Age Word Problems Practice Test Full Solutions - Free Printable

Let's solve the problem step by step using algebra.

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Problem Statement:

- Sarah is 36 years less than double Katlyn’s age.
- In 6 years, Sarah’s age will be 12 years more than half of Katlyn’s age.
- Find the current ages of Katlyn and Sarah.

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Step 1: Define Variables

Let:
- $ x $ = Katlyn’s current age
Then:
- Sarah’s current age = $ 2x - 36 $ (since she is 36 years less than double Katlyn’s age)

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Step 2: Express Ages in 6 Years

In 6 years:
- Katlyn’s age = $ x + 6 $
- Sarah’s age = $ (2x - 36) + 6 = 2x - 30 $

According to the second condition:
> In 6 years, Sarah’s age will be 12 years more than half of Katlyn’s age.

So:
$$
\text{Sarah's age in 6 years} = \frac{1}{2} \times (\text{Katlyn's age in 6 years}) + 12
$$

Substitute:
$$
2x - 30 = \frac{1}{2}(x + 6) + 12
$$

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Step 3: Solve the Equation

$$
2x - 30 = \frac{1}{2}(x + 6) + 12
$$

First, simplify the right-hand side:

$$
\frac{1}{2}(x + 6) = \frac{x}{2} + 3
$$

So:
$$
2x - 30 = \frac{x}{2} + 3 + 12
$$
$$
2x - 30 = \frac{x}{2} + 15
$$

Now eliminate fractions by multiplying every term by 2:

$$
2(2x - 30) = 2\left(\frac{x}{2} + 15\right)
$$
$$
4x - 60 = x + 30
$$

Subtract $ x $ from both sides:

$$
3x - 60 = 30
$$

Add 60 to both sides:

$$
3x = 90
$$

Divide by 3:

$$
x = 30
$$

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Step 4: Find Both Ages

- Katlyn’s age = $ x = 30 $
- Sarah’s age = $ 2x - 36 = 2(30) - 36 = 60 - 36 = 24 $

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Step 5: Check the Solution

Current ages:
- Katlyn: 30
- Sarah: 24

Is Sarah 36 years less than double Katlyn’s age?
- Double Katlyn’s age: $ 2 \times 30 = 60 $
- $ 60 - 36 = 24 $ → Yes, matches Sarah’s age ✔

In 6 years:
- Katlyn: $ 30 + 6 = 36 $
- Sarah: $ 24 + 6 = 30 $

Is Sarah’s age 12 years more than half of Katlyn’s age?
- Half of Katlyn’s age: $ \frac{36}{2} = 18 $
- $ 18 + 12 = 30 $ → Yes, matches Sarah’s age ✔

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✔ Final Answer:

- Katlyn is 30 years old.
- Sarah is 24 years old.

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