The sum of three numbers is 16. The largest number is equal to the sum of the other two, and 3 times the smallest number is 1 more than the largest.

Question

Find the three numbers.

Answer 1

Let's call the three numbers A, B, and C in increasing order (i.e. A is smallest, C is largest).

The sum of three numbers is 16

(1) A + B + C = 16

The largest number is equal to the sum of the other two

(2) C = A + B

3 times the smallest number is 1 more than the largest

(3) 3A = C + 1

Let's substitute C from equation (2) into both (1) and (3) to get:

A + B + (A + B) = 16

=> 2A + 2B = 16

=> (4) A + B = 8

and 3A = (A + B) + 1

=> (5) 2A - B = 1

With one B (but opposite signs) in both (4) and (5) we can immediately add the equations together:

A + B = 8

+

2A - B = 1

=

3A + 0B = 9

=> A = 3

Substitute found A in (4) so 3 + B = 8 => B = 5

And then substitute both values in (1) to get 3 + 5 + C = 16 => C = 8

Hence the three numbers are 3, 5, and 8.

Double-check by substituting in original equations to get 3 + 5 + 8 = 16 as required, and 3 + 5 = 8 as required, and 8 + 1 = 3(3) as required.