Ratios, Rates and Proportions

A ratio compares two amounts, and a rate compares two amounts measured in different units, like kilometres per hour or dollars per litre. Ratios and rates help you mix paint, follow a recipe for a bigger crowd, find the best deal at the grocery store, read a map, and figure out how long a trip will take. A proportion says that two ratios are equal, and it’s one of the most useful problem-solving tools in all of math.

Key ideas

Ratios

A ratio compares two (or more) quantities in the same units. The ratio of $6$ cans of blue paint to $9$ cans of yellow paint can be written three ways:

$$6 : 9 \qquad \frac{6}{9} \qquad \text{"6 to 9"}$$

Order matters. Blue to yellow is $6 : 9$, but yellow to blue is $9 : 6$.

• A part-to-part ratio compares one part to another part: blue to yellow is $6 : 9$.
• A part-to-whole ratio compares one part to the total: blue to all paint is $6 : 15$, so $\dfrac{6}{15} = \dfrac{2}{5}$ of the paint is blue.

Equivalent ratios

Multiply or divide every term by the same number (not zero) to get an equivalent ratio, just like equivalent fractions:

$$6 : 9 = 2 : 3 = 4 : 6 = 10 : 15$$

A ratio is in simplest form when its terms are whole numbers with no common factor except $1$. Here, that’s $2 : 3$.

If the quantities are in different units, convert first: $30 \text{ cm} : 2 \text{ m} = 30 \text{ cm} : 200 \text{ cm} = 3 : 20$.

Rates and unit rates

A rate compares quantities in different units: $240$ km in $3$ h, or $6.75 for $3$ kg. A unit rate is a rate “per $1$” of the second quantity. Divide to find it:

$$\frac{240 \text{ km}}{3 \text{ h}} = 80 \text{ km/h}$$

Speed (km/h), pay (dollars per hour), fuel use (L/100 km) and prices (dollars per kilogram) are all rates.

Best buys. To compare prices of different-sized packages, find the unit price (the price per gram, per $100$ g, per litre, …) of each one. The lower unit price is the better buy.

Proportions

A proportion is an equation that says two ratios are equal:

$$\frac{x}{12} = \frac{5}{8}$$

Three ways to solve it:

1. Equivalent ratios. Look for the scale factor between the terms, if it’s a friendly number.
2. Unit rate. Find the amount for $1$, then multiply.
3. Multiply both sides by the denominator that’s under the unknown. Here, multiply both sides by $12$: $x = \dfrac{5 \times 12}{8} = 7.5$.

Whatever you do, keep the quantities in the same order on both sides (for example, cups over cookies on both sides).

Proportional relations

When one quantity is always the same multiple of another, the relation is proportional. Gas that costs $1.60 per litre is an example: the cost $C$ (in dollars) of $L$ litres is $C = 1.6L$. Its graph is a straight line through the origin, and the unit rate is the slope of the line.

Cost of gas at $1.60 per litre. Every extra $10$ L costs another $16.

You’ll study relations like this, and ones that don’t go through the origin, in representing linear relations.

Scale drawings and maps

A scale is a ratio that compares a length on a drawing or map to the real length. A map scale of $1 : 50\,000$ means $1$ cm on the map stands for $50\,000$ cm in real life. Since $50\,000 \text{ cm} = 500 \text{ m}$, every centimetre on the map is half a kilometre.

Unit analysis

Treat units like numbers that can multiply and cancel. This tells you whether to multiply or divide, and it catches mistakes:

$$90 \,\frac{\text{km}}{\text{h}} \times 2.5 \text{ h} = 225 \text{ km}$$

The hours cancel, leaving kilometres. To convert units, multiply by a ratio that equals $1$, such as $\dfrac{1000 \text{ m}}{1 \text{ km}}$ or $\dfrac{1 \text{ h}}{3600 \text{ s}}$:

$$90 \,\frac{\text{km}}{\text{h}} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = \frac{90\,000 \text{ m}}{3600 \text{ s}} = 25 \text{ m/s}$$

Worked examples

Example 1: Mixing paint

A shade of green paint is made by mixing $6$ cans of blue paint with $9$ cans of yellow paint.

• (a) Write the ratio of blue to yellow in simplest form.
• (b) What fraction of the mixture is blue?
• (c) How many cans of yellow are needed with $10$ cans of blue to get the same shade?

Solution.

(a) Divide both terms by $3$: $6 : 9 = 2 : 3$.

(b) There are $6 + 9 = 15$ cans in all, so $\dfrac{6}{15} = \dfrac{2}{5}$ of the mixture is blue.

