If 1 is added to both the numerator and denominator of a fraction its value becomes 1/2 and if 1 subtracted from both the numerator and denominator of a fraction it becomes 1/3 then What is the original fraction?

2 Answers
Jan 10, 2016

3/7

Explanation:

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Feb 11, 2018

The original fraction is (3)/(7)

Explanation:

Let (x)/(y) represent the fraction

The problem specifies

(a) Adding 1 to the numerator and denominator makes the fraction equal to (1)/(2)
(b) Subtracting 1 from the top and bottom makes the fraction equal to (1)/(3)

(a)   (x + 1)/(y+1) = (1)/(2)

(b)   (x - 1)/(y-1) = (1)/(3)

color(white)(........................)―――――――

Solve for x, already defined as "the numerator of the original equation"

These equations can be solved as Ratio&Proportion problems.

1) Cross multiply both equations
(a)   2(x+1)=1(y+1)
(b)   3(x−1)=1(y−1)

2) Clear the parentheses
(a)   2x + 2 = y + 1
(b)   3x - 3 = y - 1

3) Write the equations with the variables on the left and the numbers on the right
(a)   2x - y = - 1
(b)   3x - y =     2

4) Start with Equation (b) and subtract Equation (a)
(Do it in this order to avoid negative numbers.)
(b)      3x - y =   2
(a) - (2x - y = - 1)

5) Clear the parentheses and combine to eliminate the y term
(b)     3x - y = 2
(a) -2x + y = 1
―――――――――
color(white)(..........) x color(white)(...m)= 3 larr already defined as "the numerator of the original fraction"

color(white)(........................)―――――――

Solve for y, already defined as "the denominator of the original fraction"

1) Using one of the original equations, sub in 3 in the place of x
(color(blue)(x) - 1)/(y-1) = (1)/(3)

(color(blue)(3) - 1)/(y-1) = (1)/(3)

2) Cross multiply and solve for y, already defined as "the denominator of the original fraction"
3(3-1)=1(y-1)

3) Solve inside the parentheses
3(2)=1(y-1)

4) Clear the parentheses by distributing the 3 and the 1
6 = y - 1

5) Add 1 to both sides to isolate y

6) 7 = y larr answer for y, defined as "the denominator of the original fraction"

"Answer"
The original fraction is (3)/(7)

color(white)(........................)―――――――

Check
Add 1 to the top and bottom of (3)/(7) to see if it becomes (1)/(2)

(3 + 1)/(7+1) = (4)/(8) = (1)/(2)  ✓

Subtract 1 from the top and bottom to see if it becomes (1)/(3)

(3-1)/(7-1) = (2)/(6) = (1)/(3)  ✓

Check