Question:
Alice is twice as old as Tom, but four years ago, she was three years older than Tom is now. How old is Tom now?
Alice is twice as old as Tom, but four years ago, she was three years older than Tom is now. How old is Tom now?
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In age problems, carefully distinguish between past, present, and future ages. "Four years ago" means you subtract 4 from the current age. "Tom is now" refers to the current age variable \(T\), not his age in the past. Reading each phrase carefully is key to setting up the correct equations.
Updated On: Oct 3, 2025
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept:
This is a classic age word problem that can be solved by setting up a system of linear equations based on the information given. We need to define variables for the current ages and translate the sentences into mathematical equations.
Step 2: Key Formula or Approach:
Let \(A\) be Alice's current age.
Let \(T\) be Tom's current age.
We will form two equations from the two statements in the problem and solve for \(T\).
Step 3: Detailed Explanation:
Translate the first statement into an equation:
"Alice is twice as old as Tom" can be written as:
\[ A = 2T \quad (\text{Equation 1}) \] Translate the second statement into an equation:
"four years ago, she was..." refers to Alice's age 4 years ago, which is \(A-4\).
"...three years older than Tom is now." refers to Tom's current age (\(T\)) plus 3.
So, this statement can be written as:
\[ A - 4 = T + 3 \quad (\text{Equation 2}) \] Now we have a system of two equations with two variables. We can solve this system by substitution. Substitute the expression for \(A\) from Equation 1 into Equation 2.
\[ (2T) - 4 = T + 3 \] Now, solve for \(T\). Start by getting the \(T\) terms on one side of the equation. Subtract \(T\) from both sides:
\[ 2T - T - 4 = 3 \] \[ T - 4 = 3 \] Now, get the constant terms on the other side. Add 4 to both sides:
\[ T = 3 + 4 \] \[ T = 7 \] So, Tom is currently 7 years old.
Verification (Optional):
If Tom is 7, then Alice is \(A = 2T = 2(7) = 14\).
Four years ago, Alice was \(14 - 4 = 10\).
Is 10 three years older than Tom is now (7)? Yes, \(10 = 7 + 3\). The conditions are met.
Step 4: Final Answer:
Tom is 7 years old now.
This is a classic age word problem that can be solved by setting up a system of linear equations based on the information given. We need to define variables for the current ages and translate the sentences into mathematical equations.
Step 2: Key Formula or Approach:
Let \(A\) be Alice's current age.
Let \(T\) be Tom's current age.
We will form two equations from the two statements in the problem and solve for \(T\).
Step 3: Detailed Explanation:
Translate the first statement into an equation:
"Alice is twice as old as Tom" can be written as:
\[ A = 2T \quad (\text{Equation 1}) \] Translate the second statement into an equation:
"four years ago, she was..." refers to Alice's age 4 years ago, which is \(A-4\).
"...three years older than Tom is now." refers to Tom's current age (\(T\)) plus 3.
So, this statement can be written as:
\[ A - 4 = T + 3 \quad (\text{Equation 2}) \] Now we have a system of two equations with two variables. We can solve this system by substitution. Substitute the expression for \(A\) from Equation 1 into Equation 2.
\[ (2T) - 4 = T + 3 \] Now, solve for \(T\). Start by getting the \(T\) terms on one side of the equation. Subtract \(T\) from both sides:
\[ 2T - T - 4 = 3 \] \[ T - 4 = 3 \] Now, get the constant terms on the other side. Add 4 to both sides:
\[ T = 3 + 4 \] \[ T = 7 \] So, Tom is currently 7 years old.
Verification (Optional):
If Tom is 7, then Alice is \(A = 2T = 2(7) = 14\).
Four years ago, Alice was \(14 - 4 = 10\).
Is 10 three years older than Tom is now (7)? Yes, \(10 = 7 + 3\). The conditions are met.
Step 4: Final Answer:
Tom is 7 years old now.
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