(c) Set up a proportion with blue over yellow on both sides:

$$\frac{2}{3} = \frac{10}{y}$$

The numerator went from $2$ to $10$, so it was multiplied by $5$. Do the same to the denominator: $y = 3 \times 5 = 15$. You need $15$ cans of yellow. Check: $10 : 15 = 2 : 3$. ✓

Example 2: Best buy

A $525$ g box of cereal costs $4.99. A $750$ g box of the same cereal costs $6.79. Which is the better buy?

Solution. Find each unit price per $100$ g. Divide the price by the mass, then multiply by $100$:

$$\text{small: } \frac{4.99}{525} \times 100 \approx 0.95 \qquad\qquad \text{large: } \frac{6.79}{750} \times 100 \approx 0.91$$

The small box costs about $0.95 per $100$ g and the large box about $0.91 per $100$ g. The large box is the better buy (as long as you’ll use it all before it goes stale).

Example 3: Solving proportions

Solve.

• (a) $\dfrac{x}{15} = \dfrac{4}{6}$
• (b) $\dfrac{7}{x} = \dfrac{21}{12}$
• (c) $3$ cups of flour make $24$ cookies. How much flour do you need for $60$ cookies?

Solution.

(a) Multiply both sides by $15$:

$$x = \frac{4 \times 15}{6} = \frac{60}{6} = 10$$

Check: $\dfrac{10}{15} = \dfrac{2}{3}$ and $\dfrac{4}{6} = \dfrac{2}{3}$. ✓

(b) Look for a scale factor: $7 \times 3 = 21$, so the numerator was multiplied by $3$. Then $x \times 3 = 12$, so $x = 4$. Check: $\dfrac{7}{4} = 1.75$ and $\dfrac{21}{12} = 1.75$. ✓

(c) Unit rate: $3$ cups for $24$ cookies is $\dfrac{3}{24} = \dfrac{1}{8}$ cup per cookie. For $60$ cookies:

$$60 \times \frac{1}{8} = \frac{60}{8} = 7.5 \text{ cups}$$

You need $7\dfrac{1}{2}$ cups of flour.

Example 4: Reading a map

On a map with a scale of $1 : 250\,000$, two towns are $6.4$ cm apart.

• (a) How far apart are the towns in real life, in kilometres?
• (b) How long would it take to cycle between them in a straight line at $20$ km/h?

Solution.

(a) Each centimetre on the map is $250\,000$ cm in real life:

$$6.4 \times 250\,000 = 1\,600\,000 \text{ cm}$$

Convert to kilometres. There are $100$ cm in a metre and $1000$ m in a kilometre, so $100\,000$ cm in a kilometre:

$$1\,600\,000 \text{ cm} \times \frac{1 \text{ km}}{100\,000 \text{ cm}} = 16 \text{ km}$$

(b) Time is distance divided by speed. Check the units: km divided by km/h gives hours.

$$16 \text{ km} \div 20 \,\frac{\text{km}}{\text{h}} = 0.8 \text{ h}$$

$0.8 \text{ h} \times 60 \,\dfrac{\text{min}}{\text{h}} = 48$ min.

Common mistakes

Writing the terms in the wrong order. If the question asks for “yellow to blue”, the yellow amount goes first. Read the wording and keep the order the same all the way through.

Adding instead of multiplying to get an equivalent ratio. $2 : 3$ is not the same as $4 : 5$ (adding $2$ to each term). Equivalent ratios come from multiplying or dividing every term by the same number: $2 : 3 = 4 : 6$.

Comparing quantities in different units. $30 \text{ cm} : 2 \text{ m}$ is not $15 : 1$. Convert to the same units first: $30 : 200 = 3 : 20$.

Comparing total prices instead of unit prices. A bigger package costs more, but that doesn’t mean it’s a worse deal. Find the price per unit of each, and don’t round until the end.

Setting up a proportion with mismatched sides. In $\dfrac{3 \text{ cups}}{24 \text{ cookies}} = \dfrac{x \text{ cups}}{60 \text{ cookies}}$, both sides are cups over cookies. Writing cups over cookies on one side and cookies over cups on the other gives a wrong answer.

Forgetting to convert map units. A map scale usually gives real distances in centimetres. Convert to metres or kilometres before you answer: $100$ cm $= 1$ m, $100\,000$ cm $= 1$ km.

Practice

1. (Warm-up) Write each ratio in simplest form.

• (a) $15 : 25$
• (b) $1.2 : 0.8$
• (c) $45 \text{ cm} : 1.5 \text{ m}$

Solution

(a) Divide by $5$: $3 : 5$.

(b) Multiply by $10$ to get whole numbers, $12 : 8$, then divide by $4$: $3 : 2$.

