[Smart Math] Arithmetic Problem 13

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

Arithmetic

Problem

In one of the following sets of fractions, the values increase progressively. Which is the one?

$\frac{5}{7}$ , $\frac{8}{11}$ , $\frac{11}{15}$ , $\frac{13}{17}$ , $\frac{7}{9}$
$\frac{5}{7}$ , $\frac{7}{9}$ , $\frac{13}{17}$ , $\frac{11}{15}$ , $\frac{8}{11}$
$\frac{5}{7}$ , $\frac{11}{15}$ , $\frac{8}{11}$ , $\frac{13}{17}$ , $\frac{7}{9}$
$\frac{5}{7}$ , $\frac{11}{15}$ , $\frac{8}{11}$ , $\frac{7}{9}$ , $\frac{13}{17}$
$\frac{5}{7}$ , $\frac{13}{17}$ , $\frac{11}{15}$ , $\frac{7}{9}$ , $\frac{8}{11}$

The Usual Method

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Equate the denominators using the LCM of the denominators as follows:

LCM of 7, 11, 15, 17, and 9 = 7 x 11 x 9 x 5 x 17 = 58905

Hence, $\frac{5}{7}$ can be written as $\frac{5}{7}$ = $\frac{42075}{58905}$

$\frac{8}{11}$ can be written as $\frac{8}{11}$ = $\frac{42840}{58905}$

$\frac{11}{15}$ can be written as $\frac{11}{15}$ = $\frac{43197}{58905}$

$\frac{13}{17}$ can be written as $\frac{13}{17}$ = $\frac{45045}{58905}$

$\frac{7}{9}$ can be written as $\frac{7}{9}$ = $\frac{45815}{58905}$

As can be seen,

$\frac{42075}{58905}$ < $\frac{42840}{58905}$ < $\frac{43197}{58905}$ < $\frac{45045}{58905}$ < $\frac{45815}{58905}$

Hence, $\frac{5}{7}$ < $\frac{8}{11}$ < $\frac{11}{15}$ < $\frac{13}{17}$ < $\frac{7}{9}$

Since, option ‘1’ gives this arrangement, it is the correct answer.

(Ans: 1)

Estimated Time to arrive at the answer = 200 seconds.

Using Technique

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We already know the value of the reciprocals of 7, 11, 15, 17 and 9 as:

$\frac{1}{7}$ = 0.1482… $\therefore \frac{5}{7}=5\times 0.1428\approx 0.714$

$\frac{1}{11}$ = 0.09090… $\therefore \frac{8}{11}=8\times 0.0909\approx 0.7272$

$\frac{1}{15}$ = 0.0666… $\therefore \frac{11}{15}=11\times 0.066\approx 0.7337$

$\frac{1}{17}$ = 0.0588… $\therefore \frac{13}{17}=13\times 0.0588=0.764$

$\frac{1}{9}$ = 0.1111… $\therefore \frac{7}{9}=7\times 0.1111=0.7777$

From above it is clear that,