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Algebra Smart Math

[Smart Math] Algebra Problem 11

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

Algebra

Problem

What is the value of a and b in 5\frac{1}{a}\times b\frac{3}{4} = 20?

  1. 2, 1
  2. 4, 1
  3. 1, 5
  4. 3, 3
  5. 4, 3

The Usual Method

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One may substitute each option to see which satisfies the equation as follows:

2, 1 => 5\frac{1}{2}\times 1\frac{3}{4}=\frac{11}{3}\times \frac{7}{4}=\frac{77}{12}\ne 20 (does not satisfy hence not the answer)

4, 1 => 5\frac{1}{4}\times 1\frac{3}{4}=\frac{21}{4}\times \frac{7}{4}=\frac{147}{16}\ne 20 (does not satisfy hence not the answer)

1, 5 => 5\frac{1}{1}\times 5\frac{3}{4}=5\times \frac{23}{4}=\frac{115}{4}\ne 20 (does not satisfy hence not the answer)

3, 3 => 5\frac{1}{3}\times 3\frac{3}{4}=\frac{16}{3}\times \frac{15}{4}=4\times 5=20 (satisfies hence is the answer)

No need to check as option ‘4’ satisfies the condition.

(Ans: 4)

Estimated Time to arrive at the answer = 60 seconds.

Using Technique

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In 5\frac{1}{a}\times b\frac{3}{4} = 20, ‘ 5\frac{1}{a}\times b\frac{3}{4}’ are mixed fractions which when converted to improper fractions will become \frac{5a+1}{a}\times \frac{4b+3}{4}.

20 being an integer, \frac{5a+1}{a}\times \frac{4b+3}{4} should yield an integer, which is possible only if 5a+1 is a multiple of 4 or ‘1’. Since, we do not know the value of ‘1’, we look at the option check for that value of ‘1’, which makes 5a+1 a multiple of 4. The only value possible from the options to make 5a+1 a multiple of 4 is 3. Hence, a = b = 3.

(Ans: 4)

Estimated Time to arrive at the answer = 15 seconds.
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