[tex]\underline{\underline{\large{\red{\mathcal{✒GIVEN:}}}}}[/tex]
The denominator of a fraction is 5 more than its numerator. If both the numerator and denominator are increased by 3, the value of the fraction is 2/3.
[tex]\underline{\underline{\large{\red{\mathcal{REQUIRED:}}}}}[/tex]
Find the original fraction.
[tex]\underline{\underline{\large{\red{\mathcal{SOLUTION:}}}}}[/tex]
Devise a Plan:
Let us denote the numerator of the fraction as x.
The denominator is then x + 5. The original fraction is:
[tex]\tt{ \dfrac{x}{x + 5} }[/tex]
According to the problem, if both the numerator and the denominator are increased by 3, the new fraction is:
[tex]\tt{ \dfrac{x + 3}{x + 5 + 3} = \dfrac{2}{3} }[/tex]
[tex]\tt{ \dfrac{x + 3}{x + 8} = \dfrac{2}{3} }[/tex]
Carry out the plan:
To solve for [tex]\rm{x}[/tex], we cross-multiply:
[tex]\tt{3(x+3)=2(x+8)}[/tex]
[tex]\tt{3x+9=2x+16}[/tex]
[tex]\tt{3x-2x=16-9}[/tex]
[tex]\large{\tt{x=7}}[/tex]
Thus, the numerator of the original fraction is 7. The denominator, being 5 more than the numerator, is:
[tex]\tt{x+5}[/tex]
[tex]\tt{7+5}[/tex]
[tex]\tt{=12}[/tex]
So, the original fraction is [tex]\large{\boxed{\tt{\purple{ \dfrac{7}{12}} }}}[/tex]
Final Answer:
Hence, the original fraction is [tex]\large{\rm{\purple{ \dfrac{7}{12} }}}[/tex].
