points A farmhouse shelters 10 animals. Some are goats and some are ducks. Altogether there are 36 legs. How many ducks and goats are there?
The Correct Answer and Explanation is:
To solve this problem, let’s define:
- Let x be the number of goats.
- Let y be the number of ducks.
We know two things:
- The total number of animals is 10:
x + y = 10 - Goats have 4 legs and ducks have 2 legs. The total number of legs is 36:
4x + 2y = 36
We can solve this system of equations step by step.
Step 1: Solve the first equation for y
From x + y = 10, we get:
y = 10 – x
Step 2: Substitute into the second equation
Now plug y = 10 – x into the leg equation:
4x + 2(10 – x) = 36
Simplify:
4x + 20 – 2x = 36
2x + 20 = 36
2x = 16
x = 8
Step 3: Find y
Since x = 8, then y = 10 – 8 = 2
Final Answer:
- Goats: 8
- Ducks: 2
Explanation:
This problem is a classic example of solving a system of linear equations. The scenario involves a farmhouse with 10 animals—some goats and some ducks—and a total of 36 legs. Each duck has 2 legs, and each goat has 4 legs. We aim to determine how many of each animal are present.
We begin by letting variables represent the unknowns: x for goats and y for ducks. The first key detail tells us that there are 10 animals in total, giving us the equation x + y = 10. The second detail refers to the number of legs. Since goats have 4 legs and ducks have 2, we express the total number of legs as 4x + 2y = 36.
Solving such problems often involves substitution or elimination. We isolate y in the first equation to get y = 10 – x. Then we substitute this expression into the second equation. This transforms the problem into a single-variable equation: 4x + 2(10 – x) = 36. Distributing and combining like terms simplifies the equation to 2x + 20 = 36. Subtracting 20 from both sides gives 2x = 16, and dividing by 2 yields x = 8.
We then substitute x back into the first equation to find y. Since x = 8, y = 10 – 8 = 2. So, there are 8 goats and 2 ducks.
We can double-check our solution: 8 goats have 32 legs, and 2 ducks have 4 legs, totaling 36 legs. This confirms our answer is accurate.
