[Smart Math] Arithmetic Problem 20

Here’s and example of a SMART MATH problem for ARITHMETIC.

Arithmetic

Problem

$\frac{\sqrt{6}+\sqrt{7}}{\sqrt{5}-\sqrt{6}}=$ ?

$6+\sqrt{30}+\sqrt{35}+\sqrt{42}$
$6-\sqrt{30}+\sqrt{35}+\sqrt{42}$
$6+\sqrt{30}-\sqrt{35}+\sqrt{42}$
$6-\sqrt{30}-\sqrt{35}-\sqrt{42}$
$6+\sqrt{30}+\sqrt{35}-\sqrt{42}$

The Usual Method

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Using Rationalization:

$\frac{\sqrt{6}+\sqrt{7}}{\sqrt{5}-\sqrt{6}}=$ $\frac{\left( \sqrt{6}+\sqrt{7} \right)}{\left( \sqrt{5}-\sqrt{6} \right)}\times \frac{\left( \sqrt{5}+\sqrt{6} \right)}{\left( \sqrt{5}+\sqrt{6} \right)}$

= $\frac{\sqrt{30}+6+\sqrt{35}+\sqrt{42}}{5-6}=\sqrt{30}+6+\sqrt{35}+\sqrt{42}$

(Ans: 1)

Estimated Time to arrive at the answer = 30 seconds.

Using Technique

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Since the denominator is $\left( \sqrt{5}-\sqrt{6} \right)$ , its conjugate is $\left( \sqrt{5}+\sqrt{6} \right)$ which is multiplied to it and the numerator $\left( \sqrt{6}+\sqrt{7} \right)$ . Since both the terms ( $\left( \sqrt{5}+\sqrt{6} \right)$ and $\left( \sqrt{6}+\sqrt{7} \right)$ ) are positive terms, their product too would be positive and there would not be any negative sign. The only option that does not have a negative sign is option ‘1’.

(Ans: 1)

Estimated Time to arrive at the answer = 5 seconds.
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Problem

The Usual Method

Using Technique

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