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What is Proportion?

“Proportion is the statement that identifies the fact that two ratios have equal values. For instance, consider the following.”

\(\dfrac{a}{b} = \dfrac{u}{v}\)

In the above mathematical statement, it shows that the value of a/b (“a” divided by “b”) is the same as the value of u/v (“u” divided by “v”).  Suppose that if the value of \(\Big(\dfrac{a}{b}\Big)\) is 10 then \(\Big(\dfrac{u}{v}\Big)\) would have a value of 10 as well.

A proportion between ratios can be expressed using 2 forms.

1.    The fraction form

The proportion between a,b and u,v would appear as follows if the fraction layout is used.

\(\dfrac{a}{b} = \dfrac{u}{v}\)

In the fraction form, a forward slash sign “/” is used between each pair of numbers.

2.    Ratio Form

The proportion \(c,d\) and \(e,f\) would appear in the following way if the ratio layout is used.

\(c:d = e:f\)

In the ratio form, a colon sign “:” appears between every pair of variables instead of the forward slash. 

Proportion Formula

The proportion formula is given below for pairs of variables \((a,b)\) and \((c,d)\)

\(\text{Proportion} = \dfrac{a}{b} = \dfrac{c}{d}\)

The proportion concept is used to determine the value of the unknown variable X. Consider that the value of X needs to be determined in the equation given below.

\(\dfrac{36}{6} = \dfrac{X}{10}\)

The solution is given as follows

\(\dfrac{36}{6} = \dfrac{X}{10}\)

\(6 = \dfrac{X}{10}\)

\(X = 6 \times10\)

\(X = 60\)

Understanding the steps of calculation

Here are the steps which have been performed in the above question

  1. Here, we have two ratios and one of them has an unknown value “x”. Through proportion concepts, the value of X has to be determined.  Here you need to know the terms “extremes” and “means”. In the above example, we have four values \(36, 6, X\) and \(10\). Extremes are the values that form a downward slope (36 and 10 in this case). However, the values making an upward slope are (6 and X).
  2. Multiply the two extreme values and two mean values with each other respectively. This would give you the following equation.

    \(6 \times 10 = X \times 1\)
    \(60 = X\)
    \(X = 60\)

Cross Multiplication for Solving Proportions

When you are solving a proportion, the concept of cross multiplication is applied. What is cross multiplication? When you have two ratios with values expected to be equal, certain steps of simplification are performed. The first of these steps is cross multiplication. For instance, consider that we have the following two ratios which are considered to be equal.

  1. \(\dfrac{c}{p}\)
  2. \(\dfrac{d}{q}\)

As these two ratios are equal, they can be used in the form of proportion. In other words, we can write the following statement to elaborate on this point.

\(\dfrac{c}{p} = \dfrac{d}{q}\)

Now, we need to perform cross multiplication to proceed with the process of calculating proportion. In the above sets of ratios, “c” will be multiplied by “q” and “d” will be multiplied by “p”. This form of multiplication is called cross-multiplication because values are multiplied in the form of two diagonals which appears like a cross. Let us proceed with the implication steps.

\(\dfrac{c}{p} = \dfrac{d}{q}\)

\(c \times q = p \times d\)

Cross Multiplication to check proportion between two unknown variables

If you have two unknown variables, the cross multiplication concept can be used to check the proportion between two unknown variables. For instance, consider that we have the following ratios

  1. \(\dfrac{A}{8}\)
  2. \(\dfrac{B}{4}\)

Let us consider that these two ratios are equal which means that they are in proportion. Hence, the following statement would be constructed after equating them.

\(\dfrac{A}{8} = \dfrac{B}{4}\)

Performing the cross multiplication step. This would give us

\(A \times4 = B \times 8\)

\(4A = 8B\)

Now, divide the right-hand side of the equation to determine the value of A in terms of B

\(A = \dfrac{8B}{4}\)

\(A = 2B\)

According to the above resultant statement, the value of A would be two times the value of B. If “B” has a value of 4, A would be 8.