10 years ago the age of Karishma was 1/3rd of the age of Babita. 14 years hence the ratio of ages of Karishma and Babita will be 5: 9. Find the ratio of their ages :

Options

A. 0.56180555555556
B. 0.47708333333333
C. 29:17
D. 0.55902777777778

Answer

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

### Step 1: Define Variables
Let the current age of Karishma be \( K \) and the current age of Babita be \( B \).

### Step 2: Set Up the First Equation
According to the problem, 10 years ago, Karishma's age was \( \frac{1}{3} \) of Babita's age. Therefore, we can write the equation:
\[ K - 10 = \frac{1}{3}(B - 10) \]

### Step 3: Simplify the First Equation
Multiply both sides by 3 to eliminate the fraction:
\[ 3(K - 10) = B - 10 \]
Expanding this gives:
\[ 3K - 30 = B - 10 \]
Rearranging it, we get:
\[ B = 3K - 20 \quad \text{(Equation 1)} \]

### Step 4: Set Up the Second Equation
The problem states that 14 years from now, the ratio of their ages will be \( 5:9 \). Thus, we can write:
\[ \frac{K + 14}{B + 14} = \frac{5}{9} \]

### Step 5: Cross Multiply the Second Equation
Cross multiplying gives:
\[ 9(K + 14) = 5(B + 14) \]
Expanding both sides:
\[ 9K + 126 = 5B + 70 \]
Rearranging this gives:
\[ 9K - 5B = -56 \quad \text{(Equation 2)} \]

### Step 6: Substitute Equation 1 into Equation 2
Substituting \( B \) from Equation 1 into Equation 2:
\[ 9K - 5(3K - 20) = -56 \]
Expanding this gives:
\[ 9K - 15K + 100 = -56 \]
Combining like terms results in:
\[ -6K + 100 = -56 \]

### Step 7: Solve for \( K \)
Now, isolate \( K \):
\[ -6K = -56 - 100 \]
\[ -6K = -156 \]
Dividing both sides by -6:
\[ K = 26 \]

### Step 8: Find \( B \)
Now, substitute \( K \) back into Equation 1 to find \( B \):
\[ B = 3(26) - 20 \]
\[ B = 78 - 20 = 58 \]

### Step 9: Find the Ratio of Their Ages
Now we can find the ratio of their current ages:
\[ \text{Ratio} = \frac{K}{B} = \frac{26}{58} \]
Simplifying the ratio by dividing both numbers by 2:
\[ \text{Ratio} = \frac{13}{29} \]

### Final Answer
Thus, the ratio of the ages of Karishma and Babita is \( 13:29 \).

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