if numerator is increased by 8 and denominator is doubled then the resulting fraction is 1/2 find the equation in two variables to find the fraction in above statement
Answers
The answer is given below :
SOLUTION :
Let us consider the fraction as a/b,
where both a and b are real numbers.
By the given condition, the numerator is multiplied by 2 and the denominator is reduced by 5, and thus we get 6/5.
⇒ 2a/(b - 5) = 6/5
⇒ 10a = 6b - 30
⇒ 5a = 3b - 15
⇒ 5a - 3b = - 15 .....(i)
Also given, the numerator is increased by 8 and the denominator is doubled, and thus we get 2/5.
⇒ (a + 8)/2b = 2/5
⇒ 5a + 40 = 4b
⇒ 5a - 4b = - 40 .....(ii)
Thus, we get two equations with variables a and b as follows :
5a - 3b = - 15 .....(i)
5a - 4b = - 40 .....(ii)
On subtraction, we get
(5a - 3b) - (5a - 4b) = - 15 - (- 40)
⇒ 5a - 3b - 5a + 4b = - 15 + 40
⇒ b = 25
Now, putting b = 25 in (i), we get
5a - (3 × 25) = - 15
⇒ 5a - 75 = - 15
⇒ 5a = - 15 + 75
⇒ 5a = 60
⇒ a = 12
So, the required fraction is = 12/25.
CONFIRMATION :
The fraction is = 12/25
For the first condition,
(12 × 2)/(25 - 5)
= 24/20
= (4 × 6)/(4 × 5)
= 6/5
For the second condition,
(12 + 8)/(25 × 2)
= 20/50
= (10 × 2)/(10 × 5)
= 2/5
Hence, confirmed.
Thank you for your question.
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x+8/2y = 1/2
2(x+8) = 2y
2x+16-2y = 0
x-y+16=0 or x-y=-16