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Arithmetic Smart Math

[Smart Math] Arithmetic Problem 18

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

Arithmetic

Problem

The sum of cubes of three numbers is 8072 and the ratio of the first to the second as also the second to the third is 3 : 2. What is the second number?

  1. 2
  2. 4
  3. 6
  4. 9
  5. 12

The Usual Method

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Let the three numbers be ‘1’, ‘2’ and ‘3’.

Hence a : b = 3 : 2 and b : c = 3 : 2

\therefore       a :           b :           c

=          3          :           2

3          :           2

9          :           6          :           4

Hence, a = 9x,

b = 6x

c = 4x

Now, (9x)^{3}+(6x)^{3}+(4x)^{3}= 8072

\therefore 729x^{3}+216x^{3}+64x^{3}=8072

\therefore 1009x^{3}=8072

\therefore x^{3}=\frac{8072}{1009}=8

\therefore x=2

Hence the second number is 6x = 6 x 2 = 12

(Ans: 5)

Estimated Time to arrive at the answer = 75 seconds.

Using Technique

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Simply by knowing that the ratios are in the form 3 : 2 between the first & second and 3 : 2 between the second & third number, we can know that the second number is a multiple of 6. (Since a : b : c = 9 : 6 : 4)

This means that the answer from amongst the options is either 6 or 12. All other options are eliminated.

Now assuming that b = 6, we will have a = 9 and c = 4 (Since a : b : c = 9 : 6 : 4)

Now simply add the last digits of the cubes 9, 6 and 4 i.e. 9 + 6 + 4 = 19 (Last digit of cubes of 9, 6 and 4 are 9, 6 and 4 respectively). Here the last digit is 9 (from 19). But the last digit of the sum of cubes of these number is actually 2 (from 8 072). Hence the middle number is not 6 but 12.

(Ans: 5)

Estimated Time to arrive at the answer = 30 seconds.
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