For three consecutive integers, half the sum of the largest and smallest numbers plus the middle number is equal to three times the middle number. What are the three numbers?

Answer 1

Hi Tessa,

Taking this question part by part is the best approach. Then, we can put all the parts back together to get our answer.

Part 1: "Three Consecutive Integers"

Consecutive means "in a row", and integers means positive or negative whole numbers (not fractions). This can be 2, 3, 4, or -1, -2, -3, or 900, 901, 902. The first number can be anything, so we'll use the placeholder "x". Then, the next two numbers must therefore be (x+1) and (x+2).

Part 2: "Half the sum of the largest and smallest numbers..."

Regardless of what x is, x+2 will always be greater than x and x+1. Therefore the largest number is x+2 and the smallest is x.

We can write this part as "1/2 * (x + (x+2))..."

Part 3: "...plus the middle number is equal to three times the middle number"

"...+ (x+1) = 3 * (x+1)"

Now, to put all the parts together, let's solve for what "x" is:

(x+(x+2)) / 2 + (x+1) = 3(x+1) # adding the numerator and subtracting (x+1) from both sides...

(2x + 2) / 2 = 2(x+1) # simplifying the fraction by dividing 2 from the top and bottom.....

x+1 = 2(x+1) # final simplification

x+1 = 2x + 2

x = -1

Make sure to always answer what the question is asking!!!!

The question asks for the values of the three numbers, not for what x (i.e., the smallest number) is. Therefore, our answer will be (-1), ((-1)+1), and ((-1)+2), or -1, 0, and 1.

You can check if this is correct by plugging the values back into x. " Half the sum of the largest and smallest number plus the middle number is equal to three times the middle number."

(1/2 * (1 + -1) ) + 0 = 3 * 0

0 + 0 = 0

That checks out!

Hope this helps,

John Fonte