Answer:
[tex]\dfrac{7}{6}[/tex]
Step-by-step explanation:
Let [tex]\dfrac{a}{b}[/tex] be the original fraction.
If the numerator of a fraction is increased by six, the value of the fraction will increase by one. Hence,
[tex]\dfrac{a+6}{b}=\dfrac{a}{b}+1[/tex]
If the denominator of the original fraction is increased by 36, the value of the original fraction will decrease by one. Hence,
[tex]\dfrac{a}{b+36}=\dfrac{a}{b}-1[/tex]
Add these two equalities:
[tex]\dfrac{a+6}{b}+\dfrac{a}{b+36}=\dfrac{a}{b}+1+\dfrac{a}{b}-1\\ \\\dfrac{a+6}{b}+\dfrac{a}{b+36}=2\dfrac{a}{b}\\ \\\dfrac{a}{b+36}=\dfrac{2a-(a+6)}{b}\\ \\\dfrac{a}{b+36}=\dfrac{a-6}{b}\\ \\ab=(a-6)(b+36)\\ \\ab=ab+36a-6b-216\\ \\36a-6b-216=0\\ \\6a-b-36=0\\ \\b=6a-36[/tex]
Substitute it into the first equation:
[tex]\dfrac{a+6}{6a-36}=\dfrac{a}{6a-36}+1\\ \\\dfrac{a+6}{6(a-6)}=\dfrac{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[/tex]
Thus, the initial fraction was
[tex]\dfrac{7}{6}[/tex]
