The ratio of age of A and B is x:y. If A's age is increased by 3 years and B's age is incresed by 2 years then new ratio of their ages becomes 24:25. Given that the sum of their actual ages is 93 years. Find the actual ratio of their ages.

Options

A. `21:22`
B. `42:45`
C. `45:48`
D. can't be determined

Answer

To solve the problem step by step, we need to set up equations based on the information given in the question.

### Step 1: Define the variables
Let A's age be \( a \) and B's age be \( b \). According to the problem, we know:
1. The ratio of A's age to B's age is \( x:y \).
2. The sum of their ages is \( a + b = 93 \).

### Step 2: Set up the equations based on the new age ratios
After increasing A's age by 3 years and B's age by 2 years, the new ages will be \( a + 3 \) and \( b + 2 \). The new ratio of their ages is given as \( 24:25 \). Therefore, we can write the equation:
\[ \frac{a + 3}{b + 2} = \frac{24}{25} \]
Cross-multiplying gives us:
\[ 25(a + 3) = 24(b + 2) \]
Expanding this, we have:
\[ 25a + 75 = 24b + 48 \]
Rearranging the equation gives:
\[ 25a - 24b = -27 \quad \text{(Equation 1)} \]

### Step 3: Use the sum of ages to create a second equation
From the sum of their ages, we have:
\[ a + b = 93 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations
We will solve Equations 1 and 2 simultaneously. From Equation 2, we can express \( a \) in terms of \( b \):
\[ a = 93 - b \]
Now substitute this expression for \( a \) into Equation 1:
\[ 25(93 - b) - 24b = -27 \]
Expanding this gives:
\[ 2325 - 25b - 24b = -27 \]
Combining like terms:
\[ 2325 - 49b = -27 \]
Now, add \( 49b \) to both sides:
\[ 2325 = 49b - 27 \]
Adding 27 to both sides:
\[ 2352 = 49b \]
Now, divide by 49:
\[ b = \frac{2352}{49} = 48 \]

### Step 5: Find A's age
Now that we have \( b \), we can find \( a \):
\[ a = 93 - b = 93 - 48 = 45 \]

### Step 6: Find the actual ratio of their ages
The actual ratio of A's age to B's age is:
\[ \frac{a}{b} = \frac{45}{48} \]
To simplify this ratio, we can divide both numbers by their greatest common divisor, which is 3:
\[ \frac{45 \div 3}{48 \div 3} = \frac{15}{16} \]

### Final Answer
Thus, the actual ratio of their ages is \( 15:16 \).

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