Algebra Problem 1

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

Find the fraction such that if numerator is multiplied by 2 and 2 is added to the denominator, the fraction equals 2. If the numerator is squared and 2 is added to the denominator, the fraction equals 7.

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The Usual Way

Let the original fraction be x/y.

$\therefore$ New Fraction = $\frac{2x}{y+2}$ = 2

$\therefore$ 2x = 2y + 4 or x = y + 2 …… Eq. 1

Also; $\frac{x^{2}}{y+2}$ = 7

$\therefore$ $\frac{x^{2}}{7}$ = y + 2 …… Eq. 2
From Eq. 1 and 2;

$\frac{x^{2}}{7}$ = x

$\therefore$ x = 7 and y = 5

$\therefore$ Fraction = 7/5

(Ans: 3)

Estimated Time to arrive at the answer = 60 seconds

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The Smart Way

Simply read the first part of the question and find those options that satisfy that condition as shown below:

$\frac{5}{3}$ => $\frac{5\times 2}{3+2}$ = $\frac{10}{5}$ = 2 (Satisfies the condition)

$\frac{4}{8}$ => $\frac{4\times 2}{8+2}$ = $\frac{8}{10}$ = $\frac{4}{5}$ (does not satisfy the condition)

$\frac{7}{5}$ => $\frac{7\times 2}{5+2}$ = $\frac{14}{7}$ = 2 (Satisfies the condition)

$\frac{3}{2}$ => $\frac{3\times 2}{2+2}$ = $\frac{6}{4}$ = $\frac{3}{2}$ (does not satisfy the condition)

$\frac{8}{5}$ => $\frac{8\times 2}{5+2}$ = $\frac{16}{7}$ (does not satisfy the condition)

This leaves us with options ‘1’ and ‘3’ only. Now, look at the 2nd condition. Apply this to options ‘1’ and ‘3’ only as follows:

$\frac{5}{3}$ => $\frac{5^{2}}{3+2}$ = $\frac{25}{5}$ = 5 (does not satisfy the condition)
Hence, answer is option ‘3’

Just to verify…c.
$\frac{7}{5}$ => $\frac{7^{2}}{5+2}$ = $\frac{49}{7}$ = 7 (Satisfies the condition and hence is the answer)

(Ans: 3)

Estimated Time to arrive at the answer = 15 seconds

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Algebra Problem 1