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Algebra Smart Math

[Smart Math] Algebra Problem 6

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

Algebra

Problem

The difference of squares of two numbers is 24 and one of the numbers is 5/7th of the other. Find the numbers.

  1. 7, 5
  2. 7, 9
  3. 10, 14
  4. 7, 12
  5. 2, 5

The Usual Method

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The ratio of one of the numbers to another is \frac{5}{7}.

Let the numbers be ‘1’ and ‘2’.

\therefore \frac{a}{b}=\frac{5}{7}

\therefore a=\frac{5b}{7}

Also, given that a^{2}-b^{2}=24

\therefore \left( \frac{5b}{7} \right)^{2}-b^{2}=24

\therefore \frac{25b^{2}}{49}-b^{2}=24

\therefore 25b^{2}-49b^{2}=24\times 49

\therefore b^{2}=49

\therefore b=7

\therefore a=\frac{5\times 7}{7}=5

(Ans: 1)

Estimated Time to arrive at the answer = 45 seconds.

Using Technique

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Just look at the options and see which numbers are in the ratio of 5 : 7. Thus all options other than ‘1’ and ‘3’ are eliminated.

Also, for option ‘3’: 14^{2}-10^{2}\ne 24

Thus answer is option ‘1’: 7^{2}-5^{2}=24

(Ans: 1)

Estimated Time to arrive at the answer = 10 seconds.

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