Consecutive Integer Word Problems 3 - YouTube - Free Printable

Let's solve the problem step by step.

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Problem Statement:

Zach found four consecutive odd integers such that:

> 5 times the sum of the first and the third was 22 greater than the product of 8 and the sum of the second and the fourth.

We are to find these four integers.

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Step 1: Define the variables

Let the four consecutive odd integers be:

- First: $ x $
- Second: $ x + 2 $
- Third: $ x + 4 $
- Fourth: $ x + 6 $

Since odd numbers are 2 apart, this is correct.

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Step 2: Translate the condition into an equation

> 5 times the sum of the first and the third

That is:
$$
5(x + (x + 4)) = 5(2x + 4)
$$

> Was 22 greater than the product of 8 and the sum of the second and the fourth

Sum of second and fourth:
$$
(x + 2) + (x + 6) = 2x + 8
$$

Product of 8 and that sum:
$$
8(2x + 8)
$$

So the full condition is:
$$
5(2x + 4) = 8(2x + 8) + 22
$$

Wait — let’s double-check the wording:

> "5 times the sum of the first and the third was 22 greater than the product of 8 and the sum of the second and the fourth."

So:
$$
\text{Left side} = \text{Right side} + 22
$$

So:
$$
5(x + (x + 4)) = 8((x + 2) + (x + 6)) + 22
$$

Now simplify both sides.

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Step 3: Simplify the equation

Left side:
$$
5(x + x + 4) = 5(2x + 4) = 10x + 20
$$

Right side:
$$
8((x + 2) + (x + 6)) + 22 = 8(2x + 8) + 22 = 16x + 64 + 22 = 16x + 86
$$

Now set up the equation:
$$
10x + 20 = 16x + 86
$$

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Step 4: Solve for $ x $

Subtract $ 10x $ from both sides:
$$
20 = 6x + 86
$$

Subtract 86 from both sides:
$$
20 - 86 = 6x \Rightarrow -66 = 6x
$$

Divide by 6:
$$
x = -11
$$

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Step 5: Find the four consecutive odd integers

- First: $ x = -11 $
- Second: $ -11 + 2 = -9 $
- Third: $ -9 + 2 = -7 $
- Fourth: $ -7 + 2 = -5 $

So the integers are: -11, -9, -7, -5

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Step 6: Verify the condition

Check:
> 5 times the sum of first and third = ?

First + third = $-11 + (-7) = -18$

5 × (-18) = -90

> Product of 8 and sum of second and fourth = ?

Second + fourth = $-9 + (-5) = -14$

8 × (-14) = -112

Now, is -90 equal to -112 + 22?

-112 + 22 = -90 → ✔ Yes!

So the condition is satisfied.

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✔ Final Answer:

The four consecutive odd integers are:
$$
\boxed{-11,\ -9,\ -7,\ -5}
$$