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## Ratio and Proportion -> IMPORTANT FACTS AND FORMULAE

 Ratio : The ratio of two quantities a and b in the same units, is the fraction a/b and we write it as a : b. Proportion : The equality of two ratios is called proportion. Mean Proportional : Mean Proportional between a and b is √ab. Fourth Proportional : If a : b = c : d, then d is called the fourth proportional to a,b,c. Third Proportional : If a : b = b : c, then c is called the third proportional to a and b. Duplicate ratio : of (a : b) is (a2 : b2). Sub-duplicate ratio : of (a : b) is (√a : √b). Triplicate ratio : of (a : b) is (a3 : b3). Variation : (I). We say that x is directly proportional to y, if x = ky for some constant k and we write, x∝y. (II). We say that x is inversely proportional to y, if xy = k for some constant k and we write, x∝1/y.

## Ratio and Proportion -> SOLVED EXAMPLES

1. Divide Rs. 1162 among A, B, C in the ratio 35 : 28 : 20. Sol. Sum of ratio terms = (35 + 28 + 20) = 83. A's share = Rs.[1162 x 35/83] = Rs. 490. B's share = Rs.[1162 x 28/83] = Rs. 392. C's share = Rs.[1162 x 20/83] = Rs. 280. If a : b = 5 : 9 and b : c = 4 : 7, find a : b : c. Sol.a : b = 5 : 9 and b : c = 4 : 7 = (4x9/4) : (7x9/4) = 9 : 63/4 ⇒ a : b : c = 5 : 9 : 63/4 = 20 : 36 : 63.

## Ratio and Proportion -> EXERCISE

16. 60 kg of an alloy A is mixed with 100 kg of alloy B. If alloy A has lead and tin in the ratio 3 : 2 and alloy B has tin and copper in the ratio 1 : 4, then the amount of tin in the new alloy is

• A. 36 kg
• B. 40 kg
• C. 44 kg
• D. 53 kg
 Ans: C. Sol. Quantity of tin in 60 kg of A = (60x2/5) kg = 24 kg Quantity of tin in 100 kg of B = (100x1/5) kg = 20 kg Quantity of tin in the new alloy = (24 + 20) kg = 44 kg.

17. The ratio of three numbers is 3 : 4 : 5 and the sum of their squares is 1250. The sum of the numbers is

• A. 20
• B. 40
• C. 60
• D. 80
 Ans: C. Sol. Let the numbers be 3x, 4x and 5x. Then, ⇔ 9x2 + 16x2 + 25x2 = 1250 ⇔50x2 = 1250 ⇔x2 = 25 ⇔x = 5. ∴ Sum of numbers = (3x + 4x + 5x) = 12x = 12x5 = 60.

18. In a ratio, which is equal to 3 : 4, if the antecedent is 12, then the consequent is

• A. 6
• B. 9
• C. 16
• D. 10
 Ans: C. Sol. We have 3/4 = 12/x ⇔ 3x = 48 ⇔ x = 16. ∴ Consequent = 16.