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The denominator of a fraction is 5 more than its numerator. If both numerator and denominator are increased by 3, the value of the fraction is 2/3. Find the original fraction.

Devise a plan:
Carry out the plan:
Answer to the Question?

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Sagot :

Answer:

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Step-by-step explanation:

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View image Johnreystarosa46
View image Johnreystarosa46

[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].