440}{1.08} = 18{,}000. Check: 18{,}000 \times 1.08 = 19{,}440.
About 5{,}714 visitors. The multiplier is 0.7, and \displaystyle \frac{4{,}000}{0.7} = 5{,}714.28\ldots, which rounds to 5{,}714. The answer is not whole because the 30\% was itself rounded.
160 goats, since the multiplier is 0.75 and \displaystyle \frac{120}{0.75} = 160. Check: 160 \times 0.75 = 120. The 25\% fall was 25\% of the old size, 160, which is 40 goats. Coralie took 25\% of the new size, 120, which is only 30 goats, so she put back too few. Her calculation is 120 \times 1.25 = 150, and the change is undone by dividing by its own multiplier, 0.75, not by multiplying by 1.25.
A decrease of r has multiplier 1 - r. A decrease of less than 100\% has r between 0 and 1, so 1 - r is less than 1 (something is taken off) and more than 0 (not everything is). A multiplier of 1 leaves every amount as it was, no change. A multiplier of 0 sends every amount to 0, a decrease of 100\%.
\displaystyle \frac{1}{1.25} = \frac{100}{125} = \frac{4}{5} = 0.8, so dividing by 1.25 is multiplying by 0.8. So after a 25\% rise the original is 0.8 = 80\% of the new amount, which is 20\% smaller than the new amount. A rise of 25\% measured from the original is a gap of only 20\% measured from the new amount, because the new amount is the bigger whole.