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# [Smart Math] Algebra Problem 11

Here’s and example of a SMART MATH problem for 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

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

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.