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?
Let's start by assigning variables to the unknowns:
- Let's call Paul's age "P"
- Let's call Patricia's age "P+2" (since we know Paul is two years younger)
- Let's call Daniel's current age "D"
From the problem, we also know that:
- D = 1.25(P+2) (since Daniel is 25% older than Patricia)
- 10 years ago, D-10 = 1.5(P+2-10) (since 10 years ago, Daniel was 50% older than Patricia)
Simplifying the second equation:
- D-10 = 1.5P - 15
- D = 1.5P - 5
Now we can substitute the second equation into the first:
- 1.5P - 5 = 1.25(P+2)
- 1.5P - 5 = 1.25P + 2.5
- 0.25P = 7.5
- P = 30
So Paul is currently 30 years old.
Let's solve this step-by-step.
Let's denote Patricia's age as P.
According to the given information, Paul is two years younger than Patricia, so Paul's age can be denoted as P - 2.
The next piece of information states that Daniel is 25% older than Patricia.
Since percent means "per hundred," we can express this mathematically as:
Daniel's age = Patricia's age + (25/100) * Patricia's age (which is the same as Patricia's age multiplied by 0.25)
Simplifying this equation, we get:
Daniel's age = Patricia's age + 0.25 * Patricia's age
Given that ten years ago, Daniel was 50% older than Patricia, we can calculate their ages ten years ago.
Let's denote Patricia's age ten years ago as X.
Therefore, Daniel's age ten years ago can be expressed as X + (50/100) * X (which is the same as X multiplied by 0.50).
Now, let's summarize the ages ten years ago:
Patricia's age ten years ago: X
Paul's age ten years ago: X - 2
Daniel's age ten years ago: X + (50/100) * X
We can now set up an equation to relate the ages of Patricia, Paul, and Daniel, ten years ago:
Daniel's age ten years ago = Patricia's age ten years ago + 10
Substituting the values we found, we get:
X + (50/100) * X = X + 10
Simplifying this equation, we have:
1.5X = X + 10
Subtracting X from both sides, we get:
0.5X = 10
Divide both sides by 0.5:
X = 20
Now, we know that Patricia's age ten years ago was 20. Therefore, her current age is 20 + 10 = 30.
Using this information, we can calculate the current ages of Paul and Daniel:
Paul's age = Patricia's age - 2 = 30 - 2 = 28
Daniel's age = Patricia's age + (25/100) * Patricia's age = 30 + (0.25 * 30) = 30 + 7.5 = 37.5
Therefore, Paul is currently 28 years old.
To find Paul's current age, we need to first determine Patricia's age. Let's proceed step-by-step:
Let's assume Patricia's current age is P.
According to the given information, Paul is two years younger than Patricia. Hence, Paul's current age would be P - 2.
Daniel is 25% older than Patricia, which we can express as:
Daniel's current age = P + (25/100) * P
Ten years ago, Daniel was 50% older than Patricia. We need to calculate Patricia's age ten years ago and Daniel's age ten years ago, based on their current ages.
For Patricia ten years ago, her age would be P - 10.
For Daniel ten years ago, his age would be (P + (25/100) * P) - 10.
According to the information, Daniel's age ten years ago was 50% older than Patricia's age ten years ago:
(P + (25/100) * P) - 10 = (P - 10) + (50/100) * (P - 10)
Simplifying the equation, we have:
(P + (25/100) * P) - 10 = (P - 10) + (1/2) * (P - 10)
Multiplying through by 100 to remove the decimals, we get:
(100P + 25P) - 1000 = (100P - 1000) + (50P - 500)
Simplifying the equation further:
125P - 1000 = 150P - 1500
Moving variables to one side and constants to the other side:
125P - 150P = -1000 + 1500
-25P = 500
Dividing through by -25:
P = -500/25
P = 20
Therefore, Patricia's current age is 20 years old.
Now we can find Paul's current age:
Paul's current age = Patricia's age - 2
Paul's current age = 20 - 2
Paul's current age = 18
Hence, Paul is currently 18 years old.