Number System
- The product of two numbers is 24 times the difference of these two numbers. If the sum of these numbers is 14, the larger number is
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Let the first number be x.
∴ Second number = 14–x
∴ x (14 – x ) = 24 (x – 14 + x)
⇒ x (14 – x ) = 24 (2x – 14)
⇒ 14x – x2 = 48x – 336
⇒ x2 + 34x – 336 = 0
⇒ x2 + 42x – 8x – 336 = 0
⇒ x (x + 42) – 8 (x + 42) = 0
⇒ (x + 42) (x – 8) = 0
∴ x = 8 as x ≠ – 42
∴ Second number = 14 – 8 = 6
∴ Larger number = 8
Note : : It is preferable to solve it
by oral calculation with the help
of given alternatives.Correct Option: B
Let the first number be x.
∴ Second number = 14–x
∴ x (14 – x ) = 24 (x – 14 + x)
⇒ x (14 – x ) = 24 (2x – 14)
⇒ 14x – x2 = 48x – 336
⇒ x2 + 34x – 336 = 0
⇒ x2 + 42x – 8x – 336 = 0
⇒ x (x + 42) – 8 (x + 42) = 0
⇒ (x + 42) (x – 8) = 0
∴ x = 8 as x ≠ – 42
∴ Second number = 14 – 8 = 6
∴ Larger number = 8
Note : : It is preferable to solve it
by oral calculation with the help
of given alternatives.
- Five times of a positive integer is equal to 3 less than twice the square of that number. The number is
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Let the positive integer be x.
∴ 2x2 – 5x = 3
⇒ 2x2 – 3 = 0
⇒ 2x2 – 6x + x – 3 = 0
⇒ 2x (x – 3) + 1 (x – 3) = 0
⇒ (x – 3) (2x + 1) = 0∴ x = 3 and x = – 1 2
is not admissible.Correct Option: A
Let the positive integer be x.
∴ 2x2 – 5x = 3
⇒ 2x2 – 3 = 0
⇒ 2x2 – 6x + x – 3 = 0
⇒ 2x (x – 3) + 1 (x – 3) = 0
⇒ (x – 3) (2x + 1) = 0∴ x = 3 and x = – 1 2
is not admissible.
- The sum of the digits of a two digit number is 10. The number formed by reversing the digits is 18 less than the original number. Find the original number.
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Let the two digit number be 10x + y where x > y.
Here, x + y = 10 ...(i)
and, 10x + y – 10y – x = 18
⇒ 9x – 9y = 18
⇒ 9 (x – y) = 18
⇒ x – y = 2 ...(ii)
Solving equations (i) and (ii),
x = 6 and y = 4
∴ Number = 10 × 6 + 4 = 64Correct Option: C
Let the two digit number be 10x + y where x > y.
Here, x + y = 10 ...(i)
and, 10x + y – 10y – x = 18
⇒ 9x – 9y = 18
⇒ 9 (x – y) = 18
⇒ x – y = 2 ...(ii)
Solving equations (i) and (ii),
x = 6 and y = 4
∴ Number = 10 × 6 + 4 = 64
- Of the three numbers, the second is twice the first and is also thrice the third. If the average of these three numbers is 44, the largest number is
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Let the third number be x.
∴ Second number = 3xFirst number = 3 x 2
According to the question,3x + 3x + x = 44 × 3 2 ⇒ 3x + 2x + 6x = 44 × 3 2
⇒ 11x = 88 × 3⇒ x = 88 × 3 = 24 11
∴ The largest number
= 3x = 3 × 24 = 72
Correct Option: C
Let the third number be x.
∴ Second number = 3xFirst number = 3 x 2
According to the question,3x + 3x + x = 44 × 3 2 ⇒ 3x + 2x + 6x = 44 × 3 2
⇒ 11x = 88 × 3⇒ x = 88 × 3 = 24 11
∴ The largest number
= 3x = 3 × 24 = 72
- The sum and product of two numbers are 12 and 35 respectively. The sum of their reciprocals will be
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Let the numbers be x and y respectively.
∴ x + y = 12 and xy = 35∴ 1 + 1 = x + y = 12 x y xy 35 Correct Option: A
Let the numbers be x and y respectively.
∴ x + y = 12 and xy = 35∴ 1 + 1 = x + y = 12 x y xy 35
