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# [Smart Math] Arithmetic Problem 12

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

### Problem

$\frac{409}{4+\frac{3}{4+\frac{3}{4+\frac{3}{4}}}}$ = ?

1. 1
2. 88
3. $\frac{88}{9}$
4. $\frac{409}{88}$
5. 409

### The Usual Method

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We have to simplify starting from the denominator value of $4+\frac{3}{4}$

$\therefore \frac{409}{4+\frac{3}{4+\frac{3}{4+\frac{3}{4}}}}=\frac{409}{4+\frac{3}{4+\frac{3}{{}^{19}\!\!\diagup\!\!{}_{4}\;}}}=\frac{409}{4+\frac{3}{4+\frac{12}{19}}}=\frac{409}{4+\frac{3}{\frac{88}{19}}}=\frac{409}{4+\frac{57}{88}}=\frac{409}{\frac{352+57}{88}}=\frac{409\times 88}{409}=88$

(Ans: 2)

Estimated Time to arrive at the answer = 120 seconds.

### Using Technique

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You can simply approximate the expression $4+\frac{3}{4+\frac{3}{4+\frac{3}{4}}}\approx 5(-)$ (Note, that the actual value will be less than 5 and hence the ‘–’ sign in brackets)

Hence, $\frac{409}{5}\approx 82$

Since, the actual denominator is less than 5, the final answer will be greater than 82. The only option coming close to this figure is 88, i.e option ‘2’.

(Ans: 2)

Estimated Time to arrive at the answer = 10 seconds.
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