Ratio and Proportion - MathsTips.com

Ratio

In comparing two quantities of the same kind, the fraction, which expresses by how many times the first quantity is greater or smaller than the second quantity is called the ‘ratio’ between the first quantity and the second quantity.

The mathematical symbol of ratio is ‘:‘

It is written as say, 1:4 and read as 1 “is to” 4. The first of the two quantities forming a ratio is called the antecedent and the second is called the consequent of the ratio. The two together are called the terms of the ratio.

Points to Note:

1. Since the quotient of two quantities of the same kind is an abstract number, the ratio involving two quantities of the same kind is abstract. It has no unit.

2. In determining the ratio, the quantities are to be expressed in the same unit.

For example, the ratio between 2Kg and 1 tonne is 2 Kg : 1000 Kg = 1 : 500.

3. The ratio of two quantities of different kinds is not possible.

For example, 4 metres : Rs 6 is inadmissible as the comparison is not possible.

4. Inverse ratio: If two ratios, the antecedent and the consequent of one are respectively the consequent and antecedent of the other, they are said to be ‘inverse ratio’ or ‘reciprocal’ to one another.

For example, the inverse ratio of $3: 4$ is $4: 3$ and the inverse ratio of $4: 3$ is $3: 4$ . The reciprocal of $\dfrac{2}{3}$ is $\dfrac{3}{2}$ and the reciprocal of $3$ is $\dfrac{1}{3}$ .

Corollary: The product of a ratio and its inverse is always unity.

Types of Ratio

Ratios are mainly of two types:

Simple Ratio
Compound Ratio

The ratio between two quantities of the same kind is called ‘simple ratio‘.

Example, $Rs \: 4 : Rs \: 5 = 4 : 5$

When the product of the antecedents of two or more simple ratios is considered as the antecedent and the product of their consequents is considered as the consequent, the ratio thus formed is known as a ‘compound ratio‘.

Thus, the compound ratio of $4 : 5$ , $6 : 7$ and $5 : 6$ is $(4 \times 6 \times 5 ) : ( 5 \times 7 \times 6 ) = 4 : 7$

There are three kinds of simple ratios:

Ratio of greater inequality
Ratio of equality and
Ratio of lesser inequality

1. In ratio of greater inequality, the antecedent is greater than the consequent.

For example, 20 : 13. Its value is always greater than 1.

2. In the ratio of equality, the antecedent is equal to the consequent.

For example, 3 : 3. Its value is always equal to 1 .

3. In the ratio of lesser inequality, the antecedent is smaller than the consequent.

For example, 5 : 18. Its value is always less than 1 .

Ratio of greater inequality and ratio of lesser inequality are called ‘ratios of inequality’. Evidently, inverse ratio or reciprocal of a ratio of greater inequality is a ratio of lesser inequality.

Summing up important points

The ratio of two quantities $= \dfrac{Antecedent}{Consequent}$
Antecedent and Consequent are similar quantities and in determining the ratio, they are to be expressed in the same unit.
Ratio may be a whole number or a fraction.
The ratio is an abstract number, it has no unit.
Since a ratio is expressed as a fraction, its value is not changed when both its antecedent and consequent are multiplied or divided by the same number except 0.
Ratio can be expressed in reduced form.
Conversion into same denominator, etc. can also be applied in the case of ratios.

Proportion

When the values of two ratios are equal, they are said to be in ‘proportion’ and one is called proportional to the other.
Four quantities are said to be in proportion or are proportional when the ratio of the first to the second is equal to the ratio of the third to the fourth.

For example, $4 : 6, 10 : 15$ are in proportion and the four numbers $4, 6, 10, 15$ are proportional. This is because $4 : 6=2 : 3$ and $10 : 15=2 : 3$

Mathematical symbol of proportion is ‘: :’. The symbol ‘: :’ is used in place of ‘=’

Therefore, $4 : 6 :: 10 : 15$ i.e., $4 : 6 = 10: 15$

If a ratio is equal to the reciprocal of the other, then either of them is in ‘inverse proportion’ of the other.

Continued Proportion

Three quantities of the same kind are said to be in ‘Continued Proportion ‘when the ratio of the first to the second is equal to the ratio of the second to the third. The second quantity is called mean proportional between the first and the third and the third quantity is called third proportional to the first and second. Thus $2, 4, 8$ are in continued proportion for $2 : 4 :: 4 : 8$ . Here, $4$ is the mean proportion between $2$ and $8$ and $8$ is the third proportional to $2$ and $4$ . There may be continued proportion in case of more than three quantities of the same kind. In general, for any odd number of quantities of the same kind, continued proportion will exist among the quantities.

For example, $1 : 2 : : 2 : 4 : : 4 : 8 : : 8 : 16$ .

$\therefore$ $2, 4, 8, 16$ are in continued proportion.

