1) A fraction becomes if 2 is added to both the numerator and the denominator 11 5 If, 3 is added to both the numerator and the denominator it becomes6 fraction. Find the
Answers
Correct question :
A fraction becomes 9/11 if 2 is added to both the numerator and the denominator. If 3 is added to both the numerator and the denominator, it becomes 5/6. Find the original fraction.
Answer :
The required fraction is 7/9.
Step–by–step explaination :
Let the original fraction be a/b, wherein :
➵ a = numerator
➵ b = denominator
Given that, fraction becomes 9/11 if 2 is added to both the numerator and the denominator.
➵ (a + 2)/(b + 2) = 9/11
Cross multiplying them :
➵ 11 (a + 2) = 9 (b + 2)
➵ 11a + 22 = 9b + 18
➵ 11a – 9b = 18 – 22
➵ 11a – 9b = – 4 . . . . . . . ❶
Also, if 3 is added to both the numerator and the denominator, it becomes 5/6.
➵ (a + 3)/(b +3) = 5/6
Cross multiplying them :
➵ 6 (a + 3) = 5 (b + 3)
➵ 6a + 18 = 5b + 15
➵ 6a – 5b = 15 – 18
➵ 6a – 5b = – 3 . . . . . . . . ❷
Getting the value of a from ❶ :
➵ 11a – 9b = – 4
➵ 11a = – 4 + 9b
➵ a = (– 4 + 9b)/11 . . . . . . . . ❸
Substituting this value in ❷ to get b :
➵ 6a – 5b = – 3
➵ 6 [(– 4 + 9b)/11] – 5b = – 3
Multiplying 11 on both sides :
➵ {11 (6[(–4 + 9b)/11])}–{5b × 11} = –3 (11)
➵ 6 (– 4 + 9b) – 55b = – 33
➵ – 24 + 54b – 55b = – 33
➵ – b = – 33 + 24
➵ – b = – 9
➵ b = 9
Substituting the value of b in ❶ to get a :
➵ 11a – 9b = – 4
➵ 11a – 9 (9) = – 4
➵ 11a – 81 = – 4
➵ 11a = – 4 + 81
➵ 11a = 77
➵ a = 77/11
➵ a = 7
Substituting the values of a and b in the fraction form :
➵ a/b
➵ 7/9
Therefore, the original fraction is 7/9.
Verification :
Given that, when 2 is added to both the numerator and denominator, the fraction becomes 9/11.
➵ (a + 2)/(b + 2) = 9/11
➵ (7 + 2)/(9 + 2) = 9/11
➵ 9/11 = 9/11
➵ LHS = RHS
➵ Hence, verified!
Also, if 3 is added to both the numerator and the denominator, the fraction becomes 5/6.
➵ (a + 3)/(b+3) = 5/6
➵ (7 + 3)/(9 + 3) = 5/6
➵ 10/12 = 5/6
➵ 5/6 = 5/6
➵ LHS = RHS
➵ Hence, verified!