Three times Diana's age is 17years more than twice Jim's age. The sum of their ages is 13 years less than their father's age which is three times Jim's age. What are the children ages?

Options

A. Diana 21 years, Jim 16 years
B. Diana 15 years, Jim 14 years
C. Diana 15 years, Jim 16 years
D. Diana 20 years, Jim 14 years

Answer

To solve the problem, we need to set up two equations based on the information given in the question. Let's define the variables first:

Let:
- \( D \) = Diana's age
- \( J \) = Jim's age

### Step 1: Set up the first equation
The first part of the problem states that "Three times Diana's age is 17 years more than twice Jim's age." We can express this as:
\[ 3D = 2J + 17 \]

### Step 2: Set up the second equation
The second part of the problem states that "The sum of their ages is 13 years less than their father's age which is three times Jim's age." This can be expressed as:
\[ D + J = 3J - 13 \]

### Step 3: Simplify the second equation
We can rearrange the second equation:
\[ D + J + 13 = 3J \]
\[ D + 13 = 2J \]
Thus, we can express \( D \) in terms of \( J \):
\[ D = 2J - 13 \]

### Step 4: Substitute the expression for \( D \) into the first equation
Now, we can substitute \( D \) from the second equation into the first equation:
\[ 3(2J - 13) = 2J + 17 \]

### Step 5: Expand and simplify
Expanding the left side:
\[ 6J - 39 = 2J + 17 \]

### Step 6: Rearranging the equation
Now, we will move all terms involving \( J \) to one side and constant terms to the other side:
\[ 6J - 2J = 17 + 39 \]
\[ 4J = 56 \]

### Step 7: Solve for \( J \)
Dividing both sides by 4:
\[ J = 14 \]

### Step 8: Substitute back to find \( D \)
Now that we have \( J \), we can find \( D \) using the equation \( D = 2J - 13 \):
\[ D = 2(14) - 13 \]
\[ D = 28 - 13 \]
\[ D = 15 \]

### Conclusion
Thus, the ages of the children are:
- Diana's age \( D = 15 \) years
- Jim's age \( J = 14 \) years

### Summary of the solution:
- Diana's age is 15 years.
- Jim's age is 14 years.