What are proportions?

A proportion is a statement in which two or more ratios are equivalent to each other.

If four real numbers are such that the ratio of the first two is equal to the ratio of the last two then the four numbers in that order are said to be proportional or said to be in proportion. If four real numbers $$a$$, $$b$$, $$c$$ and $$d$$ $$\left (b\neq 0, d\neq 0 \right)$$ be in proportion. We write $$a:b::c:d$$, where $$a$$ and $$d$$ are called extreme terms and $$b$$, $$c$$ are called the middle terms.

Now we find the relation among them. Suppose $$a$$, $$b$$, $$c$$ and $$d$$ are proportional.

Therefore, we can write it as $$a:b::c:d$$.

That is, $$\frac{a}{b} = \frac{c}{d}$$.

Therefore, $$ad=bc$$

We observe that, if four numbers are in proportion, then the product of the extreme terms is equal to the product of the middle terms. Here, $$a$$ is called the antecedent and $$b$$ is called the consequent of the ratio $$a:b$$.

If $$a=b$$ in the ratio $$a:b$$ that is the antecedent and the consequent are equal, the ratio is called the ratio of equality and if $$a\neq b$$ that is antecedent and the consequent are not equal then the ratio is called the ratio of inequality.

For example, $$3:4$$ is a ratio of equality and $$2:2$$ is a ratio of inequality.

There are two types of proportion:

Direct proportion

If the ratio of two variables is constant, the variables are directly proportional to each other. This means the value of these variables will either increase or decrease together. By the above definition, if $$x$$ and $$y$$ are connected by a direct proportion, then we have $$\frac{x}{y}=\text{constant}$$.

Inverse proportion

If two variables are connected by inverse or reciprocal proportion, the value of one variable increases and the other decreases or vice-versa.

That means if $$x$$ and $$y$$ are connected by inverse or reciprocal proportion, then $$x \times y=\text{constant}$$.

E1.11: Demonstrate an understanding of ratio and proportion.

The concept of ratio and proportion is fundamental to the understanding of the various topics in mathematics like numbers, geometry, speed, distance and time, etc. In our daily life, we also use the concept of ratio and proportion.

For example, A shopkeeper mixed higher grad of tea and lower grad of tea in the ratio $$3:1$$. If he has a total $$60\;\text{kg}$$ of tea, find the quantity of the higher grade and the lower grade tea in the mixed tea.

Let the quantity of the higher grad tea in the mixed tea be $$3x$$

and the quantity of lower grad tea in the mixed tea be $$x$$.

After mixing these two types of tea, the total quantity of tea be $$3x+x=4x$$.

From the problem, $$4x=60$$.

So, we have $$x=15$$.

Therefore, the quantity of the higher grad of tea in the mixed tea is $$3\times 15$$

That is $$45\;\text{kg}$$.

And the quantity of the lower grad of tea in the mixed tea is $$15\;\text{kg}$$.

Therefore, the quantity of the higher grad and the lower grad tea in the mixed tea are $$45\;\text{kg}$$ and $$15\;\text{kg}$$.

Suppose in a classroom there are some boys and girls. The ratio of boys and girls is $$7:5$$. If the number of students in the classroom are $$132$$. Let us calculate the number of boys and girls in the classroom.

Let's look at another example.

Let the number of boys and girls in the classroom be $$7x$$ and $$5x$$.

Then, the total number of boys and girls in the classroom is $$7x+5x$$.

From the given problem $$7x+5x=132$$.

Solve further to get, $$12x=132$$.

Divide both the sides by $$12$$, we get $$x=11$$

Therefore, the numbers of boys and girls in the class are $$77$$ and $$55$$.

Worked examples

Example 1: If $$\frac{A}{B}=\frac{2}{3}$$.Find the value of $$\frac{4A-B}{A+B}$$.

Step 1: Put the given proportion equal to $$k$$.

$$\frac{A}{B}=\frac{2}{3}=k$$.

Step 2: Make two separate equations from the above expression.

$$\frac{A}{2}=k$$

$$\frac{B}{3}=k$$

Therefore, $$A=2k$$ and $$B=3k$$

Step 3: Substitute the value of $$A$$ and $$B$$ in $$\frac{4A-B}{A+B}$$ and solve it.

$$\frac{4A-B}{A+B}=\frac{4\times2k-3k}{2k+3k}$$

$$\frac{8k-3k}{5k}$$

$$\frac{5k}{5k} = 1$$

Step 4: Answer in the preferred notation.

The value of the given fraction is $$1$$. 

Example 2: If $$X:Y=4:5$$ and $$Y:Z=6:7$$. Find $$X:Z$$.

Step 1: Convert the ratios into fraction.

$$\frac{X}{Y}=\frac{4}{5}$$ and $$\frac{Y}{Z}=\frac{6}{7}$$

Step 2: Multiply the above two fractions.

$$\frac{X}{Y}\times\frac{Y}{Z}$$

$$\frac{4}{5}\times \frac{6}{7}$$

$$\frac{X}{Z}=\frac{24}{35}$$

Step 3: Write the answer in the preferred notation.

Therefore, $$X:Z=24:35$$.