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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

7. The fourth proportional to 5, 8, 15 is

• A. 18
• B. 19
• C. 20
• D. 24
 Ans: D. Sol. Let the fourth proportional to 5, 8, 15 be x. Then, 5 : 8 : : 15 : x ⇔ 5x = (8x15) ⇔ x = (8x15) / 5 = 24.

8. In a carton containing a dozen mirrors is dropped, which of the following cannot be the ratio of broken mirrors to unbroken mirrors?

• A. 2 : 1
• B. 3 : 1
• C. 3 : 2
• D. 7 : 5
 Ans: C. Sol. For dividing 12 into two whole numbers, the sum of the ratio terms must be a factor of 12. So, they cannot be in the ratio 3 : 2.

9. The speed of three cars in the ratio 5 : 4 : 6. The ratio between the time taken by them to travel the same distance is

• A. 5 : 4 : 6
• B. 6 : 4 : 5
• C. 12 : 15 : 10
• D. 10 : 12 : 15
 Ans: C. Sol. Ratio of time taken = 1 / 5 : 1/4 : 1/6 = 12 : 15 : 10.