Points to Note:

When four abstract numbers are in proportion, the product of the extremes is equal to the product of the means. That is, $(first \hspace{1mm} number) \times (fourth \hspace{1mm} number) = (second \hspace{1mm} number) \times (third \hspace{1mm} number)$ . Thus, $3 : 5 :: 12 : 20$ gives us $3 \times 20 = 5 \times 12$ . We basically have to express the ratios as fractions and multiply both the fractions by the product of the two consequents. This sort of multiplication is known as ‘Cross Multiplication’.
If three quantities be in continued proportion, the product of the first and the third is equal to the square of the second.

Examples

1. If $A : B = 2 : 3, B : C = 4 : 5$ and $C : D = 6 : 7$ , find i) $A : D$ and ii) $A : B : C : D$

Solution:

i. $\dfrac{A}{B} \times \dfrac{B}{C} \times \dfrac{C}{D} = \dfrac{2}{3} \times \dfrac{4}{5} \times \dfrac{6}{7}$

$\: or, \dfrac{A}{D} = 16: 35$

$\: or, A: D = 16: 35$

ii. $A : B = 2 : 3$

$B: C = 4: 5 = 1: \dfrac{5}{4} = (1 \times 3) : (\dfrac{5}{4} \times 3) = 3: \dfrac{15}{4}$

$C: D = 6: 7 = 1: \dfrac{7}{6} = (1 \times \dfrac{15}{4}) : (\dfrac{7}{6} \times \dfrac{15}{4}) = \dfrac{15}{4} : \dfrac{35}{8}$

$\therefore$ $A : B : C : D = 2 : 3 : \dfrac{15}{4} : \dfrac{35}{8} = 16 : 24 : 30 : 35$

2. If $\dfrac{2}{3} A = 75\% \: of \: B = 0.6 \: of \: C$ , find $ latex A : B : C $

Solution:

$A \times \dfrac{2}{3} = B \times \dfrac{75}{100} = \dfrac{6}{10} \times C$

$or, A \times \dfrac{2}{3} = B \times \dfrac{3}{4} = C \times \dfrac{3}{5} = k \: (say)$

$or, A = \dfrac{3k}{2}, B = \dfrac{4k}{3}, C = \dfrac{5k}{3}$

$\therefore$ $A: B: C = \dfrac{3k}{2} : \dfrac{4k}{3} : \dfrac{5k}{3} = \dfrac{3}{2} : \dfrac{4}{3} : \dfrac{5}{3} = 9: 8: 10$

3. Find the fourth proportional to $5, 15 \: and \: 8$ .

Solution:

Let the fourth proportion be ‘ $x$ ‘. Here, $\dfrac{5}{15} = {8}{x}$

Hence, $5 \times x = 15 \times 8$

$or, x = \dfrac{15 \times 8}{5} = 24$ .

$\therefore$ the fourth proportion is $24$ .

4. In a continued proportion of four quantities, the first and the second proportional are $6 \: and \: 18$ . Find the fourth quantity.

Solution:

The four quantities are in continued proportion.

$\therefore$ $\dfrac{first \: quantity}{second \: quantity} = \dfrac{second \: quantity}{third \: quantity} = \dfrac{third \: quantity}{fourth \: quantity}$

Now, $\dfrac{first \: quantity}{fourth \: quantity} = \dfrac{first \: quantity}{second \: quantity} \times \dfrac{second \: quantity}{third \: quantity} \times \dfrac{third \: quantity}{fourth \: quantity}$

$or, \dfrac{first \: quantity}{fourth \: quantity} =\dfrac{first \: quantity}{second \: quantity} \times \dfrac{first \: quantity}{second \: quantity} \times \dfrac{first \: quantity}{second \: quantity}$

$= (\dfrac{first \: quantity}{second \: quantity})^3 = (\dfrac{6}{18})^3 = (\dfrac{1}{3})^3 = \dfrac{1}{27}$

$\therefore$ , $required \: fourth \: quantity \: = \: first \: quantity \times 27 = 6 \times 27 = 162$

5. The ratio of daily wages of two workers is $4: 3$ and one gets daily $Rs \: 9$ more than the other. What are their daily wages?

Solution:

Let one worker get $Rs \: 4x$ and the other $Rs \: 3x$ .

According to the problem, $4x-3x=9$ $\Rightarrow x=9$

Hence, one worker gets daily $Rs \: 4 \times 9 = Rs \: 36$ and the other one gets $Rs \: 3 \times 9 = Rs \: 27$

Exercise

Find the third proportional of $12$ and $18$ .
The ratio of two numbers is $3 : 5$ and their L.C.M. is $105$ . Find the numbers and their H.C.F.
A lantern factory received an order to supply $165$ pieces of lantern every day. The ratio of number of workers and the number of lanterns they produce everyday is $3 : 11$ . How many workers should be employed to maintain the order?
If $3x = 2y = 4z$ , find $x:y:z$
The product of three continued proportional numbers is $64$ . Find the mean proportional.

Ratio

Types of Ratio

Summing up important points

Proportion

Continued Proportion

Examples

Exercise

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