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# [Smart Math] Arithmetic Problem 9

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

### Problem

If $g(x)=\frac{3a-5}{2a+1}$, then what is the value of $\frac{5g(1)-3g(2)+10g(-1)}{8}$?

1. 1141
2. $\frac{\text{1141}}{\text{15}}$
3. $\frac{\text{1141}}{\text{120}}$
4. $\frac{\text{1141}}{\text{60}}$
5. $\frac{\text{1141}}{\text{20}}$

### The Usual Method

First get the values of:

$g(1)=\frac{3(1)-5}{2(1)+1}=\frac{-2}{3}$

$g(2)=\frac{3(2)-5}{2(2)+1}=\frac{1}{5}$

$g(-1)=\frac{3(-1)-5}{2(-1)+1}=\frac{-8}{-1}=8$

Now, substitute these values in the given equation, we get:

$\frac{5\left( \frac{-2}{3} \right)-3\left( \frac{1}{5} \right)+10\left( 8 \right)}{8}$

$=\frac{\frac{-10}{3}-\frac{3}{5}+80}{8}$

$=\frac{\frac{-50-9+1200}{15}}{8}$

$=\frac{1141}{15}\times \frac{1}{8}=\frac{1141}{120}$

(Ans: 3)

Estimated Time to arrive at the answer = 75 seconds.

### Using Technique

Observe from the equation $\frac{5g(1)-3g(2)+10g(-1)}{8}$ has ‘8’ in its denominator. This 8 cannot be reduced in further operations as 5, 3 and 10 are relatively prime to 8 and hence have an HCF of 1. Should the HCF of these numbers be 2 or any multiple of 2, then the denominator ‘8’ will get to 4, 2 or 1 accordingly. But in this particular case, it is not so. Hence, the answer should have 8 or a multiple of 8 in its denominator. The only option satisfying this criterion is options ‘3’.