Find a fraction which reduces to 3 by 4 when both the numerator and denominator are each decreased by 1 and reduces to 5 by 6 when both the numerator and denominator are each increased by 3
Hello , does anybody know the answer
n = numerator
d = deniminator
reduces to 3 by 4 when both the numerator and denominator are each decreased by 1 means:
( n - 1 ) / ( d - 1 ) = 3 / 4
Cross multiply
4 ( n - 1 ) = 3 ( d - 1 )
4 n - 4 = 3 d - 3
Add 4 to both sides
4 n = 3 d + 1
Divide both sides by 4
n = ( 3 d + 1 ) / 4
reduces to 5 by 6 when both the numerator and denominator are each increased by 3 means:
( n + 3 ) / ( d + 3 ) = 5 / 6
Cross multiply
6 ( n + 3 ) = 5 ( d + 3 )
6 n + 18 = 5 d + 15
Subtract 18 to both sides
6 n = 5 d - 3
Divide both sides by 6
n = ( 5 d - 3 ) / 6
n in first equation us equal n in second equation.
n = n
( 3 d + 1 ) / 4 = ( 5 d - 3 ) / 6
Cross multiply
6 ( 3 d + 1 ) = 4 ( 5 d - 3 )
18 d + 6 = 20 d - 12
Subtract 18 d to both sides
6 = 2 d - 12
Add 12 to bith sides
18 = 2 d
2 d = 18
d = 18 / 2
d = 9
Put this value in equation:
n = ( 3 d + 1 ) / 4
n = ( 3 • 9 + 1 ) / 4
n = ( 27 + 1 ) / 4
n = 28 / 4
n = 7
Youur fraction is:
n / d = 7 / 9
Chesk result.
when both the numerator and denominator are each decreased by 1
( 7 - 1 ) / ( 9 - 1 ) = 6 / 8 = 3 • 2 / 3 • 4 = 3 / 4
when both the numerator and denominator are each increased by 3
( 7 + 3 ) / ( 9 + 3 ) = 10 / 12 = 2 • 5 / 2 • 6 = 5 / 6
To find the fraction, let's assume the numerator of the original fraction is represented by "x" and the denominator is represented by "y".
According to the given conditions, when both the numerator and denominator are each decreased by 1, the new fraction becomes (x-1)/(y-1), which is equal to 3/4. Similarly, when both the numerator and denominator are each increased by 3, the new fraction becomes (x+3)/(y+3), which is equal to 5/6.
We can now set up a system of equations to solve for x and y:
(x - 1) / (y - 1) = 3/4 ----(1)
(x + 3) / (y + 3) = 5/6 ----(2)
To eliminate the fractions, we can cross multiply and solve the resulting equations:
4(x - 1) = 3(y - 1) ----(3)
6(x + 3) = 5(y + 3) ----(4)
Expanding equations (3) and (4) gives us:
4x - 4 = 3y - 3 ----(5)
6x + 18 = 5y + 15 ----(6)
Now let's solve the system of equations (5) and (6) for x and y:
From equation (5), rearrange it to solve for x:
4x - 3y = 1 ----(7)
From equation (6), rearrange it to solve for x:
6x - 5y = -3 ----(8)
Multiply equation (7) by 6 and equation (8) by 4 to eliminate x:
24x - 18y = 6 ----(9)
24x - 20y = -12 ----(10)
Subtract equation (10) from equation (9) to eliminate x:
-2y = 18 ----(11)
Solve equation (11) for y:
y = -9
Substitute the value of y back into equation (7) or (8) to solve for x:
4x - 3(-9) = 1
4x + 27 = 1
4x = 1 - 27
4x = -26
x = -26/4
x = -13/2
Hence, the fraction is -13/2.