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# [Smart Math] Arithmetic Problem 24

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

### Problem

The sum of 5 successive number is 100. What is the product of the first and last number?

1. 246
2. 282
3. 396
4. 484
5. 555

### The Usual Method

For 5 consecutive numbers, to have their sum = 100, we have the numbers as:

$(x-2)$, $(x-1)$, $x$, $(x+1)$ and $(x+2)$

$\therefore (x-2)+(x-1)+x+(x+1)+(x+2)=100$

$\therefore 5x=100$

$\therefore x=20$

Hence the numbers are:

$(x-2)$ = 18

$(x-1)$ = 19

$x$ = 20

$(x+1)$ = 21

$(x+2)$ = 22

Hence product of the first and last number is 18 x 22 = 396

(Ans: 3)

Estimated Time to arrive at the answer = 60 seconds.

### Using Technique

One can intuitively know that to have sum of 5 consecutive numbers to be 100, we need $\frac{100}{5}=20$ as the middle number. We also know that in Arithmetic Progression, the product of the two consecutive numbers equidistant from a central number is always less than the square of the central number. Since 20 is the central number, the product of the first and last number (equidistant from the central number) will be less than $20^{2}=400$. Also since the numbers are close to each other (being consecutive), the product will not be ‘too’ less than 400. Only 396 satisfies this condition.