Percentages · Road to 800

Unit 1: Increases, decreases and successive changes

Key points

An increase of $20\%$ multiplies by $1.20$ . A decrease of $20\%$ multiplies by $0.80$ .
In general, the multiplier is $1 + r$ for an increase and $1 - r$ for a decrease, with $r$ as a decimal.
For several changes in a row, multiply all the multipliers. Never add the percents.
‘ $30\%$ of’ and ‘ $30\%$ more than’ are different: the first multiplies by $0.30$ , the second by $1.30$ .

See it solved

Example

A jacket costs $\$60$ . Its price is raised by $20\%$ . Later, the new price is cut by $25\%$ . What is the final price of the jacket?

Worked by hand

The plan

Convert each percent to a multiplier
Determine the final value

Desmos: Desmos does not choose the multipliers for you. It only keeps the chain on one row, as a check.

Step 1: Convert each percent to a multiplier
- Up $20\%$ : $1 + 0.20 = 1.20$ .
- Down $25\%$ : $1 - 0.25 = 0.75$ .
Step 2: Determine the final value
- After the rise: $60 \cdot 1.20 = 72$ dollars.
- After the cut: $72 \cdot 0.75 = 54$ dollars.
- The final price is $\$54$ .

The correct answer is

\$54

.

The same example in Desmos

new graph
Desmos shows: $54$ , at the right of the row: the start times both multipliers.

Desmos expression list: 60·1.2·0.75 — Type

Made with Desmos

Open Desmos

Your turn

A town has $5{,}000$ people. Its population grows by $10\%$ one year and by $10\%$ again the next year. What is the population after the two years?

Show the answerHide the answer

Answer

The correct answer is $6{,}050$ .

Step 1: Convert each percent to a multiplier
- Up $10\%$ : $1 + 0.10 = 1.10$ . This happens twice.
Step 2: Determine the final value
- After the first year: $5{,}000 \cdot 1.10$ $= 5{,}500$ .
- After the second year: $5{,}500 \cdot 1.10$ $= 6{,}050$ .

Traps

Adding the percents. A $20\%$ rise and a $25\%$ cut are not a $5\%$ cut. $60 \cdot 0.95 = 57$ is a wrong answer, because the $25\%$ is taken from the new price, $\$72$ .
Doubling the percent for two equal changes. Two $10\%$ rises are not one $20\%$ rise. $5{,}000 \cdot 1.20$ $= 6{,}000$ misses the growth on the extra $500$ people.

Practice this lesson56 questions

Unit 2: Finding the original number

Key points

After a change, final value = original value $\times$ multiplier.
So original value = final value $\div$ multiplier.
Tax and tips are increases: an $8\%$ tax multiplies by $1.08$ .
Do not take the percent of the final value. The percent was taken of the original.

See it solved

Example

After a $20\%$ discount, a bike costs $\$240$ . What was the price of the bike before the discount?

Worked by hand

The plan

Convert percent to a multiplier
Determine the original value

Desmos: Desmos does not help you choose the multiplier. Once the equation is written, typing it is only a check.

Step 1: Convert percent to a multiplier
- A $20\%$ discount: $1 - 0.20 = 0.80$ .
Step 2: Determine the original value
- Original $\times\ 0.80 = 240$ .
- Original $= \dfrac{240}{0.80} = 300$ dollars.
- Check: $20\%$ of $300$ is $0.20 \cdot 300 = 60$ , and $300 - 60 = 240$ . It matches.

The correct answer is

\$300

.

The same example in Desmos

new graph
Desmos shows: A vertical line, with $x$ for the original price.
zoom out, then click where the line meets the x-axis
Desmos shows: $(300, 0)$ : the original price was $\$300$ .

Desmos graph of 0.8x=240, with (300, 0) labelled — Look

Made with Desmos

Open Desmos

Your turn

A meal costs $\$54$ , which includes an $8\%$ tax. What was the price of the meal before the tax?

Show the answerHide the answer

Answer

The correct answer is $\$50$ .

Step 1: Convert percent to a multiplier
- An $8\%$ tax is an increase: $1 + 0.08 = 1.08$ .
Step 2: Determine the original value
- Original $\times\ 1.08 = 54$ .
- Original $= \dfrac{54}{1.08} = 50$ dollars.
- Check: $8\%$ of $50$ is $0.08 \cdot 50 = 4$ , and $50 + 4 = 54$ . It matches.

Traps

Taking the percent of the final price. $20\%$ of $240$ is $48$ , and $240 + 48 = 288$ . But the discount was $20\%$ of the original price, not of $240$ . $\$288$ is a wrong answer.
Multiplying instead of dividing. $240 \cdot 0.80 = 192$ takes the discount off a second time, a wrong answer.

