Ratio and Proportions

Lessons

Senior 1 · Mathematics

Ratio and Proportions

5 lessons

Lesson 3 of 5

Proportion and direct proportion

25 min

Let's begin

Suppose a shop in Nyabugogo sells sugar at 1,600 Frw per kg: 2 kg cost 3,200 Frw and 5 kg cost 8,000 Frw. Buy twice as much and you pay twice as much; buy five times as much and you pay five times as much. The ratio of cost to mass never changes. Quantities that grow together in this exact way are in proportion, and it is the most useful idea in everyday arithmetic.

1. Proportion

Definition

Proportion: a statement that two ratios are equal, $a : b = c : d \quad\text{or}\quad \frac{a}{b} = \frac{c}{d}.$ $a$ and $d$ are the extremes (the outer terms) and $b$ and $c$ are the means (the inner terms). If one term is unknown, it is called the fourth proportional.

Example: $2 : 3 = 8 : 12$ is a proportion; 2 and 12 are the extremes, 3 and 8 the means.

Important

Properties of a proportion. If $a : b = c : d$ , then

$a d = b c$ : the product of the extremes equals the product of the means;
$b : a = d : c$ (both ratios turned over);
$a : c = b : d$ (the means swapped).

Why property 1 holds. Start from $\frac{a}{b} = \frac{c}{d}$ and multiply both sides by $b d$ : $\frac{a}{b} \times bd = \frac{c}{d} \times bd$ , which simplifies to $a d = b c$ . This is "cross-multiplying". Properties 2 and 3 follow by dividing $a d = b c$ by suitable terms.

Worked example

(a) Is $4 : 6 = 10 : 15$ a proportion? (b) Find $x$ if $3 : 5 = 12 : x$

(a) Extremes: $4 \times 15 = 60$ . Means: $6 \times 10 = 60$ . The products are equal, so yes.

(b) Product of the extremes equals product of the means: $3x = 5 \times 12 = 60$ , so $x = 20$ . Check: $3 : 5 = 12 : 20$ , since both simplify to $3 : 5$ .

2. Direct proportion

Definition

Direct proportion: two quantities $x$ and $y$ are directly proportional if their ratio is always the same number $k$ : $\frac{y}{x} = k \quad\Longleftrightarrow\quad y = kx.$ $k$ is the constant of proportionality. We write $y \propto x$ , read " $y$ is proportional to $x$ ".

Example: cost $C$ and mass $w$ of sugar: $C = 1,600 w$ , so $k = 1,600$ Frw per kg.

Mass $w$ (kg)	1	2	3	4	5
Cost $C$ (Frw)	1,600	3,200	4,800	6,400	8,000
$\frac{C}{w}$	1,600	1,600	1,600	1,600	1,600

Three tests for direct proportion (each one is enough):

Table: $\frac{y}{x}$ gives the same number for every pair.
Formula: it can be written $y = k x$ , with no number added on.
Graph: a straight line through the origin. The gradient of the line is $k$ .

Why through the origin? If $y = k x$ , then $x = 0$ gives $y = 0$ : no sugar costs nothing. A line of the form $y = m x + c$ with $c \ne 0$ misses the origin, so its $\frac{y}{x}$ keeps changing.

Definition

Unitary method: solving a proportion problem by first finding the value of one unit, then multiplying.

Example: if 5 kg of beans cost 4,000 Frw, then 1 kg costs $4{,}000 \div 5 = 800$ Frw.

Worked example

5 kg of beans cost 4,000 Frw. What do 8 kg cost? Solve it in two ways

Unitary method: 1 kg costs $4{,}000 \div 5 = 800$ Frw, so 8 kg cost $8 \times 800 = 6{,}400$ Frw.

Proportion: the ratio of cost to mass is fixed, so $\frac{4{,}000}{5} = \frac{x}{8} \;\Rightarrow\; 5x = 32{,}000 \;\Rightarrow\; x = 6{,}400.$ Both give 6,400 Frw.

Worked example

A recipe for 4 people uses 300 g of rice. How much rice is needed for 10 people?

Rice is directly proportional to the number of people. For 1 person: $300 \div 4 = 75$ g. For 10 people: $10 \times 75 = 750$ g.

Worked example

Rider A charges 300 Frw plus 100 Frw per km. Rider B charges 150 Frw per km. Which fare is directly proportional to the distance?

Distance (km)	2	4	6
Rider A (Frw)	500	700	900
Rider A: fare $\div$ km	250	175	150
Rider B (Frw)	300	600	900
Rider B: fare $\div$ km	150	150	150

Rider B's ratio is constant ( $k = 150$ ), so Rider B's fare is directly proportional to the distance. Rider A's fare also grows with distance, but its ratio changes, because of the fixed 300 Frw. On the graph, only Rider B's line passes through the origin.

Warning

"Both go up" is not enough. Rider A's fare rises as the distance rises, yet it is not directly proportional: 6 km does not cost three times as much as 2 km. Always test the ratio, or check for a line through the origin.

Tip

When you write a proportion, put the same kind of quantity in the same place on both sides: $\frac{\text{cost}}{\text{mass}} = \frac{\text{cost}}{\text{mass}}$ . Mixing them up, as in $\frac{\text{cost}}{\text{mass}} = \frac{\text{mass}}{\text{cost}}$ , is the usual source of wrong answers.

Math Antics (mathantics): "Proportions". Watch how cross-multiplying finds the missing term, and why it works.

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Practice

Show your method.

Is $6 : 9 = 8 : 12$ a proportion?
Find $x$ if $4 : 7 = x : 21$ .
Find $x$ if $\frac{x}{15} = \frac{6}{9}$ .
3 pens cost 750 Frw. What do 8 pens cost?
A car uses 6 litres of fuel to travel 75 km. How far can it travel on 10 litres at the same rate?
$y$ is directly proportional to $x$ , and $y = 18$ when $x = 4$ . Find $k$ , then $y$ when $x = 10$ .
In a table, $x = 2, 4, 6$ and $y = 7, 14, 20$ . Is $y$ directly proportional to $x$ ? Explain.

Show answersHide answers

Yes: $6 \times 12 = 72$ and $9 \times 8 = 72$ .
$7 x = 84$ , so $x = 12$ .
$9 x = 90$ , so $x = 10$ .
1 pen costs 250 Frw, so 8 pens cost 2,000 Frw.
$75 \div 6 = 12.5$ km per litre, so $12.5 \times 10 = 125$ km.
$k = \frac{18}{4} = 4.5$ , so $y = 4.5 x$ and $y = 45$ when $x = 10$ .
No: $\frac{7}{2} = 3.5$ and $\frac{14}{4} = 3.5$ , but $\frac{20}{6} \approx 3.33$ . The ratio is not constant.

Sources and credits

8.7: Solve Proportion and Similar Figure Applications, from Elementary Algebra by Lynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis, OpenStax, provided by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
8.9: Use Direct and Inverse Variation, from Elementary Algebra by Lynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis, OpenStax, provided by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Grade 7, Unit 2: Introducing Proportional Relationships, IM 6–8 Math by Illustrative Mathematics, originally developed by Open Up Resources (© 2017–2019 Open Up Resources; adaptations © 2019 Illustrative Mathematics), provided by Kendall Hunt, licensed under CC BY 4.0. Adapted by Ganzaa.
Video: Math Antics - Proportions, mathantics, embedded from YouTube.

The sugar, beans and moto-fare examples, the graphs, practice items and quiz questions were written by Ganzaa. The prices are models for practice.

11. Proportion

22. Direct proportion

1. Proportion

2. Direct proportion