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Arithmetic Smart Math

[Smart Math] Arithmetic Problem 21

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

Arithmetic

Problem

Find the sum of reciprocal of two numbers whose sum is 12 and product is 32.

  1. \frac{5}{3}
  2. \frac{8}{3}
  3. 3
  4. \frac{3}{8}
  5. 8

The Usual Method

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Let one of the numbers be ‘x’.

Hence the other number is (12 – x)

Also \left( 12-x \right)\times x=32

\therefore 12x-x^{2}=32

\therefore x^{2}-12x+32=0

\therefore \left( x-8 \right)\times \left( x-4 \right)

\therefore x=8 and x=4

Hence one of the numbers is 4 and the other number is 8.

Hence sum of reciprocals = \frac{1}{4}+\frac{1}{8}=\frac{3}{8}

(Ans: 4)

Estimated Time to arrive at the answer = 30 seconds.

Using Technique

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Since the sum and product of two numbers are integers, the two numbers are also integers. Reciprocals of integers will result in fractions and so will the sum of the reciprocals. This eliminates options ‘3’ and ‘5’.

Since both the sum and product of the two numbers is even, the two numbers individually are also even. Hence the sum of their reciprocals will have their denominator a multiple of 2 x 2 = 4. The only option satisfying these conditions is option ‘4’.

(Ans: 4)

Estimated Time to arrive at the answer = 10 seconds.
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