Practice this lesson16 questions

Unit 3: Percent of, and what percent

Key points

Percent means ‘out of $100$ ’: $p\% = \dfrac{p}{100}$ .
$p\%$ of $n$ is $\dfrac{p}{100} \cdot n$ .
‘ $x$ is what percent of $y$ ?’ is $\dfrac{x}{y} \cdot 100$ . The number after ‘of’ goes on the bottom.
For a percent of a percent, multiply the two.

See it solved

Example

A school has $640$ students. $45\%$ of them ride a bus to school. How many students ride a bus to school?

Worked by hand

The plan

Convert percent to a decimal
Determine the part

Desmos: Desmos is quicker here: type $45\%$ and then $640$ . Desmos writes the word ‘of’ for you.

Step 1: Convert percent to a decimal
- $45\%$ $= \dfrac{45}{100}$ $= 0.45$ .
Step 2: Determine the part
- $45\%$ of $640$ is $0.45 \cdot 640$ .
- $45 \cdot 640$ $= 28{,}800$ , and $\dfrac{28{,}800}{100} = 288$ .
- So $288$ students ride a bus.

The correct answer is

288

.

The same example in Desmos

new graph
Keys: 45%640
Desmos shows: $288$ , at the right of the row.

Desmos expression list: 45%operatorname{of}640 — Type

Made with Desmos

Open Desmos

Your turn

A class has $25$ students. $8$ of them play an instrument. What percent of the class plays an instrument?

Show the answerHide the answer

Answer

The correct answer is $32\%$ .

Step 1: Determine the part and the whole
- The part is $8$ . The whole, the number after ‘of’, is $25$ .
Step 2: Determine the percent
- $\dfrac{8}{25} = 0.32$ .
- $0.32 \cdot 100 = 32$ . So $32\%$ of the class plays an instrument.

Traps

Dividing the wrong way round. $\dfrac{25}{8} \cdot 100 = 312.5$ . A part cannot be more than $100\%$ of its whole, so $312.5\%$ is a wrong answer.
Using the percent as a whole number. $45 \cdot 640$ $= 28{,}800$ is far more than the whole school. Divide by $100$ : $45\%$ is $0.45$ .

Practice this lesson32 questions

Unit 4: Percent change

Key points

Percent change $= \dfrac{\text{new} - \text{old}}{\text{old}} \cdot 100$ .
Always divide by the old value: the one you started from.
A positive result is an increase. A negative result is a decrease.

See it solved

Example

A plant grows from a height of $40$ centimeters to a height of $52$ centimeters. By what percent did its height increase?

Worked by hand

The plan

Determine the old value and new value
Determine the percent change

Desmos: Desmos does not tell you which value is the old one. It can only keep the calculation on one row, as a check.

Step 1: Determine the old value and new value
- The old value is $40$ . The new value is $52$ .
Step 2: Determine the percent change
- New minus old: $52 - 40 = 12$ .
- Divide by the old value: $\dfrac{12}{40} = 0.30$ .
- $0.30 \cdot 100 = 30$ . The height increased by $30\%$ .

The correct answer is

30\%

.

The same example in Desmos

new graph
Keys: (52-40)/40 → *100
Desmos shows: $30$ , at the right of the row.

Desmos expression list: frac{52-40}{40}·100 — Type

Made with Desmos

Open Desmos

Your turn

The price of a phone falls from $\$250$ to $\$200$ . By what percent did the price decrease?

Show the answerHide the answer

Answer

The correct answer is $20\%$ .

Step 1: Determine the old value and new value
- The old value is $250$ . The new value is $200$ .
Step 2: Determine the percent change
- New minus old: $200 - 250 = -50$ .
- Divide by the old value: $\dfrac{-50}{250} = -0.20$ .
- $-0.20 \cdot 100$ $= -20$ . The result is negative, so the price decreased by $20\%$ .

Traps

Dividing by the new value. $\dfrac{12}{52}$ is about $0.23$ , which gives about $23\%$ , a wrong answer. Divide by the old value, $40$ .
Reporting the change as the percent. The height went up by $12$ centimeters, but that is not $12\%$ . Divide by the old value first.

Practice this lesson12 questions

Unit 5: Mixtures and concentrations

Key points

The amount of the pure substance is the volume times the concentration: $20\%$ of $50$ liters is $10$ liters.
When two solutions are mixed, the amounts of pure substance add, and the volumes add.
Write: amount in the first + amount in the second = amount in the mix.
The concentration of the mix is always between the two starting concentrations.

