5 years ago, Peeta was twice as old as his daughter, Katniss. In ten years, the sum of their ages will equal 90. Find their current ages.
Let's start by assigning variables to their current ages.
Let 'P' represent Peeta's current age and 'K' represent Katniss's current age.
According to the information given, 5 years ago, Peeta was twice as old as Katniss, so we can create the equation:
(P - 5) = 2(K - 5)
In ten years, the sum of their ages will equal 90, so we can create the equation:
(P + 10) + (K + 10) = 90
Now, we have a system of equations:
(P - 5) = 2(K - 5)
(P + 10) + (K + 10) = 90
First, let's solve the first equation for P in terms of K.
Expand:
P - 5 = 2K - 10
P = 2K - 5
Now, substitute this expression for P in the second equation:
(2K - 5 + 10) + (K + 10) = 90
2K + K + 15 + 10 = 90
3K + 25 = 90
Subtract 25 from both sides:
3K = 65
Divide both sides by 3:
K = 65 / 3 = 21.6667
Katniss's current age, when rounded to the nearest whole number, is 22.
Now, substitute this value back into the first equation to solve for P:
P = 2K - 5
P = 2(22) - 5
P = 44 - 5
P = 39
Therefore, Peeta's current age is 39 and Katniss's current age is 22.
No.
Apologies for the incorrect solution.
Let's start by assigning variables to their current ages: Let P represent Peeta's current age and K represent Katniss's current age.
According to the information given, 5 years ago Peeta was twice as old as Katniss. So, we can create the equation: P - 5 = 2(K - 5).
In ten years, the sum of their ages will equal 90. So, we can create the equation: (P + 10) + (K + 10) = 90.
Now, we have a system of equations:
P - 5 = 2(K - 5)
(P + 10) + (K + 10) = 90.
Let's solve this system of equations:
From the first equation, we can simplify: P - 5 = 2K - 10.
By rearranging terms, we have P - 2K = -5.
Next, we'll rewrite the second equation: P + K + 20 = 90.
Now we have a system of equations:
P - 2K = -5
P + K = 70.
To solve this system, you can use substitution or elimination method. We'll use the elimination method here.
Multiply the second equation by 2 to eliminate the P term: 2(P + K) = 2(70) -> 2P + 2K = 140.
Now, we can sum the two equations: (P - 2K) + (2P + 2K) = -5 + 140.
This simplifies to 3P = 135.
Divide both sides by 3: P = 45.
Substitute the value of P back into the second equation:
45 + K = 70.
Subtract 45 from both sides: K = 25.
Therefore, Peeta's current age is 45 and Katniss's current age is 25.
Let's assume that Peeta's current age is represented by P, and his daughter Katniss' current age is represented by K.
According to the given information, "5 years ago, Peeta was twice as old as his daughter, Katniss." This can be written as:
P - 5 = 2(K - 5) ---(equation 1)
In ten years, the sum of their ages will equal 90. This can be written as:
(P + 10) + (K + 10) = 90 ---(equation 2)
We can now solve these two equations simultaneously to find the values of P and K, representing their current ages:
From equation 1, we can simplify it as follows:
P - 5 = 2K - 10
P = 2K - 10 + 5
P = 2K - 5 ---(equation 3)
Now, substitute equation 3 into equation 2:
(2K - 5 + 10) + (K + 10) = 90
2K + 5 + K + 10 = 90
3K + 15 = 90
3K = 90 - 15
3K = 75
K = 75/3
K = 25
Now substitute the value of K back into equation 3 to find P:
P = 2K - 5
P = 2(25) - 5
P = 50 - 5
P = 45
Therefore, Peeta's current age (P) is 45, and Katniss' current age (K) is 25.