When the numerator of a fraction is increased by 7 and denominator is increased by 13, the resulting number is equivalent to the obtained when the numerator is decreased by 2 and denominator is decreased by 11. The sum of numerator and denominator is 24. Find the fraction
When the numerator of a fraction is increased by 7 and denominator is increased by 13, the resulting number is equivalent to the obtained when the numerator is decreased by 2 and denominator is decreased by 11. The sum of numerator and denominator is 24. Find the fraction Correct Answer <span class="math-tex">\(\frac{5}{{19}}\)</span>
Given:
When the numerator of a fraction is increased by 7 and denominator is increased by 13, the resulting number is equivalent to the obtained when the numerator is decreased by 2 and denominator is decreased by 11.
The sum of numerator and denominator is 24.
Calculation:
Let the numerator of a fraction be N, and the denominator of a fraction be D.
The fraction = N/D
N + D = 24
D = 24 – N ---- (1)
(N + 7)/(D + 13) = (N – 2)/(D – 11)
⇒ (N + 7) × (D – 11) = (N – 2) × (D + 13)
⇒ ND – 11N + 7D – 77 = ND + 13N – 2D – 26
⇒ 7D + 2D = 13N + 11N – 26 + 77
⇒ 9D = 24N + 51
⇒ 3D = 8N + 17 ---- (2)
Add value of D from eq (1) in eq (2)
⇒ 3(24 – N) = 8N + 17
⇒ 72 – 3N = 8N + 17
⇒ 8N + 3N = 72 – 17
⇒ 11N = 55
⇒ N = 5
Add value of N in eq (1)
D = 24 – N
⇒ D = 24 – 5
⇒ D = 19
∴ The fraction is 5/19.