See it solved

Example

How many liters of a $50\%$ acid solution must be added to $30$ liters of a $20\%$ acid solution to make a $30\%$ acid solution?

Worked by hand

The plan

Turn the words into an equation
Convert each percent solution to an amount
Solve the equation

Desmos: Writing the equation is the hard part, and it is done by hand. Desmos then solves it as a check: type it and click the vertical line.

Step 1: Turn the words into an equation
- Let $x$ be the liters of the $50\%$ solution.
- Acid in the first solution + acid that is added = acid in the mix.
Step 2: Convert each percent solution to an amount
- The first solution holds $0.20 \cdot 30 = 6$ liters of acid.
- The added solution holds $0.50x$ liters of acid.
- The mix has $30 + x$ liters, and $30\%$ of it is acid: $0.30(30 + x)$ liters.
- So $6 + 0.50x$ $= 0.30(30 + x)$ .
Step 3: Solve the equation
- $0.30 \cdot 30 = 9$ , so $6 + 0.50x$ $= 9 + 0.30x$ .
- Subtract $0.30x$ from both sides: $6 + 0.20x = 9$ .
- Subtract $6$ from both sides: $0.20x = 3$ .
- $x = \dfrac{3}{0.20} = 15$ liters.
- Check: the acid is $6 + 0.50 \cdot 15$ $= 6 + 7.5$ $= 13.5$ liters out of $30 + 15 = 45$ liters, and $\dfrac{13.5}{45} = 0.30$ .

The correct answer is

15

.

The same example in Desmos

new graph
Desmos shows: A vertical line, with $x$ for the liters added.
zoom out, then click where the line meets the x-axis
Desmos shows: $(15, 0)$ : add $15$ liters.

Desmos expression list: 0.2·30+0.5x=0.3(30+x) — Type

Desmos graph of 0.2·30+0.5x=0.3(30+x), with (15, 0) labelled — Look

Made with Desmos

Open Desmos

Your turn

$40$ grams of a $15\%$ salt solution are mixed with $60$ grams of a $40\%$ salt solution. What percent of the mix is salt?

Show the answerHide the answer

Answer

The correct answer is $30\%$ .

Step 1: Convert each percent solution to an amount
- The first solution holds $0.15 \cdot 40 = 6$ grams of salt.
- The second solution holds $0.40 \cdot 60 = 24$ grams of salt.
Step 2: Determine the percent
- Salt in all: $6 + 24 = 30$ grams.
- Mix in all: $40 + 60 = 100$ grams.
- $\dfrac{30}{100} \cdot 100 = 30$ . The mix is $30\%$ salt.

Traps

Averaging the two percents. $\dfrac{15 + 40}{2} = 27.5$ ignores that there is more of the $40\%$ solution. $27.5\%$ is a wrong answer.
Forgetting that the volume grows. The mix has $30 + x$ liters, not $30$ . Using $0.30 \cdot 30$ for the mix gives $6 + 0.50x = 9$ and $x = 6$ , a wrong answer.

Practice this lesson16 questions

1Unit 1: Increases, decreases and successive changes

Key points

See it solved

Worked by hand

Step 1: Convert each percent to a multiplier

Step 2: Determine the final value

The same example in Desmos

Your turn

Step 1: Convert each percent to a multiplier

Step 2: Determine the final value

Traps

2Unit 2: Finding the original number

Key points

See it solved

Worked by hand

Step 1: Convert percent to a multiplier

Step 2: Determine the original value

The same example in Desmos

Your turn

Step 1: Convert percent to a multiplier

Step 2: Determine the original value

Traps

3Unit 3: Percent of, and what percent

Key points

See it solved

Worked by hand

Step 1: Convert percent to a decimal

Step 2: Determine the part

The same example in Desmos

Your turn

Step 1: Determine the part and the whole

Step 2: Determine the percent

Traps

4Unit 4: Percent change

Key points

See it solved

Worked by hand

Step 1: Determine the old value and new value

Step 2: Determine the percent change

The same example in Desmos

Your turn

Step 1: Determine the old value and new value

Step 2: Determine the percent change

Traps

5Unit 5: Mixtures and concentrations

Key points

See it solved

Worked by hand

Step 1: Turn the words into an equation

Step 2: Convert each percent solution to an amount

Step 3: Solve the equation

The same example in Desmos

Your turn

Step 1: Convert each percent solution to an amount

Step 2: Determine the percent

Traps

Practice by lesson

Mixed practice: every lesson

Easy set 1

800-level question

What tripped you up

Unit 1: Increases, decreases and successive changes

Unit 2: Finding the original number

Unit 3: Percent of, and what percent

Unit 4: Percent change

Unit 5: Mixtures and concentrations