MTH1W Ratios, Rates and Proportions: Notes and Practice

The label on a can of frozen juice says "mix 1 can of concentrate with 3 cans of water". For a party you use 4 cans of concentrate: how much water? At the grocery store, a 525 g box of cereal and a 750 g box sit side by side with different prices: which is the better deal? On a hiking map, two campsites are 7.4 cm apart: how far is the walk? All three questions are about proportional reasoning, the idea that two quantities grow or shrink together at a fixed rate. Double one and the other doubles; take a third of one and the other drops to a third. This lesson gives you three tools for these questions: ratios, unit rates and proportions.

Ratios and equivalent ratios

A ratio compares two or more quantities measured in the same unit, written $1 : 3$ or $\frac{1}{3}$ and read "1 to 3". Order matters: concentrate to water is $1 : 3$ , but water to concentrate is $3 : 1$ . A part-to-part ratio compares pieces of a mixture ( $1 : 3$ concentrate to water); a part-to-whole ratio compares one piece with the total ( $1 : 4$ , since 1 can of concentrate is part of 4 cans of juice). Like fractions, ratios have equivalent forms: multiply or divide every term by the same non-zero number. A ratio is in simplest form when its terms are whole numbers with no common factor other than 1, so $18 : 24$ simplifies to $3 : 4$ by dividing both terms by 6.

Cans of concentrate	Cans of water	Total cans of juice	Concentrate : water
1	3	4	1 : 3
2	6	8	2 : 6 = 1 : 3
4	12	16	4 : 12 = 1 : 3
5	15	20	5 : 15 = 1 : 3

Equivalent ratios: every row tastes the same because concentrate to water stays 1 to 3

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A bag holds 12 red marbles and 8 blue marbles. What is the ratio of blue to red, in simplest form?

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A drink is mixed 2 : 5, syrup to water. What fraction of the drink is syrup? (Type it like 3/4.)

Rates and unit rates

A rate compares two quantities measured in different units: 240 km in 3 hours, $7.50 for 3 kg, 7.5 L of fuel per 100 km. A unit rate is a rate for one unit of the second quantity: $80$ km per hour, $2.50$ dollars per kilogram. You find a unit rate by dividing, and once you have it, any other amount is one multiplication away. A double number line makes this visible: one line for each quantity, matched up so that whatever you do to one line (divide by 3, multiply by 2) you do to the other.

Fig 1Double number line for 3 kg of apples costing 7.50 dollars. Dividing both lines by 3 gives the unit rate of 2.50 dollars per kg; multiplying by 2 gives 6 kg for 15 dollars.

Box of cereal	Price	Unit price (per 100 g)
350 g	$3.99	$3.99 \div 3.5 = 1.14$ , so about $1.14
525 g	$5.49	$5.49 \div 5.25 \approx 1.046$ , so about $1.05
750 g	$8.29	$8.29 \div 7.5 \approx 1.105$ , so about $1.11

Best buy: the 525 g box has the lowest unit price, even though it is not the biggest box

Finding the best buy

Divide each price by its quantity, using the same unit for every item (all per 100 g, or all per kg, or all per litre). The lowest unit price is the best value for money. Bigger packages are often cheaper per unit, but not always, so check. Value also depends on whether you will use it all before it spoils.

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A train travels 315 km in 3.5 hours. What is its unit rate, in kilometres per hour?

km/h

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Juice comes in a 2 L jug for $3.80 or a 3 L jug for $5.40. Which is the better buy?

VideoMath Antics explains how ratios and rates compare two amounts and how dividing gives a unit rate, the tool for deciding which package is the better deal.Watch on YouTube

Proportions and why cross-multiplying works

A proportion is an equation saying two ratios are equal, such as $\frac{x}{12} = \frac{5}{8}$ . You can solve it three ways. Scale factor: if you can see how one denominator turns into the other, do the same to the numerator. Unit rate: find the amount for 1, then multiply. Cross-multiplying: for $\frac{a}{b} = \frac{c}{d}$ (with $b$ and $d$ not zero), multiply both sides by $b$ and by $d$ . The left side becomes $a \times d$ and the right side becomes $c \times b$ , so $a d = b c$ . Cross-multiplying is just that shortcut: it clears both denominators in one step, which is why it always works for a proportion and why it does not apply to adding or multiplying fractions.

\frac{x}{12} = \frac{5}{8} \Rightarrow 8 \times x = 5 \times 12 \Rightarrow 8 x = 60 \Rightarrow x = \frac{60}{8} = 7.5

Fig 2Two bars of the same length: 5 of 8 parts shaded, and a bar of 12 parts shaded to the same point, 7.5 parts. Cross-multiplying gives 8x = 60, so x = 7.5.

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Solve the proportion

\frac{x}{9} = \frac{10}{6}

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Why does cross-multiplying work for a proportion

\frac{a}{b} = \frac{c}{d}

VideoMath Antics sets two equal ratios side by side and uses cross-multiplying to find the missing value.Watch on YouTube

Scale drawings and maps

A map scale is a ratio of map distance to real distance, in the same unit. A scale of $1 : 50000$ means 1 cm on the map stands for $50000$ cm on the ground, which is 500 m. To find a real distance, multiply the map distance by the scale's second number, then convert units: 3.2 cm on that map is $3.2 \times 50000 = 160000$ cm $= 1600$ m $= 1.6$ km. Going the other way (real to map), divide. Scale models and floor plans work the same way: a $1 : 24$ model car that is 18.5 cm long represents a real car $18.5 \times 24 = 444$ cm, or 4.44 m, long.

Match the units before you compare

A ratio compares like with like. Compare 750 g with 1.2 kg only after writing both in grams (750 g and 1200 g) or both in kilograms. In a proportion, the units in each ratio must line up: kilograms over dollars on one side means kilograms over dollars on the other.

ARatio

Compares quantities in the same unit
1 : 3 concentrate to water
Map scale 1 : 50 000
No units in the simplified answer

BRate

Compares quantities in different units
240 km in 3 h
2.50 dollars per kg
Units stay attached: km/h, dollars per kg

Fig 3A 1 : 50 000 map scale: a map ruler in centimetres above a ground scale where each centimetre is 500 m. Campsites 3.2 cm apart on the map are 1.6 km apart on the ground.

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On a map with scale

1 : 25000

, two lakes are 6 cm apart. How far apart are they in real life, in kilometres?

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A trail is 3 km long. How long is it on a map with scale

1 : 50000

, in centimetres?

Ratios, Rates and Proportions

Key things to remember

Ratios and equivalent ratios

Rates and unit rates

Proportions and why cross-multiplying works

Scale drawings and maps

Worked examples

Where marks get lost

Flashcards 8

Practice