Find the Fraction

Question

Write and solve a system of equations to find the fraction that reduces to 5/7 when 2 is subtracted from both the numerator and denominator, and reduces to 5/6 if the numerator and the denominator are each increased by 3.

Answer 1

So, call your unknown fraction X/Y

Translate statements into equations:

" reduces to 5/7 when 2 is subtracted from both the numerator and denominator":

(X-2) / (Y-2) = 5/7 (equation I)

"reduces to 5/6 if the numerator and the denominator are each increased by 3.:

(X+3) / (Y+3) = 5/6 (equation II)

Now we have 2 equation in 2 unknowns---these can be solve to get X and Y:

Rewriting to put all variables on top:

I: X-2 = 5Y/7 - 10/7

II: X+3 = 5Y/6 + 15/6

clean up:

I: 7X - 14 = 5Y - 10

7X - 5Y = 4

II: 6X + 18 = 5Y + 15

6X - 5Y = -3

Subtract II from I:

X = 7

Solve either I or II to get Y = 9

Answer 2

Hi Alice

This is a system of linear equations, which plays around with fractions;

Let's call the desired numerator N, and the denominator D

The first equation is: (N-2)/(D-2) = 5/7

The second equation is: (N+3)/(D+3) = 5/6

If you write these as fractions, you can cross-multiply each equation, thus:

7(N-2) = 5(D-2) and

5(N+3) = 6(D+3)

Reorganizing to use standard form, we get:

7N-5D = 4

6N-5D = -3

Look familiar? Next step is to solve by elimination.

Subtracting the equations yields N =7 and D = 9

So the fraction is 7/9

Be sure t check your answer by putting it into the original statements

Answer 3

Write and solve a system of equations to find the fraction that reduces to 5/7 when 2 is subtracted from both the numerator and denominator, and reduces to 5/6 if the numerator and the denominator are each increased by 3

Let fraction = x/y

he fraction that reduces to 5/7 when 2 is subtracted from both the numerator and denominator

(x-2)/(y-2)= 5/7

cross multiplication

7(x-2) =5(y-2)

7x -14 =5y-10

7x-5y =4 ------------eq 1

reduces to 5/6 if the numerator and the denominator are each increased by 3

(x+3)/(y+3) = 5/6

6(x+3) =5(y+3)

6x+18 =5y+15

6x-5y =-3-----------eq 2

subtract eq 2 from eq 1

x = 7

substitute x=7 in eq 1

7x-5y = 4

49-5y =4

-5y =-45

y=9

so fraction = x/y = 7/9