the ratio of kim’s age to her daughter’s age is 5:2. Four years ago, the ration of their age was 3:1. How old will kim be in 6 years time? Solve using models

Let's solve the problem using a model-based approach.

### Step 1: Define Variables
Let's denote Kim's current age as \( K \) and her daughter's current age as \( D \).

Given:
- The ratio of Kim's age to her daughter's age is \( 5:2 \).
- Four years ago, the ratio of their ages was \( 3:1 \).

### Step 2: Express Ages in Terms of a Variable
From the first condition, we know:
\[
K = 5x \quad \text{and} \quad D = 2x
\]
where \( x \) is a common factor.

### Step 3: Model the Situation Four Years Ago
Four years ago:
- Kim's age was \( K - 4 \)
- Her daughter's age was \( D - 4 \)

The problem states that the ratio of their ages four years ago was \( 3:1 \). Thus:
\[
\frac{K - 4}{D - 4} = \frac{3}{1}
\]

### Step 4: Substitute and Solve
Substituting the expressions for \( K \) and \( D \) into the equation:
\[
\frac{5x - 4}{2x - 4} = 3
\]

Cross-multiply to solve for \( x \):
\[
5x - 4 = 3(2x - 4)
\]
\[
5x - 4 = 6x - 12
\]

Rearrange the equation to solve for \( x \):
\[
5x - 6x = -12 + 4
\]
\[
-x = -8 \quad \Rightarrow \quad x = 8
\]

### Step 5: Find Their Current Ages
Using \( x = 8 \):
\[
K = 5x = 5 \times 8 = 40
\]
\[
D = 2x = 2 \times 8 = 16
\]

So, Kim is currently 40 years old, and her daughter is 16 years old.

### Step 6: Determine Kim's Age in 6 Years
To find Kim's age in 6 years:
\[
K_{\text{in 6 years}} = 40 + 6 = 46
\]

### Final Answer:
Kim will be **46 years old** in 6 years.

Would you like more details or have any questions?

Here are 8 related questions you might be interested in:
1. What if the ratio was 4:1 instead of 5:2?
2. How would the problem change if the ratio 4 years ago was 2:1?
3. Can you solve this using algebra instead of models?
4. How does the value of \( x \) influence their ages?
5. What if the difference between their ages was given instead of the ratio?
6. How do ratios help in solving age-related problems?
7. How would you verify the solution is correct?
8. What are some other real-life examples where ratios are useful?

**Tip:** When solving ratio problems, always express the quantities in terms of a common variable for clarity.

Learn how to solve age ratio problems using a model-based approach. This problem involves determining Kim's current age based on given ratios and age differences. Suitable for students in Grades 6-8.

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