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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

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.