if 1 is added to both numerator and denominator of a fraction, the fraction becomes 1/2. if 8 is added to both fractions, the fraction becomes 2/3. what is the fraction
If the fraction is a/b
(a+1)/(b+1) = 1/2
(a+8)/(b+8) = 2/3
2a+2 = b+1
(1): 2a - b = -1
3a+24 = 2b + 16
(2): 3a - 2b = -8
subtract 2x#1 from #2
3a - 2b = -8
4a - 2b = -2
-a = -6
a = 6
so, b = 13
Original fraction: 6/13
Check:
(6+1)/(13+1) = 7/14 = 1/2
(6+8)/(13+8) = 14/21 = 2/3
why cant you just add 1\2 and 2\3 and see what you get let s cee 1\2 + 2\3 is 3\5and cant you simplify that in too and that a whole it is 15 okay love yasi26
To solve this problem, let's denote the unknown fraction as `x/y`, where `x` is the numerator and `y` is the denominator.
According to the first condition, if 1 is added to both the numerator and denominator, the fraction becomes `1/2`. Therefore, the new fraction can be expressed as `(x + 1) / (y + 1) = 1/2`.
Similarly, according to the second condition, if 8 is added to both the numerator and denominator, the fraction becomes `2/3`. Therefore, the new fraction can be expressed as `(x + 8) / (y + 8) = 2/3`.
To find the original fraction `x/y`, we need to solve this system of equations simultaneously.
Let's solve the first equation `(x + 1) / (y + 1) = 1/2` for `x` in terms of `y`:
Cross multiply: 2(x + 1) = y + 1
Distribute: 2x + 2 = y + 1
Subtract 1 from both sides: 2x + 1 = y
Now substitute this value of `y` into the second equation `(x + 8) / (y + 8) = 2/3`:
Replace `y` with `2x + 1`: (x + 8) / (2x + 1 + 8) = 2/3
Simplify: (x + 8) / (2x + 9) = 2/3
To eliminate the fractions, we can cross multiply:
3(x + 8) = 2(2x + 9)
Simplify: 3x + 24 = 4x + 18
Subtract 3x from both sides: 24 = x + 18
Subtract 18 from both sides: 6 = x
Now that we have the value of `x`, we can substitute it back into one of the original equations to find the value of `y`. Let's use the first equation:
(x + 1) / (y + 1) = 1/2
(6 + 1) / (y + 1) = 1/2
7 / (y + 1) = 1/2
To eliminate the fractions, we can cross multiply:
2(7) = 1(y + 1)
14 = y + 1
Subtract 1 from both sides: 13 = y
Therefore, the original fraction is 6/13.