Important Aptitude formulas - Ratio and Proportion
RATIO:
The ratio of two quantities a and b in the same units, is the fraction a/b and we write it as x : y.
In the ratio x : y, we call ‘x’ as the first term or antecedent and ’y’, the second term or consequent.
1. A ratio x : y does not change, when both its terms are multiplied or divided by the same number, i.e., The ratio is unaltered if each term is multiplied or divided by the same number.
2. A ratio is always expressed in its lowest terms. I.e. The quantities of a ratio must be expressed in the same units.
3. x/y ≠ y/x when x ≠ y
4. If the terms of a ratio are fractions, they can be multiplied by the L.C.M of their denominators.
PROPORTION:
The equality of two ratios is called proportion.
1. If a : b = c : d, we write, a : b :: c : d and we say that a, b, c, d are in proportion . Here ‘a’ and ‘d’ are called extremes, while ‘b’ and ‘c’ are called mean terms.
2. In proportion, a : b :: c : d
Product of means = Product of extremes.
Thus, a : b :: c : d <=> (b x c) = (a x d).
3. For a proportion, a : x :: x : b, x = √ab
4. When a certain quantity ‘x’ is divided in the ratio of a:b:c, then three parts are
ax/(a + b + c), bx/(a + b + c), cx/(a + b + c)
5. If A : B = a : b and B : C = p : q, Then, A : B : C = ap : bp : bq
6. Fourth Proportional: If a : b = c : d, then d is called the fourth proportional to a, b, c.
7. Third Proportional: If a : b = b : c, then c is called the third proportional to ‘a’ and ‘b’.
8. Mean Proportional: Mean proportional between ‘a’ and ‘b’ is square root of ‘ab’
9. If two investment in the ratio a : b and period in the ratio p : q. Then profit will be shared in the ratio of ap : bq
10. If ‘p’ kg of one kind of material costing Rs A per kg is mixed with ‘q’ kg of another kind of material of costing Rs. B per kg, then the price of the mixture is (Ap + Bq)/(p + q) per kg.
11. In mean, days, hours and work ratio proportion problem, M₁D₁H₁/W₁ = M₂D₂H₂/W₂
Compounded Ratio:
The compounded ratio of the ratios (a : b), (c : d), (e : f) is (ace : bdf)
Duplicate Ratio:
1. Duplicate ratio of (a : b) is (a² : b²).
2. Sub-duplicate ratio of (a : b) is (√a : √b).
3. Triplicate ratio of (a : b) is (a3 : b3).
4. Sub-triplicate ratio of (a : b) is (a1/3 : b1/3).
5. If (a/b)=(c/d), then [(a + b)/(a - b)] = [(c + d)/(c - d)] (Componendo and dividendo)
Variation:
1. We say that ‘x’ is directly proportional to y, if x = ky for some constant ‘k’ and we write, x ∝ y.
2. We say that ‘x’ is inversely proportional to y, if xy = k for some constant ‘k’ and we write, x ∝ (1/y)
Example: 01
A bag contains Rs. 187 in the form of 1
rupee; 50 paisa and 10 paisa coins in the ratio 3 : 4 : 5 . Find the number of
each type of coins.
Solution:
Ratio of 1
rupee, 50 paisa and 10 paisa coins = 3 : 4 : 5
Ratio of
values of 1 rupee, 50 paisa and 10 paisa coins
= (3/1) : (4/2) : (5/10) = (3/1) : 2
; (1/2) = 6 : 4 : 1
∴ Sum of the ratios = 6 = 4 + 1 = 11
Value of 1 rupee coins = (6/11) × 187 =
Rs.102
Value of 50
paisa coins = (4/11) × 187 = Rs. 68
Value of 50
paisa coins = (1/11) × 187 = Rs. 17
∴ Number of 1 rupee coins = 102 × 1 = 102
Number of 50 paisa coins = 68 × 2 = 136
Number of 10 paisa coins = 17 × 10 = 170
Example: 02
If A: B = 3 : 2 and B : C = 3 : 4 then A: C
is equal to
Solution:
Given, A : B
= 3 : 2 and B : C = 3 : 4
Therefore, A
: B : C = (3 × 3) : (2 × 3) : (2 × 4) = 9 : 6 : 8
∴ A : C = 9 : 8
{or, (A/B) ×
(B/C) = (3/2) × (3/4) => A : C = 9/8}
Example: 03
An amount of Rs. 2430 is divided among A, B
and C such that if their shares be reduced by Rs. 5, Rs. 10 and Rs. 15
respectively. The remainders shall be in the ratio 3 : 4 : 5 . The share of B is:
Solution:
Remainder = Rs. [2430 - (5 + 10 + 15)] = Rs. 2400
Remainder = Rs. [2430 - (5 + 10 + 15)] = Rs. 2400
∴ B’s share = [2400 × (4/12)] + 10 = Rs.810