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# Proportion of Quantities

The comparison of two numbers or quantities by division is known as the ratio. The symbol ‘:’ is used to denote the ratio. Proportion is the quality of to ratios. In this topic, we will see the concepts of the proportion of two quantities. We will also solve many examples that may be asked from the concept of the proportion of quantities.

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The equality of two or more ratios is what we call the proportion. For example, let us say that any two quantities, a and b are in a ratio as a:b. Suppose two more quantities, c and d are also in some ratio c:d. In a ratio, the first term is what we call the antecedent (here a and c) and the second term is what we call the consequent (here b and d respectively). Now the proportion of these quantities will come into the picture if we have a:b = c:d. Thus the proportion of these quantities in terms of the fractions is equal to:

a/b = c/d. In terms of the ratio, we write that a:b :: c:d.

The quantities ‘a’ and ‘d’ are what we call the extremes, while we call the quantities ‘b’ and ‘c’ as the mean terms. Remember that the mean terms are not the average or any measure of central tendency. There is an important property that you may want to keep in mind here, the product of the means is always equal to the product of the extremes. In the language of mathematics, we can write that if a:b :: c:d then, b×c = a×d.

## Comparison Of Ratios In Proportion

Let us consider the same four quantities that are in proportion. Suppose we have a:b and b:c as two different ratios. Then we say that (a:b) > (c:d) if and only if a/b > c/d or ad>bc.

Similarly if we have (a:b) < (c:d), then we must have a/b < c/d or ad < bc. Let us see a simple example of the proportion of quantities.

Example 1: If a:b = 5:9 and b:c = 4 : 7, then the value of a:b:c will be:

A) 10: 18:32                B) 2:3:7                  C) 3:13:17                D) 20:36:63

Answer: Here only three quantities are present. In the first ratio, b is proportional to 9 (both are in the denominator) while as in the second ratio b is proportional to 4 (both are in the numerator). Let us make b proportional to one of the two terms i.e. either 9 or  4 in both the ratios. In this way, we will be able to find the ratio a:b:c.

We will try to make the quantity ‘b’ proportional to 9 in both the ratios. Therefore we may write:

b:c = 4:7 = {4×9/4} : {7×9/4} = {9 : 63/4}

Therefore we can write a:b:c = 5:9: 63/4 = 20:36:63 and hence the correct option is D) 20:36:63.

## Solved Examples

Example 2: The ratio of alcohol: water in a mixture is 4:3. If we add 5 liters of water to the mixture, this ratio becomes 4: 5. Then the quantity or concentration of alcohol in the mixture is:

A) 0.2 litres            B) 2 litres              C) 8 litres              D) 10 litres

Answer: Suppose the quantity of the alcohol and water is 5x and 3x respectively. ‘x’ is an unknown variable. Then as per the question, we have:

4x/[3x + 5] = 4/5 or 20x = 4(3x + 5)

In other words, we can write 8x = 20 or x = 2.5. Hence the quantity of alcohol in the given mixture is equal to 4×2.5 = 10 liters. Thus the correct option is D) 10 liters.

Example 3: Four people A, B, C and D work together in an office such that salary od A is less than the salary of D. Their salaries are in a proportion such that the salaries of B and C are the mean terms of the proportion. If the salary of C increases by a factor of 1/5, then what should be the change in the salary of the other people, individually, such that we maintain the proportion and the minimum money is spent?

A) Salary remains same for A and B but for D increases by a factor of 1/5.

B) Salary of each individual changes by 1/5.

C) Salary remains same for B and D, but increases by a factor of 1/5 for A.

D) Everybody’s salary decreases by a factor of 4/5 except C’s.

Answer: Let a, b, c, and d represent the individual salaries of A, B, C, and D respectively. Then as per the question, these quantities form a proportional such that:

a/b = c/d . Now when the salary of C increases by a factor of 1/5. Therefore the new salary of C = c + (1/5) c = 6c/5 or 6/5 (c).

Now in order to maintain the proportion, we can either increase the salary of d by a factor of 1/5 or of a by the same factor. We will not increase the salary of every person by a factor of 1/5 because the question asks us to select the condition in which we will have to spend the minimum amount. therefore to maintain the proportional and to spend the minimum amount, we increase the salary of A by a factor of 1/5 keeping the salaries of the other two same as before.

Learn the Simple Concept of Variation here.

## Practice Problems

Q 1: A boy carries 50 p, 25 p, and 10 p coions in the ratio of 5:9:4 respectively. the total money the boy carries is Rs. 206. Then the number of coins of each kind as per the order is:

A) 129, 673, 836            B) 200, 360, 160                    C) 364, 385, 368            D) 654, 633, 532

Ans: B) B) 200, 360, 160.

Q 2: If A:B:C = 2:3:4, then A/B:B/C:C/A will be equal to:

A) 8: 9: 24           B) 12: 33: 23          C) 12: 23: 8                D) 8: 34: 7

Ans: A) 8: 9: 24

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