Paul is two years younger than Patricia. Daniel is 25% older than Patricia. Ten years ago, Daniel was 50% older than Patricia: How old is Paul currently?
The correct answer and explanation is:
Let’s solve the problem step-by-step.
Let:
- Patricia’s current age = xx
- Paul’s current age = x−2x – 2 (since Paul is 2 years younger than Patricia)
- Daniel’s current age = x+0.25x=1.25xx + 0.25x = 1.25x (since Daniel is 25% older than Patricia)
Ten years ago:
- Patricia’s age = x−10x – 10
- Daniel’s age = 1.25x−101.25x – 10
According to the problem:
Ten years ago, Daniel was 50% older than Patricia.
This means: 1.25x−10=1.5(x−10)1.25x – 10 = 1.5(x – 10)
Solve the equation:
1.25x−10=1.5x−151.25x – 10 = 1.5x – 15
Subtract 1.25×1.25x from both sides: −10=0.25x−15-10 = 0.25x – 15
Add 15 to both sides: 5=0.25×5 = 0.25x
Divide both sides by 0.25: x=20x = 20
Now compute Paul’s age:
Paul=x−2=20−2=18\text{Paul} = x – 2 = 20 – 2 = \boxed{18}
✅ Final Answer: Paul is currently 18 years old.
✍️ Explanation (300 words):
This is a classic algebra word problem involving relationships among three people’s ages. The key is translating words into equations.
First, we let Patricia’s age be the variable xx. Paul is stated to be 2 years younger, so we define Paul’s age as x−2x – 2. Daniel is 25% older than Patricia, which we translate to a multiplier of 1.25. Thus, Daniel’s age becomes 1.25×1.25x.
The crucial piece of information is about their ages 10 years ago: “Daniel was 50% older than Patricia.” To use this, we express both of their ages 10 years ago: Daniel was 1.25x−101.25x – 10 and Patricia was x−10x – 10. If Daniel was 50% older, that means his age equaled Patricia’s age plus 50% of it — which is the same as multiplying Patricia’s age by 1.5.
Setting up the equation: 1.25x−10=1.5(x−10)1.25x – 10 = 1.5(x – 10)
We solve it step by step to isolate xx, which gives us Patricia’s age as 20. Using this, we subtract 2 to find Paul’s age: 18.
Always remember to verify the solution using the original statements. Ten years ago, Patricia was 10, Daniel was 15 — and 15 is exactly 50% more than 10. So the answer holds true.
This systematic approach ensures you solve such age problems accurately.