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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

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.