A farmer had twice as many ducks as chickens. After he had sold 413 ducks and another 19 ducks died, he had half as many ducks as chickens left. How many ducks did he have left?
d-(413+19) = 1/2 (d/2)
d = 576
Now finish it off
Let's assume the number of chickens the farmer had initially as "C" and the number of ducks as "D".
According to the given information, the farmer had twice as many ducks as chickens, so we can write the equation: D = 2C.
After he sold 413 ducks and another 19 ducks died, the number of ducks left is (D - 413 - 19).
It is also stated that after these events occurred, the number of ducks left was half the number of chickens, so we can write the equation: (D - 413 - 19) = 0.5C.
We can substitute the value of D from the first equation into the second equation to solve for C: (2C - 413 - 19) = 0.5C
Simplifying the equation: 2C - 432 = 0.5C
Combining like terms: 2C - 0.5C = 432
Simplifying further: 1.5C = 432
Dividing both sides of the equation by 1.5: C = 432 / 1.5
Calculating C: C = 288
Now, we can substitute the value of C into the first equation to find D: D = 2 * 288 = 576
Finally, let's calculate the number of ducks left after the events: (D - 413 - 19) = 576 - 413 - 19 = 144 ducks.
Therefore, the farmer has 144 ducks left.
To find the number of ducks the farmer had left, we need to follow a step-by-step reasoning process. Let's break down the problem into smaller parts and find the solution.
Let's assume the number of chickens is C and the number of ducks is D. We know that initially, the farmer had twice as many ducks as chickens, so we can write the equation D = 2C.
Now, let's calculate the number of ducks the farmer had left after selling 413 ducks and 19 more ducks died. Initially, he had D ducks, sold 413, and then 19 more died, so the remaining number of ducks is D - 413 - 19 = D - 432.
According to the problem, the farmer had half as many ducks as chickens left, so we can write another equation D - 432 = (1/2)C.
Now, we have a system of two equations:
1) D = 2C
2) D - 432 = (1/2)C
Let's solve this system of equations to find the values of C and D.
First, let's substitute the value of D from equation 1 into equation 2:
2C - 432 = (1/2)C
Next, let's simplify the equation by multiplying everything by 2 to eliminate fractions:
4C - 864 = C
Now, let's move all terms containing C to one side of the equation:
4C - C = 864
This simplifies to:
3C = 864
Finally, let's solve for C by dividing both sides of the equation by 3:
C = 864 / 3
C = 288
Now that we have the value of C (chickens), let's substitute it into equation 1 to find D (ducks):
D = 2C
D = 2 * 288
D = 576
Therefore, the farmer had 576 ducks left after selling 413 ducks and having 19 more ducks die.