 Rivka Thorpe

2021-10-22

Which of following expressions are equivalent to $\frac{3}{n}+\frac{2}{6{n}^{2}+12n}?$
a) $\frac{18n+38}{6n\left(n+2\right)}$
b) $\frac{5}{6{n}^{2}+13n}$
c) $\frac{18\left(n+2\right)}{6n\left(n+2\right)}+\frac{2}{6n\left(n+2\right)}$
d) $\frac{3\left(6{n}^{2}+12n\right)+2n}{n\left(6{n}^{2}+12n\right)}$ crocolylec

Step 1
The addition(subtraction) between two like fractions can be done by adding(subtracting) the numerators of both fractions by fixing same denominator to the result. Like fractions are the fraction with same denominator.
For the addition (subtraction) between two unlike fractions , first we have to convert both denominators of the fractions to same value. This can be done by finding the LCM(Least Common multiple) of the denominators of the fractions. We can multiply and divide both the fractions by suitable constant to produce same value in both denominators.
Step 2
We have an addition of unlike fractions. Therefore we have to convert denominators of both fractions to same denominator. Then we can add the numerators.
$\frac{3}{n}+\frac{2}{6{n}^{2}+12n}$
The LCM of the denominators is $n\left(6{n}^{2}+12n\right)$
Now convert the denominators of the fractions to the same denominator.
$\frac{3}{n}+\frac{2}{6{n}^{2}=12n}=\frac{3}{n}×\frac{6{n}^{2}+12n}{6{n}^{2}+12n}+\frac{2}{6{n}^{2}+12n}×\frac{n}{n}$
$=\frac{3\left(6{n}^{2}+12n\right)}{n\left(6{n}^{2}+12n\right\}+\frac{2n}{n\left(6{n}^{2}+12n\right)}}$
$=\frac{3\left(6{n}^{2}+12n\right)+2n}{n\left(6{n}^{2}+12n\right)}$
Hence the equivalent expression is $\frac{3\left(6{n}^{2}+12n\right)+2n}{n\left(6{n}^{2}+12n\right)}$
Hence option d) is correct.
We can further simplify the answer as,
$\frac{3\left(6{n}^{2}+12n\right)+2n}{n\left(6{n}^{2}+12n\right)}=\frac{18{n}^{2}+36+2n}{6{n}^{2}\left(n+2\right)}$
$=\frac{n\left(18n+38\right)}{6{n}^{2}\left(n+2\right)}$
$=\frac{18n+38}{6n\left(n+2\right)}$
Hence option a) is correct
Also from the expression
$\frac{3\left(6{n}^{2}+12n\right)}{n\left(6{n}^{2}+12n\right)}+\frac{2n}{n\left(6{n}^{2}+12n\right)}=\frac{18{n}^{2}+36n}{n\left(6{n}^{2}+12n\right)}+\frac{2n}{n\left(6{n}^{2}+12n\right)}$
$=\frac{n18n\left(n+2\right)}{n6n\left(n+2\right)}+\frac{2n}{n6n\left(n+2\right)}$

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