Question

If $\frac{5\sqrt{3}+\sqrt{7}}{4\sqrt{3}-2\sqrt{7}}=X+Y\sqrt{21}$, then what is the value of $10X+10Y$.

• A. $\frac{47}{10}$
• B. $47$
• C. $44$
• D. $\frac{44}{10}$

Answer 1

The correct option is C $44$

First on rationalising the denominator of $\frac{5\sqrt{3}+\sqrt{7}}{4\sqrt{3}-2\sqrt{7}}$ we get,

$\Rightarrow \frac{5\sqrt{3}+\sqrt{7}}{4\sqrt{3}-2\sqrt{7}}=\frac{5\sqrt{3}+\sqrt{7}}{4\sqrt{3}-2\sqrt{7}}\times \frac{4\sqrt{3}+2\sqrt{7}}{4\sqrt{3}+2\sqrt{7}}=\frac{(5\sqrt{3}+\sqrt{7})(4\sqrt{3}+2\sqrt{7})}{48-28}=\frac{60+10\sqrt{21}+4\sqrt{21}+14}{20}=\frac{74+14\sqrt{21}}{20}$

On simplifying the terms we get,
$\Rightarrow \frac{5\sqrt{3}+\sqrt{7}}{4\sqrt{3}-2\sqrt{7}}=\frac{74}{20}+\frac{14\sqrt{21}}{20}=\frac{37}{10}+\frac{7\sqrt{21}}{10}$

$\therefore \frac{37}{10}+\frac{7\sqrt{21}}{10}=X+Y\sqrt{21}$

On comparing we get $X=\frac{37}{10}\&Y=\frac{7}{10}$

$\Rightarrow 10X=37$ and $10Y=7$
On adding we get,
$10X+10Y=37+7=44$