Question

If the numerator of a fraction is increased by six, the value of the fraction will increase by one. If the denominator of the original fraction is increased by 36, the value of the original fraction will decrease by one. What is the original fraction?

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Answer 1

7/6

Explanation:

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Answer 2

`(7)/(6)`

Explanation:

Let
`(a)/(b)` be the original fraction.

If the numerator of a fraction is increased by six, the value of the fraction will increase by one. Hence,

`(a+6)/(b)=(a)/(b)+1`

If the denominator of the original fraction is increased by 36, the value of the original fraction will decrease by one. Hence,

`(a)/(b+36)=(a)/(b)-1`

Add these two equalities:

`(a+6)/(b)+(a)/(b+36)=(a)/(b)+1+(a)/(b)-1\\ \\(a+6)/(b)+(a)/(b+36)=2(a)/(b)\\ \\(a)/(b+36)=(2a-(a+6))/(b)\\ \\(a)/(b+36)=(a-6)/(b)\\ \\ab=(a-6)(b+36)\\ \\ab=ab+36a-6b-216\\ \\36a-6b-216=0\\ \\6a-b-36=0\\ \\b=6a-36`

Substitute it into the first equation:

`(a+6)/(6a-36)=(a)/(6a-36)+1\\ \\(a+6)/(6(a-6))=(a)/(6(a-6))+1\\ \\a+6=a+6(a-6)\\ \\a+6=a+6a-36\\ \\6a=6+36\\ \\6a=42\\ \\a=7\\ \\b=6\cdot 7-36=42-36=6`

Thus, the initial fraction was

`(7)/(6)`