# In a fraction the numerator is 1 less than the denominator. If 1 is added to the numerator and 5 to the denominator, the fractions becomes 1/2 find the original number?

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Let the numerator be *`x` *, then the denominator becomes *`x+1.` *

`therefore ` the fraction = `x/(x+1)`

According to the question, 1 is added to the numerator and 5 is added to denominator,i.e. `x+1 ` and `x+1+5,` which equals to `1/2`

`(x+1)/(x+1+5) = 1/2`

`(x+1)/(x+6) = 1/2`

By cross-multiplication you get

`2(x+1) = 1(x+6)`

`2x +2 = x+6`

Combine x terms

`2x - x = 6 - 2`

`x = 4`

`x+1 = 4+1 = 5`

`therefore ` the required fraction is `4/5`

Suppose that the denominator of the fraction is x; then, the original fraction is:

(x - 1)/x

When we add 1 and 5 in the numerator and denominator respectively, we have the following equation:

[(x - 1) + 1]/(x + 5) = 1/2

Solving for the variable x:

x/(x + 5) = 1/2

2x = x+5

x = 5

Then, the initial fraction is:

(x - 1)/x

(5 – 1)/5 = 4/5

Let's check:

[(x - 1) + 1]/(x +5) = 1/2

x/(x + 5) = 1/2

5/10 = 1/2

1/2 = 1/2

Let the numerator = x and the denominator = x + 1

`x/(x+1) `

According to the problem, we need to add 1 to the numerator and add 5 to the denominator and set it equal to 1 / 2.

`(x+1)/(x + 1 + 5) =1 / 2`

Then simplify by adding like terms in each numerator and denominator if possible.

`(x + 1)/ (x + 6) = 1 / 2`

According to a property called the means-extremes property, two fractions or ratios set equal to one another is called a proportion and can be solved by using cross products. In other words cross multiply and then solve the resulting equation for x.

2 ( x + 1) = 1 ( x + 6)

Simplify by eliminating the parentheses using the distributive property

Then solve the equation for x by isolating the variable one side of the equation (the left) and the number on the right. ` `

`2x + 2 = 1 x + 6`

`2x - 1x + 2 = 1x - 1x + 6`

`1 x + 2 - 2 = 6 - 2`

`1 x = 4`

Therefore the numerator is 4 and the denominator is x + 1 or 5.

`4 / 5`

` `