maths > commercial-arithmetics

Proportions

what you'll learn...

overview

Two ratios are said to be in proportion, if the ratios are equivalent. For example 2:4$2 : 4$ and 3:6$3 : 6$ are equivalent. Such equivalent ratios are formally represented as a proportion. The representation is 2:4::3:6$2 : 4 : : 3 : 6$.

illustrative example

A ratio of two quantities helps: to understand and to use the comparative measure of quantities.

Let us consider making dough for chappati or pizza. The recipe gives that for 400$400$ gram of flour, 150$150$ml water is used.

One person has only 200$200$ gram of flour. In that case, the person can reduce the water to $75$ml.

For $400$ gram flour $150$ml water is used. This is in $400 : 150$ ratio.

For $200$ gram flour $75$ml water is used. This is in $200 : 75$ ratio.

These two ratios can be simplified to $8 : 3$.

To denote that two ratios are identical, they are said to be in same proportion. That is $400 : 150$ is in the same proportion as $200 : 75$. This is given as $400 : 150 : : 200 : 75$.

proportion

The word "proportion" means: comparative measurement of quantities".

Pro-portion was from root word meaning "person's portion or share".

Proportion : Two ratios are said to be in proportion if the corresponding terms of ratio are identical in the simplified form.

Consider the example $2 : 3 : : 4 : 6$ proportion.

•  The numbers in the proportion are called first term, second term, third term, and fourth term in the order.

•  The first and fourth terms are called the extremes of the proportion.

•  The second and third terms are called the means of the proportion.

The word "extreme" means: farthest from the center.

The root word is from "exter" meaning outer.

The word "mean" means: average. The word is derived from a root word meaning "middle".

illustrative example

•  Fruit basket $A$ has $4$ apples and $16$ oranges. Ratio of apples to oranges is $4 : 16$ which is $1 : 4$.

•  Fruit basket $B$ has $20$ apples and $80$ oranges. Ratio of apples to oranges is $20 : 80$ which is $1 : 4$.

The proportion of apples to oranges in the two baskets is $4 : 16 : : 20 : 80$.

The proportion of apples to oranges in basket $A$ and basket $B$ is $4 : 16 : : 20 : 80$.

The word "Proportion" is also used to specify the simplified ratio, as in the following.

The proportion of apples to oranges in basket $A$ and basket $B$ is $1 : 4$.

Students are reminded to note the context in which the word "proportion" is used.

The proportion of count in basket A to count in basket B for apples and oranges is $4 : 20 : : 16 : 80$.

This proportion is given as
apples of basket $A$ to basket $B$ is in the same proportion as oranges of basket $A$ to basket $B$

•  Fruit basket $A$ has $4$ apples and $16$ oranges.

•  Fruit basket $B$ has $20$ apples and $80$ oranges.

The proportion of apples to oranges in basket $A$ and basket $B$ is $4 : 16 : : 20 : 80$.

The proportion of apples and oranges in basket $A$ to basket $B$ is $4 : 20 : : 16 : 80$.

Students are reminded to note the context in which proportion is defined.

proportion to fractions

We learned that "Ratio can be equivalently represented as a fraction.".

A basket has $3$ apples and $4$ oranges.

•  The ratio of the number of apples to number of oranges is $3 : 4$.

•  Number of apples are $\frac{3}{4}$ of the number of oranges.

The number of apples to number of oranges in basket A and B is in proportion $3 : 4 : : 6 : 8$.The following are all true.

the number of apples is $\frac{3}{4}$ of the number of oranges in basket A

the number of apples is $\frac{6}{8}$ of the number of oranges in basket B

the number of apples is $\frac{3}{4}$ of the number of oranges -- in both basket A and basket B

Note that in a proportion, the two fractions are equivalent fractions.
$\frac{6}{8}$, when simplified, is $\frac{3}{4}$.

Given the proportion $3 : 4 : : 6 : 8$,

$\frac{3}{4} = \frac{6}{8}$, that is, the two ratios given as fractions are always equal.

formula

Given a proportion, $a : b : : c : d$, it is understood that $\frac{a}{b} = \frac{c}{d}$. In that case,

$a \times d = b \times c$.

Given a proportion, $a : b : : c : d$, it is understood that $\frac{a}{b} = \frac{c}{d}$.

Given that
$\frac{a}{b} = \frac{c}{d}$

Note that $a , b , c , d$ are numbers. As per the properties of numbers, if two numbers are equal, then the numbers multiplied by another number are equal. (eg: if $4 = 2 \times 2$, then multiplying by $5$ we get $4 \times 5 = 2 \times 2 \times 5$.)
$\frac{a}{b}$ and $\frac{c}{d}$ are two numbers that are equal. On multiplying these numbers by $b d$, we get the two numbers $\frac{a}{b} \times b d$ and $\frac{c}{d} \times b d$. Simplifying these two numbers we get, $a d$ and $b c$. As per the property, these two numbers are equal.

$a d = b c$.

That is product of extremes and product of means are equal.

examples

Given the proportion $3 : 4 : : x : : 12$ find the value of $x$.

The answer is "$9$". Products of extremes equals product of means.
$3 \times 12 = 4 \times x$
$x = \frac{36}{4}$
$x = 9$

summary

Proportion : Two ratios are said to be in proportion if the corresponding terms of ratio are identical in the simplified form.

Mean-Extreme Property of Proportions : product of extremes = product of means
If $a : b : : c : d$ is a proportion, then $a d = b c$

Outline