(c) Convert to centimetres: $45 : 150$. Divide by $15$: $3 : 10$.

2. (Warm-up) Find each unit rate.

• (a) $13.50 for $6$ muffins
• (b) $336$ km on $28$ L of gas
• (c) $187 earned in $11$ hours

Solution

(a) $13.50 \div 6 = 2.25$, so $2.25 per muffin.

(b) $336 \div 28 = 12$, so $12$ km/L.

(c) $187 \div 11 = 17$, so $17 per hour.

3. (Warm-up) Do the ratios form a proportion? Explain.

• (a) $\dfrac{4}{10}$ and $\dfrac{6}{15}$
• (b) $\dfrac{3}{8}$ and $\dfrac{9}{20}$

Solution

(a) Yes. Both simplify to $\dfrac{2}{5}$.

(b) No. $\dfrac{3}{8} = 0.375$ but $\dfrac{9}{20} = 0.45$.

4. (Core) Solve each proportion.

• (a) $\dfrac{x}{9} = \dfrac{8}{12}$
• (b) $\dfrac{5}{7} = \dfrac{35}{x}$
• (c) $\dfrac{2.5}{x} = \dfrac{10}{16}$

Solution

(a) Multiply both sides by $9$: $x = \dfrac{8 \times 9}{12} = \dfrac{72}{12} = 6$.

(b) The numerator was multiplied by $7$ ($5 \times 7 = 35$), so $x = 7 \times 7 = 49$.

(c) The numerator went from $10$ to $2.5$, which is dividing by $4$. So $x = 16 \div 4 = 4$. Check: $\dfrac{2.5}{4} = 0.625$ and $\dfrac{10}{16} = 0.625$. ✓

5. (Core) Apple juice is sold in three sizes. Which is the best buy?

• a $1.89$ L bottle for $3.49
• a single $355$ mL can for $1.29
• a pack of $12$ cans ($355$ mL each) for $7.99

Solution

Find the price per litre of each. Change millilitres to litres first: $355 \text{ mL} = 0.355$ L, and $12$ cans hold $12 \times 0.355 = 4.26$ L.

$$\text{bottle: } \frac{3.49}{1.89} \approx 1.85 \qquad \text{can: } \frac{1.29}{0.355} \approx 3.63 \qquad \text{pack: } \frac{7.99}{4.26} \approx 1.88$$

The bottle is the best buy at about $1.85 per litre, just ahead of the 12-pack at about $1.88 per litre. A single can is almost twice as expensive per litre.

6. (Core) A train travels $345$ km in $2.5$ h.

• (a) Find its average speed.
• (b) At that speed, how long will it take to travel $552$ km?

Solution

(a) $345 \div 2.5 = 138$, so $138$ km/h.

(b) $552 \text{ km} \div 138 \,\dfrac{\text{km}}{\text{h}} = 4$ h.

7. (Core) On a floor plan, $1$ cm represents $0.5$ m. A bedroom measures $9.2$ cm by $7$ cm on the plan. Find the real length, width and floor area of the room.

Solution

Multiply each plan length by $0.5$ m per cm:

$$9.2 \times 0.5 = 4.6 \text{ m} \qquad\qquad 7 \times 0.5 = 3.5 \text{ m}$$

Area: $4.6 \times 3.5 = 16.1 \text{ m}^2$.

8. (Challenge) A car uses $7.2$ L of gas per $100$ km. A family drives $450$ km from Toronto to Ottawa, and gas costs $1.55 per litre. How much gas do they use, and what does it cost?

Solution

Use unit analysis so the units cancel:

$$450 \text{ km} \times \frac{7.2 \text{ L}}{100 \text{ km}} = 32.4 \text{ L}$$$$32.4 \text{ L} \times 1.55 \,\frac{\text{dollars}}{\text{L}} = 50.22 \text{ dollars}$$

They use $32.4$ L of gas, which costs $50.22.

9. (Challenge) Use unit analysis for both parts.

• (a) Convert $100$ km/h to metres per second. Round to one decimal place.
• (b) A sprinter runs at $10$ m/s. What is that in km/h?

Solution

(a)

$$100 \,\frac{\text{km}}{\text{h}} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = \frac{100\,000 \text{ m}}{3600 \text{ s}} \approx 27.8 \text{ m/s}$$

(b) Flip the conversion ratios so the units cancel the other way:

$$10 \,\frac{\text{m}}{\text{s}} \times \frac{3600 \text{ s}}{1 \text{ h}} \times \frac{1 \text{ km}}{1000 \text{ m}} = \frac{36\,000 \text{ km}}{1000 \text{ h}} = 36 \text{ km/h}$$