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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

22. Salaries of Prasanna and Sathish are in the ratio 2 : 3. If the salary of each is increased by Rs. 4000, the new ratio becomes 40 : 57. What is Sathish's present salary?
 
  • A. Rs. 15000
  • B. Rs. 18000
  • C. Rs. 38000
  • D. Rs. 43000
Ans: C.
Sol.
Let the original salaries of Prasanna and Sathish be Rs. 2x and
Rs. 3x respectively. Then,
2x + 4000 / 3x + 4000 = 40/57
⇔ 57(2x + 4000) = 40(3x + 4000)
⇔ 6x = 68000
⇔ 3x = 34000.
Sathish's present salary = (3x + 4000) = (34000 + 4000) = Rs.38000.
 
23. The mean proportional between 234 and 104 is
 
  • A. 12
  • B. 39
  • C. 54
  • D. 156
Ans: D.
Sol.
Required mean proportional = √234 x 104
= √13x9x2x13x8 = (13x3x4) = 156.
 
 
24. Which of the following ratios is greatest?
 
  • A. 7 : 15
  • B. 15 : 23
  • C. 17 : 25
  • D. 21 : 29
Ans: D.
Sol.
7/15 = 0.466,
15/23 = 0.652,
17/25 = 0.68<,br/> 21/29 = 0.724,
Clearly, 0.724 is greatest and therefore, 21 : 29 is greatest.