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

Algebra Problems

Algebra Problem 1

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.

  1. 5/3
  2. 4/8
  3. 7/5
  4. 3/2
  5. 8/5

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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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