The ratio of the ages of a father and a son at present is `5:2`. Four years hence the ratio of the ages of the son his mother will be `1:2`.What is the ratio of the present ages of the father and the mother ?

Options

A. `3:4`
B. `5:4`
C. `4:3`
D. Cannot be determined

Explanation

To solve the problem step by step, we will denote the present ages of the father, mother, and son as F, M, and S respectively.

### Step 1: Set up the equations based on the given ratios.

1. The ratio of the ages of the father and son at present is given as \(5:2\). This can be expressed as:
\[ \frac{F}{S} = \frac{5}{2} \]
From this, we can express the age of the father in terms of the age of the son:
\[ F = \frac{5}{2}S \quad \text{(Equation 1)} \]

### Step 2: Set up the second equation based on future ages.

2. Four years hence, the ratio of the ages of the son and the mother will be \(1:2\). In four years, the son's age will be \(S + 4\) and the mother's age will be \(M + 4\). This can be expressed as:
\[ \frac{S + 4}{M + 4} = \frac{1}{2} \]
Cross-multiplying gives:
\[ 2(S + 4) = M + 4 \]
Simplifying this, we get:
\[ 2S + 8 = M + 4 \]
Rearranging gives us:
\[ M = 2S + 4 \quad \text{(Equation 2)} \]

### Step 3: Substitute Equation 1 into Equation 2.

3. Now we can substitute Equation 1 into Equation 2. From Equation 1, we have \(S = \frac{2}{5}F\). Substituting this into Equation 2:
\[ M = 2\left(\frac{2}{5}F\right) + 4 \]
Simplifying this gives:
\[ M = \frac{4}{5}F + 4 \]

### Step 4: Find the ratio of the present ages of the father and mother.

4. Now we need to find the ratio of the present ages of the father and mother:
\[ \frac{F}{M} = \frac{F}{\frac{4}{5}F + 4} \]
To simplify this, we can multiply both the numerator and the denominator by 5:
\[ \frac{F}{M} = \frac{5F}{4F + 20} \]
Now, we can express this ratio:
\[ \frac{F}{M} = \frac{5}{4 + \frac{20}{F}} \]

### Step 5: Determine the ratio.

5. To find a specific numerical ratio, we need to assume a value for \(F\). Let's assume \(F = 20\) (as an example):
\[ M = \frac{4}{5}(20) + 4 = 16 + 4 = 20 \]
Thus, the ratio becomes:
\[ \frac{F}{M} = \frac{20}{20} = 1:1 \]

### Conclusion

The ratio of the present ages of the father and mother is \(1:1\).