Writing an Equation to Represent a Proportional Relationship


An expression of equality of ratios is called a proportion. The proportion expressing the equality of the ratios A:B and C:D is written A:B = C:D or A:B::C:D. This form, when spoken or written, is often expressed as

A is to B as C is to D.

A, B, C and D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means.

For example, from a table of equivalent ratios below, proportions can be written as follows 1:3::2:6 and 2:6::3:9

x y
1 3
2 6
3 9

The proportional relationship can also be written as

$\frac{y}{x} = \frac{3}{1} = \frac{6}{2} = \frac{9}{3}$

An equation to represent the proportional relationship would be

$y = 3x$

Write an equation to represent the proportional relationship given in the table.

k 3 12 15 27 36
l 7 28 35 63 84

Solution

Step 1:

The proportional relationship can be written as

$\frac{l}{k} = \frac{7}{3} = \frac{28}{12} = \frac{35}{15}... = \frac{7}{3}$

Step 2:

So, the equation representing this proportional relationship is $l = \frac{7}{3} \times \frac{k}{1} = \frac{7k}{3}$

or $l = \frac{7k}{3}$

Write an equation to represent the proportional relationship given in the table.

a 5 7 8 9 11
b 15 21 24 27 33

Solution

Step 1:

The proportional relationship can be written as

$\frac{b}{a} = \frac{15}{5} = \frac{21}{7} = \frac{24}{8}... = \frac{3}{1}$

Step 2:

So, the equation representing this proportional relationship is $b = \frac{3}{1} \times \frac{a}{1} = \frac{3a}{1} = 3a$

or $b = 3a$

Write an equation to represent the proportional relationship given in the table.

r 10 20 30 40 50
s 6 12 18 24 30

Solution

Step 1:

The proportional relationship can be written as

$\frac{s}{r} = \frac{6}{10} = \frac{12}{20} = \frac{18}{30}... = \frac{3}{5}$

Step 2:

So, the equation representing this proportional relationship is $s = \frac{3}{5} \times \frac{r}{1} = \frac{3r}{5}$

or $s = \frac{3r}{5}$

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