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Numerical Aptitude (Chapter : Ratio & Proportion)

Ratio

The ratio of two quantities a and b is the fraction a/b and is expressed as a : b. Here a is the first term or antecedent and b is the second term or consequent. Since the ratio expresses the number of times one quantity contains the other, it is an abstract (without units) quantity.

Kinds of Ratios

• Duplicate Ratio: a2 : b2 is called duplicate ratio of a : b.
• Triplicate Ratio: a3 : b3 is called triplicate ratio of a : b.
• Sub-Duplicate Ratio: is called sub-duplicate ratio of a : b.
• Sub-triplicate Ratio : is called sub-triplicate ratio of a : b.

Proportion

When two ratios are equal, they make a proportion. ie, (a/b)=(c/d) if then a, b, c and d are in proportion.
This is represented as a : b : : c : d and is read as “a is to b as c is to d”.
When a, b, c and d are in proportion, then a and d are called the Extremes and b and c are called the Means. also, Product of the Means = Product of the Extremes ie, bc = ad.

Continued Proportion

If three quantities a, b and c are such that a : b : : b : c, then b² = ac and a, b and c are in continued proportions. Also, the quantity c is called the third proportion of a and b.

Fourth Proportion

If four quantities a, b, c and x are such that a : b : : b : c, then ax = bb and x is called the fourth proportion of a, b, and c.

Mean or Second Proportion

If three quantities a, b and x are such that a : x : : x : b, then x² = ab and x is called the mean of a and b. Also, If a : b = c : d, then the following properties hold good.
(i) b : a = d : c (Invertendo)
(ii) a : c = b : d (Alter nendo)
(iii) (a + b) : b = (c + d) : d (Componendo)
(iv) (a – b) : b = (c – d) : d (Dividendo)

Variation

If two quantities x and y are related in such a way that as the quantity x changes it also brings a change in the second quantity y, then the two quantities are in variation.

Direct Variation

The quantity x is in direct variation to y if an increase in x makes y to increase proportionally. Also a decrease in x makes y to decrease proportionally it can be expressed as x = ky, where, k is called the constant of proportionality. eg, Cost is directly proportional to the number of articles bought.

Inverse Variation

The quantity x is in inverse variation to y if an increase in x makes y to decrease proportionally. Also, a decrease in x makes y to increase proportionally. It can be expressed as x=(k/y) where, k is called the constant of proportionality. eg, The time taken by a vehicle in covering a certain distance is inversely proportional to the speed of the vehicle.

:: Home Assignment for Practice ::

1. Ratio of the earnings of A and B is 4 : 7. If the earnings of A increase by 50% and those of B decrease by 25%, the new ratio of their earnings becomes 8 : 7. What are A’s earnings?

(a) Rs 25,000
(b) Rs 26,000
(c) Rs 29,000
(d) Data incorrect

2. What least number must be subtracted from each of the numbers 14, 17, 34 and 42 so that the remainders may be proportional?

(a) 4
(b) 3
(c) 2
(d) 9

3. In a mixture of 60 litres, the ratio of milk and water is 2 : 1. If this ratio is to be 1 : 2, then the quantity of water to be further added is:

(a) 30 litres
(b) 70 litres
(c) 40 litres
(d) 60 litres

4. The fourth proportional to 5, 8, 15 is :

(a) 22
(b) 24
(c) 23
(d) 20

5. The mean proportional between 234 and 104 is:

(a) 16
(b) 40
(c) 54
(d) None of these

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