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Ratio and Proportion

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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.
2. 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.