Five times the second of three consecutive even integers is six more than twice the sum of the first and third integers. Find the middle even integer.

Question

This question has too much expression and can't figure out the middle integers because can't find it and when I try it it came out with a remainder

Answer 1

As we are being asked to solve for the middle of three consecutive even integers, let this be x.  Then, the first even integer is x - 2, and the third even integer is x + 2.

Then, we are given that 5 times the second of three consecutive even integers is six more than twice the sum of the first and third integers.

Translating this into an equation, we have

5 * x = 6 + 2 * (x-2 + x+2)

5x = 6 + 2 * 2x

5x = 6 + 4x

5x - 4x = 6

1x = 6

x = 6

Check the answer; the middle even integer is 6, so the first even integer is 4 and the third is 8

 

Then, 5 times the second even integer is 5 * 6 = 30

6 + 2 * the sum of the first and third integers is

6 + 2 *(4 + 8) = 6 + 2 * 12 = 30

 

30 = 30, so it checks.

Answer 2

First, how do we express 3 consecutive, even integers?

2n, 2n + 2, and 2n + 4.

Next, what is 5 times the second of these integers?

5 (2n + 2)

What is 6 more than twice the sum of the first and third of these integers?

6 + 2(2n + 2n + 4)

Now, put it all together. Given that [5 times the second of 3 consecutive integers] is [6 more than twice the sum of the first and third of these integers], we can write:

5(2n + 2) = 6 + 2(2n + 2n + 4)

Multiply out both sides:

 

10n + 10 = 6 + 8n + 8

Isolate the 'n' terms on the left side by eliminating the 8n term on the right:

 

10n + 10 - 8n = 6 + 8n + 8 - 8n

 

2n + 10 = 6 + 8

 

Isloate the 'n' terms on the left by eliminating the '10' on the left:

 

2n + 10 - 10 = 6 + 8 - 10

 

2n = 14 - 10 = 4

 

2n = 4

 

Eliminate the coefficient in front of n by dividing it out:

 

2n / 2 = 4 / 2

 

n = 2

 

But remember that our consecutive integers were:

 

2n, 2n + 2, and 2n + 4, or, when n = 2,

 

4, 6, 8

 

So, our middle integer would be 6.

 

Now check: does 5 * 6 = 6 + 2 (4 + 8)?

 

30 = 6 + 2 (12) = 6 + 24 = 30

 

Yup!