Proportion

Look at the two ratios $18 : 45$ and $2 : 5$. They look dif­fer­ent, but $\displaystyle \frac{18}{45}$ can­cels down to $\displaystyle \frac{2}{5}$. So the two ratios are equal. When two ratios are equal, we say the four num­bers are in pro­por­tion. Pro­por­tion helps us find a miss­ing num­ber when three num­bers are known. It also gives us some handy rules for work­ing with ratios.

What a pro­por­tion is#

Pro­por­tion. A state­ment that two ratios are equal is called a pro­por­tion.

We write $a : b = c : d$, or $a : b :: c : d$. The sign $::$ is read "is as". So $18 : 45 :: 2 : 5$ is read "18 is to 45 as 2 is to 5".

When $a : b :: c : d$, we say that $a$, $b$, $c$ and $d$ are in pro­por­tion.

Means and extremes#

Take the pro­por­tion $a : b :: c : d$.

• The first and fourth terms, $a$ and $d$, are called the extremes. They are at the two ends.
• The sec­ond and third terms, $b$ and $c$, are called the means. They are in the mid­dle.

Here is the most use­ful fact about a pro­por­tion:

$$\text{product of the means} = \text{product of the extremes}$$

$$bc = ad$$

This comes straight from $\displaystyle \frac{a}{b} = \frac{c}{d}$. Mul­ti­ply both sides by $bd$ and you get $ad = bc$. This is cross mul­ti­pli­ca­tion.

For $18 : 45 :: 2 : 5$, the means give $45 \times 2 = 90$ and the extremes give $18 \times 5 = 90$. They agree.

Fourth, third and mean pro­por­tional#

Fourth pro­por­tional. If $a : b :: c : d$, then $d$ is called the fourth pro­por­tional to $a$, $b$ and $c$.

Con­tin­ued pro­por­tion. If $a : b :: b : c$, the mid­dle num­ber is used twice. Then we say:

1. $a$, $b$ and $c$ are in con­tin­ued pro­por­tion.
2. $c$ is the third pro­por­tional to $a$ and $b$.
3. $b$ is the mean pro­por­tional between $a$ and $c$.

From $a : b :: b : c$, the prod­uct of the means is $b \times b$ and the prod­uct of the extremes is $a \times c$. So

$$b^2 = ac \qquad b = \sqrt{ac}$$

The mean pro­por­tional between $a$ and $b$ is $\sqrt{ab}$.

For exam­ple, 4, 6 and 9 are in con­tin­ued pro­por­tion, because $\displaystyle \frac{4}{6} = \frac{2}{3}$ and $\displaystyle \frac{6}{9} = \frac{2}{3}$. The mean pro­por­tional between 4 and 9 is $\sqrt{4 \times 9} = \sqrt{36} = 6$.

Use­ful prop­er­ties of a pro­por­tion#

Sup­pose $a : b = c : d$, so $\displaystyle \frac{a}{b} = \frac{c}{d}$. Then each of these is also true.

Check each rule with a sim­ple pro­por­tion. Take $2 : 4 = 1 : 2$.

• Addendo: $\displaystyle \frac{2 + 1}{4 + 2} = \frac{3}{6} = \frac{1}{2}$. Yes, it is the same ratio.
• Sub­tra­hendo: $\displaystyle \frac{2 - 1}{4 - 2} = \frac{1}{2}$. Yes.

Take $1 : 2 = 2 : 4$.

• Com­po­nendo: $\displaystyle \frac{1 + 2}{2} = \frac{3}{2}$ and $\displaystyle \frac{2 + 4}{4} = \frac{6}{4} = \frac{3}{2}$. Equal.
• Div­i­dendo: $\displaystyle \frac{1 - 2}{2} = -\frac{1}{2}$ and $\displaystyle \frac{2 - 4}{4} = -\frac{2}{4} = -\frac{1}{2}$. Equal.
• Com­po­nendo and div­i­dendo: $\displaystyle \frac{1 + 2}{1 - 2} = -3$ and $\displaystyle \frac{2 + 4}{2 - 4} = -3$. Equal.

Com­po­nendo and div­i­dendo can also be writ­ten the other way up: $\displaystyle \frac{a - b}{a + b} = \frac{c - d}{c + d}$.

The equiv­a­lent ratio rule#

This rule is very impor­tant.

