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# [Smart Math] Arithmetic Problem 18

Here’s and example of a SMART MATH problem for 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

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

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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