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# [Smart Math] Algebra Problem 6

Here’s and example of a SMART MATH problem for 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

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

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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