Proportion

15 دقيقة

الشرح

A ratio compares two quantities of the same kind. The ratio of $a$ to $b$ can be written as $a:b$ or as the fraction $\frac{a}{b}$ . For example, if a recipe uses 2 cups of sugar for every 3 cups of flour, the ratio of sugar to flour is $2:3$ .

Proportion

A proportion is a statement that two ratios (or two rates) are equal, such as

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

. When this holds, the four quantities are called proportional quantities.

In the proportion $a:b = c:d$ the terms have special names. The extremes are the outer terms $a$ and $d$ ; the means are the inner terms $b$ and $c$

a:b = c:d \quad\Longleftrightarrow\quad a\times d = b\times c

Cross-multiplication: the product of the extremes equals the product of the means.

Why cross-multiplication works

Starting from

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

You can also recognise a proportion from a graph. If you plot the pairs of quantities as points, they represent a proportion when the points lie on one straight line that passes through the origin $(0,0)$ .

Proportional quantities: the points sit on a straight line through the origin.

…

A proportion: two equal ratios.

▶ Watch cross-multiplication test a proportion.

أمثلة محلولة

Worked exampleTesting a proportion

Do the ratios $\frac{6}{9}$ and $\frac{8}{12}$ form a proportion?

Worked exampleSolving a proportion

Find $x$ if $\frac{12}{16}=\frac{21}{x}$ .

Worked exampleChecking a proportion from a graph

A table shows time (s) and distance (m): $(1,6),(2,12),(3,18),(4,24)$ . Is the distance proportional to the time?

Worked exampleUnknown in the denominator (colon form)

Solve $a:36 = 25:20$ for $a$ .

Worked exampleWord problem: unit price

$8$ notebooks cost $60$ LE. At the same price, how many notebooks can be bought for $105$ LE?

Worked exampleWord problem: same rate (speed)

A car covers $320$ km in $4$ hours. Travelling at the same rate, how long does it take to cover $480$ km?