Dad is 4 times as old as his son Jim. In 10 years, Dad's age will be 20 years more than twice Jim's age. How old is Jim? If \( D= \) dad's age now and \( J= \) Jim's age now, which of the following systems could be used to solve the problem? \( D=4 J \) and \( D+10=2(4 J+10)+20 \) \( D=4 J \) and \( D+10=2(J+10)+20 \) \( D=4 J \) and \( D+10=2 J+20 \)

To solve the problem, we need to extract the known conditions and set up the equations based on the information provided. 1. **Extract Known Conditions:** - Let \( D \) be Dad's current age. - Let \( J \) be Jim's current age. - From the problem, we know: - Dad is 4 times as old as Jim: \( D = 4J \). - In 10 years, Dad's age will be 20 years more than twice Jim's age: \( D + 10 = 2(J + 10) + 20 \). 2. **Identify the Correct System of Equations:** - The first equation \( D = 4J \) is correct as it represents Dad's current age in relation to Jim's age. - The second equation must represent the condition about their ages in 10 years. Let's analyze the options provided: - **Option 1:** \( D = 4J \) and \( D + 10 = 2(4J + 10) + 20 \) - This option incorrectly substitutes \( D \) in the second equation, as it uses \( 4J \) instead of \( J \). - **Option 2:** \( D = 4J \) and \( D + 10 = 2(J + 10) + 20 \) - This option correctly represents the relationship in 10 years, as it uses Jim's age \( J \). - **Option 3:** \( D = 4J \) and \( D + 10 = 2J + 20 \) - This option is incorrect because it does not account for Jim's age in 10 years. 3. **Conclusion:** The correct system of equations to solve the problem is: \[ D = 4J \quad \text{and} \quad D + 10 = 2(J + 10) + 20 \] Now, let's solve the equations to find Jim's age \( J \). ### Step-by-Step Solution: 1. From the first equation, we have: \[ D = 4J \] 2. Substitute \( D \) in the second equation: \[ 4J + 10 = 2(J + 10) + 20 \] 3. Simplify the second equation: \[ 4J + 10 = 2J + 20 + 20 \] \[ 4J + 10 = 2J + 40 \] 4. Rearranging gives: \[ 4J - 2J = 40 - 10 \] \[ 2J = 30 \] \[ J = 15 \] Thus, Jim is currently \( 15 \) years old.