Ratios, proportions and rates

Chapter 8 · 2 lessons · 172 questions

Unit 1: Ratios and proportions

Key points

A ratio compares two amounts. ‘ $3$ to $5$ ’ can be written $3:5$ or $\dfrac{3}{5}$ .
A part-to-part ratio compares two groups. A part-to-total ratio compares one group with everything.
A proportion says two ratios are equal. Keep the same thing on top on both sides.
To solve a proportion, cross-multiply.

See it solved

Example

A paint mix uses blue paint and yellow paint in the ratio $3$ to $5$ . How many liters of blue paint are needed to make $40$ liters of the mix?

Worked by hand

The plan

Determine the ratio
Set up a proportion

Desmos: Desmos does not help you set up the proportion. Once it is written, typing it is only a check.

Step 1: Determine the ratio
- Blue to yellow is $3$ to $5$ . That is part to part.
- The mix has $3 + 5 = 8$ parts in all.
- So blue is $3$ parts out of $8$ : blue to total is $\dfrac{3}{8}$ .
Step 2: Set up a proportion
- Put blue on top and the total on the bottom, on both sides: $\dfrac{3}{8} = \dfrac{b}{40}$ .
- Cross-multiply: $8 \cdot b = 3 \cdot 40$ .
- $3 \cdot 40 = 120$ , so $8b = 120$ .
- $b = \dfrac{120}{8} = 15$ liters.
- Check: yellow is $40 - 15 = 25$ liters, and $\dfrac{15}{25} = \dfrac{3}{5}$ .

The correct answer is

15

The same example in Desmos

new graph
Keys: 3/8 → =x/40
Desmos shows: A vertical line, with $x$ for the liters of blue paint.
zoom out, then click where the line meets the x-axis
Desmos shows: $(15, 0)$ : $15$ liters of blue paint.

Desmos expression list: frac{3}{8}=frac{x}{40} — Type

Desmos graph of frac{3}{8}=frac{x}{40}, with (15, 0) labelled — Look

Made with Desmos

Open Desmos

Your turn

On a map, $2$ centimeters stand for $15$ kilometers. Two towns are $7$ centimeters apart on the map. How many kilometers apart are the towns?

Show the answerHide the answer

Answer

The correct answer is $52.5$ .

Step 1: Determine the ratio
- $2$ centimeters to $15$ kilometers: $\dfrac{2}{15}$ , with centimeters on top.
Step 2: Set up a proportion
- Keep centimeters on top on both sides: $\dfrac{2}{15} = \dfrac{7}{d}$ .
- Cross-multiply: $2 \cdot d = 15 \cdot 7$ .
- $15 \cdot 7 = 105$ , so $2d = 105$ .
- $d = \dfrac{105}{2} = 52.5$ kilometers.

Traps

Using part to part as part to total. $\dfrac{3}{5} = \dfrac{b}{40}$ treats $5$ as the total. It gives $5b = 120$ and $b = 24$ , a wrong answer.
Flipping one side of the proportion. $\dfrac{3}{8} = \dfrac{40}{b}$ has blue on top on the left and the total on top on the right. It gives $3b = 320$ , so $b$ is about $106.7$ liters, more than the whole mix.

Practice this lesson48 questions

Unit 2: Rates, unit conversions, speed and work

Key points

A rate compares two different units: miles per hour, dollars per pound.
To change units, multiply by a conversion factor written as a fraction, so that the old unit cancels.
Distance = speed $\times$ time. Average speed = total distance $\div$ total time.
When two people or machines work together, add their rates, not their times.

See it solved

Example

A car travels at $90$ kilometers per hour. What is its speed in meters per second? ( $1$ kilometer $= 1{,}000$ meters)

Worked by hand

The plan

Use conversion factors

Desmos: Setting up the chain is done by hand. Desmos only keeps the whole chain on one row, as a check on the arithmetic.

Step 1: Use conversion factors
- Start with $90$ kilometers per hour. The answer must be in meters per second.
- Kilometers to meters: multiply by $1{,}000$ , because $1$ kilometer is $1{,}000$ meters.
- $90 \cdot 1{,}000$ $= 90{,}000$ meters per hour.
- Hours to seconds: $1$ hour is $60 \cdot 60 = 3{,}600$ seconds. Divide by $3{,}600$ .
- $\dfrac{90{,}000}{3{,}600}$ $= 25$ meters per second.

The correct answer is

25

The same example in Desmos

new graph
Keys: 90*1000/3600
Desmos shows: $25$ , at the right of the row: the whole chain at once.

Desmos expression list: 90·frac{1000}{3600} — Type

Made with Desmos

Open Desmos

Your turn

Pump A can fill a pool in $6$ hours. Pump B can fill the same pool in $3$ hours. How many hours do the two pumps take to fill the pool when they work together?

Show the answerHide the answer

Answer

The correct answer is $2$ .

Step 1: Determine each rate
- Pump A fills $\dfrac{1}{6}$ of the pool each hour.
- Pump B fills $\dfrac{1}{3}$ of the pool each hour.
Step 2: Add the rates
- $\dfrac{1}{3} = \dfrac{2}{6}$ .
- Together they fill $\dfrac{1}{6} + \dfrac{2}{6} = \dfrac{3}{6} = \dfrac{1}{2}$ of the pool each hour.
Step 3: Determine the time
- Half the pool each hour means the whole pool takes $2$ hours.

Traps

Dividing by $60$ only once. An hour has $60$ minutes, and each minute has $60$ seconds. $\dfrac{90{,}000}{60}$ $= 1{,}500$ is meters per minute, a wrong answer.
Adding the times instead of the rates. $6 + 3 = 9$ hours is slower than either pump alone, so it cannot be right. Two pumps together are faster than the faster pump.

Practice this lesson124 questions

Typing tips for Desmos

In the keys lines, → means the right-arrow key.
Typing / starts a fraction, and everything after it goes into the denominator until you press →. So 42/120*100 typed straight through becomes 42 ÷ (120 × 100). Type 42/120 → *100 instead.

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