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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

10. Zinc and copper are melted together in the ratio 9 : 11. What is the weight of melted mixture, if 28.8 kg of zinc has been consumed in it?

• A. 58 kg
• B. 60 kg
• C. 64 kg
• D. 70 kg
 Ans: C. Sol. For 9 kg Zinc, mixture melted = (9+11) kg. For 28.8 kg Zinc, mixture, melted = [20/9 x 28.8] kg = 64 kg.

11. 20 litres of a mixture contains milk and water in the ratio 5 : 3. If 4 litres of this mixture be replaced by 4 litres of milk, the ratio of milk to water in the new mixture would be

• A. 2 : 1
• B. 4 : 3
• C. 8 : 3
• D. 7 : 3
 Ans: D. Sol. Quantity of milk in 16 litres of mix = [16x5/8] litres = 10 litres. Quantity of milk in 20 litres of new mix. = (10+4) litres. Quantity of water in it = (20 - 14) litres = 6 litres. ∴ Ratio of milk and water in the new mix. = 14 : 6 = 7 : 3.

12. A sum of money is to be distributed among A, B, C, D in the proportion of
5 : 2 : 4 : 3. If C gets Rs. 1000 more than D, what is B’s share?

• A. Rs. 500
• B. Rs. 1000
• C. Rs. 1500
• D. Rs. 2000
 Ans: D. Sol. Let the shares of A, B, C and D be Rs. 5x, Rs. 2x, Rs. 4x and Rs. 3x respectively. Then 4x - 3x = 1000 ⇔ x = 1000. ∴ B's share = Rs. 2x = Rs. (2x1000) = Rs. 2000.