The sum of three number is 12. The first number is twice the sum of the second and third. The third number is 5 less than the first. Find the numbers.

Question

The sum of three number is 12. The first number is twice the sum of the second and third. The third number is 5 less than the first. Find the numbers.

Answer 1

Hi, Debra. We can set up three equations with three unknowns based on the info given in the problem. Let x = 1st number, y = 2nd number, and z = 3rd number. Then we know that: 

 

x + y + z = 12

 

We also know that: x = 2 (y + z) = 2y + 2z and finally, 

 

z = x - 5 

 

Let's solve the first equation for x so that we can eventually set up a system of two linear equations with two unknowns, which is easier to solve. 

 

x = 12 - y - z 

 

We have two expressions for x, so we can set them equal to each other and solve for one of the variables. 

 

2y + 2z = 12 - y - z

3y + 3z = 12 (when we combine like terms and isolate the variables on the left-hand side of the equation)

y + z = 4 (by dividing each side by 3)

 

Remember, that x + y + z = 12, and z = x - 5. 

If z = x - 5, then x = z + 5

 

We can re-write the first equation as (z + 5) + y + z = 12 or

2z + y + 5 = 12

y + 2z = 7

 

Now we have a system of two equations with two unknowns:

 

y + z = 4

y + 2z = 7

 

We can subtract these two from each other:

 

   y + z = 4

- (y + 2z = 7)

 

-z = -3 or z = 3

 

If z = 3, then 3 = x - 5 (from our equation above)

x = 8

 

Lastly, since we know y + z = 4 (from above), then y + 3 = 4 or y = 1

 

Hence, the three numbers are 8, 1, and 3. Please feel free to contact me if you have any questions. 

 

 

Answer 2

If the first number is twice the sum of the 2nd and third, then it has to be half of 12, or 6.

 

The sum of the 2nd and 3rd is 3 and the 3rd number is 6 - 5 = 1

 

6, 2, 1