If both numerator and denominator of a fraction are increased by one, the fraction formed is 2/3. if both numerator and denominator are decreased by two, the fraction formed is 3/5. what is the fraction formed, if both numerator and denominator are increased by four?
Please help me
let the original fraction be x/y
(x+1)/(y+1) = 2/3
3x + 3 = 2y + 2 ----> 3x - 2y = -1 , #1
(x-2)/(y-2) = 3/5
5x - 10 = 3y - 6 ----> 5x - 3y = 4 , #2
#1 times 3 ---> 9x - 6y = -3
#2 times 2 ---> 10x - 6y = 8
subtract them:
x = 11
in #1:
33 - 2y = -1
-2y = -34
y = 17
so the original fraction was 11/17
if both top and bottom are increased by 4 we would have
15/21 or 5/7
Oh God! I made a big mistake at this part
33 - 2y = -1
-2y = 32
I should improve
happens to everybody
To solve this problem, we can first set up two equations based on the information given. Let's call the original fraction "n/d" (with n representing the numerator and d representing the denominator).
According to the first condition, if both the numerator and denominator of the fraction are increased by one, the new fraction is 2/3. We can write this as:
(n + 1)/(d + 1) = 2/3
Similarly, according to the second condition, if both the numerator and denominator are decreased by two, the new fraction is 3/5. We can write this as:
(n - 2)/(d - 2) = 3/5
Now we can solve these two equations simultaneously to find the values of n and d.
First, let's solve the first equation for n:
3(n + 1) = 2(d + 1)
3n + 3 = 2d + 2
3n = 2d - 1 (Equation 1)
Now let's solve the second equation for n:
5(n - 2) = 3(d - 2)
5n - 10 = 3d - 6
5n = 3d + 4 (Equation 2)
We have a system of two linear equations (Equation 1 and Equation 2). We can solve this system by substitution or elimination method.
Let's multiply Equation 1 by 5 and Equation 2 by 3 to eliminate d:
15n = 10d - 5 (Equation 3)
15n = 9d + 12 (Equation 4)
Now we can subtract Equation 4 from Equation 3:
15n - 15n = 10d - 5 - (9d + 12)
0 = d - 17
Therefore, d = 17.
Now substitute the value of d into Equation 1 to find the value of n:
3n = 2(17) - 1
3n = 34 - 1
3n = 33
n = 11
So the original fraction is 11/17.
Now, to find the fraction formed when both the numerator and denominator are increased by four, we need to add 4 to both n and d:
New fraction = (11 + 4)/(17 + 4)
New fraction = 15/21
Therefore, the fraction formed when both the numerator and denominator are increased by four is 15/21.