Number riddle

RIDDLE

There are 5 positive whole numbers. The first is the square of the
second. The sum of the 2nd and the 3rd is 10. The 4th is greater than
the 2nd by 1. The sum of the 3rd and the 5th is 14. The sum of all the
numbers is 30. Find the values of the numbers.

SOLUTION

Represent the 5 numbers by any 5 letters of the alphabet you fancy. I
am using a, b, c, d and e for the 1st, 2nd, 3rd, 4th and 5th numbers
respectively.

From the statements in the question,

a = b² ................ (1)
b + c = 10 ................(2)
d = b + 1 ................(3)
c + e = 14 ................(4)
a + b + c + d + e = 30................(5).

* Express any 4 of the unknowns in terms of the 5th unknown in
equations (1) to (4).

a = b² (equation 1).
c = 10 - b (from equation 2).
d = b + 1 (equation 3).
e = 14 - c (from equation 4) = 14 - (10 - b) (from equation 2) = 14
- 10 + b = b + 4.

* Substitute these values into equation 5.

a + b + c + d + e = 30.

b² + b + (10 - b) + (b + 1) + (b + 4) = 30.

b² + 2b + 15 = 30.

b² + 2b + 15 - 30 = 0.

b² + 2b - 15 = 0.

(b + 5) (b - 3) = 0.

Either b + 5 = 0 or b - 3 = 0.

b + 5 = 0 ➡ b = - 5.

b - 3 = 0 ➡ b = 3.

Therefore, b = 3 (positive number).

Hence, a = b² = 3² = 9 ;
c = 10 - b = 10 - 3 = 7;
d = b + 1 = 3 + 1 = 4;
e = b + 4 = 3 + 4 = 7.

Therefore, the 1st, 2nd, 3rd, 4th and 5th numbers are 9, 3, 7, 4 and
7 respectively.

NOTE: * If you express the other unknowns in terms of a, substitution
of the values will lead you to an equation involving surds which you
can manipulate into a quadratic equation (by isolating the surd on one
side of the equation and then squaring both sides of the equation).
You will get 2 values of a (a = 9 or a = 25). Use the smaller value of
a , or your
a + b + c + d + e will be greater than 30 (violating equation 5).