a fraction become 8/11 if 3 is added to both numarator and denominator become 2/5 if 3 is subtracted from both numerator and denominator find fraction
Answers
Answer:
Given:-
a fraction becomes 8/11 if three is both numerator and denominator
when 3 is subtracted from the numerator and denominator it becomes 2/5
To find :-
The fraction.
Solution :-
Let the numerator of the fraction be x.
Let the denominator of the fraction be y.
Hence, the fraction = \bf\large\frac{x}{y}
y
x
As per the first condition,
a fraction becomes 8/11 if three is both numerator and denominator
\bf\large\frac{x + 3}{y + 3}
y+3
x+3
= \bf\large\frac{8}{11}
11
8
Cross multiplying,
11 (x + 3) = 8 (y + 3)
11x + 33 = 8y + 24
Transporting the terms,
11x - 8y = 24 - 33
11x - 8y = - 9 ------> 1
As per the second condition,
if 3 is subtracted from the numerator and denominator it becomes 2/5
Numerator = x
Denominator = y
If 3 is subtracted from both,then,
Numerator = x - 3
Denominator = y - 3
Fraction = \bf\large\frac{2}{5}
5
2
\bf\large\frac{x - 3}{y - 3}
y−3
x−3
= \bf\large\frac{2}{5}
5
2
Cross multiplying,
5(x - 3) = 2 (y - 3)
5x - 15 = 2y - 6
Transporting the terms,
5x - 2y = -6 + 15
5x - 2y = 9 ---->2
Multiply equation 2 by 4
20x - 8y = 36 -----> 3
Solve equations 1 and 3 simultaneously by elimination method,
.......20x - 8y = 36 ----> 3
- .....11x - 8y = -9 ----->1
........9x = 45
x = \bf\large\frac{45}{9}
9
45
x = 5
Substitute the value of x in equation 2
5x - 2y = 9 ---->2
5 × 5 - 2y = 9
25 - 2y = 9
- 2y = 9 - 25
- 2y = - 16
y = \bf\large\frac{-16}{-2}
−2
−16
y = 8
\bf{\large{\underline{\boxed{\rm{\blue{Solution :\: (x, y) = (5,8)}}}}}}
Solution:(x,y)=(5,8)
Fraction formed = \bf\large\frac{5}{8}
8
5
Step-by-step explanation:
For first case :-
a fraction becomes 8/11 if three is added to both numerator and denominator
Numerator = 5
Denominator = 8
If three is added,
Numerator = 5 + 3 = 8
Denominator = 8 + 3 = 11
As per the question, the fraction becomes 8/11, hereby, by adding 3 to both the numerator and denominator we satisfy the first condition.
\bf\large\frac{5+3}{8 + 3}
8+3
5+3
= \bf\large\frac{8}{11}
11
8
\bf\large\frac{8}{11}
11
8
= \bf\large\frac{8}{11}
11
8
LHS = RHS.
For second case :-
if 3 is subtracted from the numerator and denominator it becomes 2/5
Numerator = 5 - 3 = 2
Denominator = 8 - 3 = 5
Fraction = \bf\large\frac{2}{5}
5
2
We subtracted 3 from both numerator and denominator and hence satisfied the condition of fraction becoming 2/5.
\bf\large\frac{5-3}{8-3}
8−3
5−3
= \bf\large\frac{2}{5}
5
2
\bf\large\frac{2}{5}
5
2
= \bf\large\frac{2}{5}
5
2
LHS = RHS
Hence verified!
Note :- In the first part of the question, a fraction becomes 8/11 if three is both numerator and denominator, the word added is missing. I carried on with the question just with an assumption of this word. So the first condition is a fraction becomes 8/11 if three is " added" both numerator and denominator and we were even able to verify the answer, hence our assumption was right