Ex 3.3, 3
Form the pair of linear equations for the following problems and find their solution by substitution method.
(v) 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 fraction.
Let Numerator be x
& Denominator be y
So, fraction is 𝒙/𝒚
Given that
If 2 is added to both the numerator and the denominator, fraction becomes 9/11
(𝑵𝒖𝒎𝒆𝒓𝒂𝒕𝒐𝒓 + 𝟐)/(𝑫𝒆𝒏𝒐𝒎𝒊𝒏𝒂𝒕𝒐𝒓 + 𝟐)=𝟗/𝟏𝟏
(𝑥 + 2)/(𝑦 + 2)=9/11
11(x + 2) = 9(y + 2)
11x + 22 = 9y + 18
11x – 9y = 18 − 22
11x – 9y = −4
Also,
Given that if 3 is added to both the numerator and the denominator, fraction becomes 5/6
(𝑵𝒖𝒎𝒆𝒓𝒂𝒕𝒐𝒓 + 𝟑)/(𝑫𝒆𝒏𝒐𝒎𝒊𝒏𝒂𝒕𝒐𝒓 + 𝟑)=𝟓/𝟔
(𝑥 + 3)/(𝑦 + 3)=5/6
6(x + 3) = 5(y + 3)
6x + 18 = 5y + 15
6x – 5y = 15 − 18
6x – 5y = −3
Hence, our equations are
11x – 9y = −4 …(1)
6x – 5y = −3 …(2)
From (1)
11x – 9y = −4
11x = 9y – 4
x = ((𝟗𝒚 − 𝟒)/𝟏𝟏)
Putting x in (2)
6x – 5y + 3 = 0
6 ((9𝑦 − 4)/11) – 5y + 3 = 0
Multiply both sides by 11
11 ×(6 (9𝑦 − 4))/11 −5𝑦×11+3×11=0×11
6(9y – 4 ) – 55y + 33 = 0
6(9y) – 6(4) – 55y + 33 = 0
54y – 24 – 55y + 33 = 0
–y + 9 = 0
y = 9
Putting y = 9 in (1)
11x – 9y = −4
11x – 9(9) = −4
11x – 81 = −4
11x = −4 + 81
11x = 77
x = 77/11
x = 7
Therefore x = 7, y = 9
So,
Numerator = x = 7
Denominator = y = 9
Hence,
Original fraction = 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟/𝐷𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟
"= " 𝑥/𝑦
=𝟕/𝟗

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.