Ratio, Proportion
- If the square of the sum of two numbers is equal to 4 times of their product, then the ratio of these numbers is :
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(x + y)2 = 4xy
⇒ x2 + y2 + 2xy – 4xy = 0
⇒ (x – y)2 = 0 ⇒ x = y
⇒ x : y = 1 : 1Correct Option: C
(x + y)2 = 4xy
⇒ x2 + y2 + 2xy – 4xy = 0
⇒ (x – y)2 = 0 ⇒ x = y
⇒ x : y = 1 : 1
- The ratio of number of balls in bags x,y is 2 : 3. Five balls are taken from bag y and are dropped in bag x. Number of balls are equal in each bag now. Number of balls in each bag now is
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Number of balls in bag x and y respectively = 2a and 3a
∴ 3a – 5 = 2a + 3
⇒ a = 5 + 3 = 8
∴ Total number of balls
= 5a = 40
∴ Balls in each bag = 20Correct Option: B
Number of balls in bag x and y respectively = 2a and 3a
∴ 3a – 5 = 2a + 3
⇒ a = 5 + 3 = 8
∴ Total number of balls
= 5a = 40
∴ Balls in each bag = 20
- The ratio between a two – digit number and the sum of the digits of that number is 4 : 1. If the digit at the unit’s place is 3 more than the digit at the ten's place, then the number is
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Check through options
Number = 36
3 + 6 = 9; 6 – 3 = 3
and 36 : 9 = 4 : 1
Otherwise– 10x + y : x + y = 4 : 1
⇒ 10x + (x + 3) : x + (x + 3) = 4 : 1⇒ 11x + 3 = 4 2x + 3 1
⇒ 11x + 3 = 8x + 12
⇒ 3x = 9 ⇒ x = 3 and y = 6
Then the number is 36.Correct Option: C
Check through options
Number = 36
3 + 6 = 9; 6 – 3 = 3
and 36 : 9 = 4 : 1
Otherwise– 10x + y : x + y = 4 : 1
⇒ 10x + (x + 3) : x + (x + 3) = 4 : 1⇒ 11x + 3 = 4 2x + 3 1
⇒ 11x + 3 = 8x + 12
⇒ 3x = 9 ⇒ x = 3 and y = 6
Then the number is 36.
- Of three positive numbers, the ratio of 1st and 2nd is 8 : 9, that of 2nd and 3rd is 3:4. The product of 1st and 3rd is 2400. The sum of the three numbers is
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A : B = 8 : 9
B : C = 3 : 4 = 9 : 12
∴ A : B : C = 8 : 9 : 12
∴ Numbers = 8x, 9x and 12x
∴ 8x × 12x = 2400⇒ x2 = 2400 = 25 8 × 12
∴ x = √25 = 5
∴ A + B + C = 8x + 9x + 12x
= 29x
= 29 × 5 = 145Correct Option: A
A : B = 8 : 9
B : C = 3 : 4 = 9 : 12
∴ A : B : C = 8 : 9 : 12
∴ Numbers = 8x, 9x and 12x
∴ 8x × 12x = 2400⇒ x2 = 2400 = 25 8 × 12
∴ x = √25 = 5
∴ A + B + C = 8x + 9x + 12x
= 29x
= 29 × 5 = 145
- What number should be added to each of 6, 14, 18 and 38 so that the resulting numbers make a proportion ?
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6 + x = 18 + x 14 + x 38 + x
From the given alternatives6 + 2 = 18 + 2 14 + 2 38 + 2 ⇒ 1 = 1 2 2
Second Method :
Here, a = 6, b = 14, c = 18, d = 38Required number x = bc − ad (a + d) − (b + c) = 14 × 18 − 6 × 38 (6 + 38) − (14 + 18) = 252 − 228 44 − 32 = 24 = 2 12 Correct Option: B
6 + x = 18 + x 14 + x 38 + x
From the given alternatives6 + 2 = 18 + 2 14 + 2 38 + 2 ⇒ 1 = 1 2 2
Second Method :
Here, a = 6, b = 14, c = 18, d = 38Required number x = bc − ad (a + d) − (b + c) = 14 × 18 − 6 × 38 (6 + 38) − (14 + 18) = 252 − 228 44 − 32 = 24 = 2 12