1. If $\displaystyle \frac{a}{b} = \frac{c}{d}$, then for any num­bers $l$ and $m$, $\displaystyle \frac{a}{b} = \frac{c}{d} = \frac{al + cm}{bl + dm}$.
2. If $\displaystyle \frac{a}{b} = \frac{c}{d} = \frac{e}{f}$, then $\displaystyle \frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \frac{ap + cq + er}{bp + dq + fr}$.

Addendo and sub­tra­hendo are spe­cial cases. Take $l = m = 1$ for addendo, and $l = 1$, $m = -1$ for sub­tra­hendo.

For exam­ple, $\displaystyle \frac{1}{2} = \frac{3}{6}$. Take $l = 2$ and $m = 5$:

$$\displaystyle \frac{1 \times 2 + 3 \times 5}{2 \times 2 + 6 \times 5} = \frac{17}{34} = \frac{1}{2}$$

Exam­ple 1: Using com­po­nendo and div­i­dendo#

If $\displaystyle \frac{a}{b} = \frac{7}{3}$, find $\displaystyle \frac{a + b}{a - b}$.

By com­po­nendo and div­i­dendo,

$$\displaystyle \frac{a + b}{a - b} = \frac{7 + 3}{7 - 3} = \frac{10}{4} = \frac{5}{2}$$

Check with num­bers: take $a = 7$ and $b = 3$. Then $\displaystyle \frac{a + b}{a - b} = \frac{10}{4}$, the same.

Find­ing a miss­ing term#

Every prob­lem below uses one idea: the prod­uct of the means equals the prod­uct of the extremes.

Exam­ple 2#

The first, sec­ond and fourth terms of a pro­por­tion are 5, 15 and 90. Find the third term.

Let the third term be $x$. Then $5 : 15 :: x : 90$.

Prod­uct of the extremes $=$ prod­uct of the means:

$$\begin{aligned} 5 \times 90 &= 15 \times x \\ 450 &= 15x \\ x &= 30 \end{aligned}$$

You can also think: $\displaystyle \frac{5}{15} = \frac{1}{3}$, and one third of 90 is 30.

Exam­ple 3#

The ratio of the length to the width of a sheet of paper is $5 : 3$. The width is 18 cm. Find the length.

First method. Let the length be $l$. Then $l : 18 = 5 : 3$, so $\displaystyle \frac{l}{18} = \frac{5}{3}$ and $\displaystyle l = \frac{5 \times 18}{3} = 30$ cm.

Sec­ond method. Let the length be $5k$ and the width be $3k$. Then $3k = 18$, so $k = 6$. The length is $5 \times 6 = 30$ cm.

Exam­ple 4#

If 81, $x$, $x$ and 256 are in pro­por­tion, find $x$.

$$\displaystyle \frac{81}{x} = \frac{x}{256}$$

The prod­uct of the means equals the prod­uct of the extremes:

$$x^2 = 81 \times 256 = 9^2 \times 16^2 = (9 \times 16)^2 = 144^2$$

So $x = 144$.

Exam­ple 5#

Find the mean pro­por­tional between 8 and 98.

$$\sqrt{8 \times 98} = \sqrt{8 \times 2 \times 49} = \sqrt{16 \times 49} = 4 \times 7 = 28$$

You can also say $8 \times 98 = 784$ and $\sqrt{784} = 28$.

Exam­ple 6#

Find the mean pro­por­tional between 234 and 104.

Break both num­bers into fac­tors that pair up. $234 = 2 \times 9 \times 13$ and $104 = 2 \times 4 \times 13$.

$$\sqrt{234 \times 104} = \sqrt{(2 \times 2) \times 9 \times 4 \times (13 \times 13)} = 2 \times 3 \times 2 \times 13 = 156$$

Check: $156 \times 156 = 24336$ and $234 \times 104 = 24336$.

Prac­tice#

1. Are 4, 6, 10 and 15 in pro­por­tion?
2. Are 3, 5, 7 and 9 in pro­por­tion?
3. Find the fourth pro­por­tional to 3, 8 and 12.
4. Find the third pro­por­tional to 4 and 10.
5. Find the mean pro­por­tional between 9 and 25.
6. Find the mean pro­por­tional between 12 and 75.
7. If $2 : x :: 6 : 21$, find $x$.
8. If 16, $x$, $x$ and 49 are in pro­por­tion, find $x$.
9. If $\displaystyle \frac{a}{b} = \frac{5}{3}$, find $\displaystyle \frac{a + b}{a - b}$.
10. The ratio of the length to the width of a flag is $3 : 2$. The flag is 1.5 m long. How wide is it?

Answers#