The sum of the numerator and denominator of a fraction is 11. If we add 2 to both numerator and denominator, the fraction will be increased by 1/24. What is the difference between the numerator and denominator of that fraction?a)1b)2c)6d)8Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Let the numerator of the fraction be x, so the denominator will be 11 - x

So, the fraction = x/11 - x

On adding 2 in both numerator and denominator, according to the question, the fraction will be -

(x + 2)/(11 - x + 2) = (x)/(11 - x) + 1/24

(x + 2)/(13 - x) - (x)/(11 - x) = 1/24

[11x + 22 - x² -2x - 13x + x²]/(13 - x) (11 - x) = 1/24

After solving the above equation, we will get

=> 528 - 96x = 143 - 24x + x²

x² + 72x - 385 = 0

(x + 77) (x - 5) = 0

So, x =5

So, numerator = 5

and, denominator = 11 - 5 = 6

Difference between both is = 6 - 5 = 1

Answer

Let's assume the fraction is represented as x/y, where x is the numerator and y is the denominator.

1. Determine the equation for the sum of the numerator and denominator:
x + y = 11

2. Determine the equation for the increased fraction:
(x + 2)/(y + 2) = x/y + 1/24

3. Solve the equation system:
From equation (1), we can express x as x = 11 - y.
Substituting x in equation (2), we get:
(11 - y + 2)/(y + 2) = (11 - y)/y + 1/24

4. Simplify the equation:
(13 - y)/(y + 2) = (11 - y)/y + 1/24

5. Cross-multiply to eliminate the denominators:
(y + 2)((11 - y)/y + 1/24) = (13 - y)
(y + 2)(11 - y) + (y + 2)(1/24) = (13 - y)y

6. Expand and simplify the equation:
11y - y^2 + 22 - 2y + (y + 2)/24 = 13y - y^2
11y - y^2 + 22 - 2y + 1/24 + y/24 = 13y - y^2

7. Combine like terms:
11y - y^2 + 22 - 2y + 1/24 + y/24 = 13y - y^2
11y - 2y + y + 22 + 1/24 + y/24 = 13y
10y + 22 + 1/24 + y/24 = 13y

8. Combine the constant terms:
10y + 22 + 1/24 + y/24 = 13y
10y - 13y + 22 + 1/24 + y/24 = 0
-3y + 22 + 1/24 + y/24 = 0
-3y + y/24 + 22 + 1/24 = 0

9. Combine the y terms:
(-3y + y/24) + 22 + 1/24 = 0
(-72y + y)/24 + 22 + 1/24 = 0
(-71y)/24 + 22 + 1/24 = 0

10. Combine the constant terms:
(-71y)/24 + 22 + 1/24 = 0
(-71y + 22 + 1)/24 = 0
(-71y + 23)/24 = 0

11. Solve for y:
-71y + 23 = 0
-71y = -23
y = -23/-71
y = 23/71

12. Calculate the numerator:
x = 11 - y
x = 11 - 23/71
x = 11*71/71 - 23/71
x = (781 - 23)/71
x = 758/71

13. Calculate the difference between the numerator and denominator:
Difference = |x - y|
Difference = |758/71 - 23/71|
Difference = |(758

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