At the end of the year 2020, the ratio of the ages of Rita and Gita is 1 : 2. Some whole number of years later, it will be 5 : 9. At the end of which of the following years can the ratio of their ages never be 4 : 7? Assume that their ages are integers.

Let's denote the age of Rita at the end of the year 2020 as \( R \) and the age of Gita as \( G \).

### Step 1: Establish the initial ratio
Given that the ratio of their ages in 2020 is 1:2, we can write:
\[
R : G = 1 : 2 \implies G = 2R
\]

### Step 2: Establish the second ratio
Some whole number of years later, the ratio will be 5:9. Let \( n \) be the number of years after 2020 when this occurs. At that time, the ages of Rita and Gita will be \( R+n \) and \( G+n \), respectively. The ratio can be expressed as:
\[
\frac{R+n}{G+n} = \frac{5}{9}
\]
Substituting \( G = 2R \):
\[
\frac{R+n}{2R+n} = \frac{5}{9}
\]
### Step 3: Solve for \( n \)
Cross-multiplying gives:
\[
9(R+n) = 5(2R+n)
\]
Expanding both sides:
\[
9R + 9n = 10R + 5n
\]
Rearranging terms:
\[
9n - 5n = 10R - 9R \implies 4n = R
\]
Thus, \( R = 4n \).

### Step 4: Use the relationship to find \( G \)
Since \( G = 2R \), we get:
\[
G = 2(4n) = 8n
\]

### Step 5: Check if the ratio 4:7 is achievable
We need to check when \( \frac{R+m}{G+m} = \frac{4}{7} \), where \( m \) is the number of years after 2020. Substituting \( R = 4n \) and \( G = 8n \):
\[
\frac{4n+m}{8n+m} = \frac{4}{7}
\]
Cross-multiplying gives:
\[
7(4n+m) = 4(8n+m)
\]
Expanding both sides:
\[
28n + 7m = 32n + 4m
\]
Rearranging terms:
\[
7m - 4m = 32n - 28n \implies 3m = 4n
\]
Therefore:
\[
m = \frac{4n}{3}
\]

Since \( m \) must be a whole number, \( n \) must be a multiple of 3 for \( m \) to be an integer.

### Step 6: Conclusion
If \( n \) is not a multiple of 3, the ratio 4:7 cannot occur. Thus, we need to check for years where \( n \) is not a multiple of 3. 

The years that satisfy this condition will be those where \( n \) (years after 2020) is not divisible by 3. Hence, the ratio of their ages can never be 4:7 at the end of any such year.

### Follow-up Questions:
1. What would happen if the initial ratio was different?
2. How does changing the ratios affect the solution method?
3. Can this problem be solved using another algebraic method?
4. How would the solution change if we knew their exact ages in 2020?
5. How can this approach be applied to real-life age problems?

**Tip:** Always verify that conditions hold true for integer solutions when working with ratios and age problems.

Solve a challenging age ratio problem for Rita and Gita with a step-by-step solution. Learn how to determine when their age ratio cannot be 4:7, using algebra and integer properties. Suitable for high school students in Grades 10-12.

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