The number of dogs and chickens on a farm add up to 14. The number of legs between them is 36. How many dogs and how many chickens are on the farm if there are at least twice as many chickens as dogs?
Well, it sounds like we've got ourselves a good old-fashioned farm math problem! Let's put on our thinking caps.
Let's call the number of dogs "x" and the number of chickens "y." According to the problem, we know that x + y = 14 (since the number of dogs and chickens on the farm adds up to 14).
Now, let's talk about the number of legs. Dogs have 4 legs, and chickens have 2 legs. So, the total number of legs can be calculated as 4x + 2y = 36.
We also know that there are at least twice as many chickens as dogs. This means y ≥ 2x.
Now, let the fun begin! Let's crack this mystery wide open, step by step.
First, let's rearrange the equation y ≥ 2x to solve for x. We get x ≤ (1/2)y.
Now, substitute this value of x into the equation x + y = 14, giving us (1/2)y + y = 14. Simplify that naughty equation to get (3/2)y = 14.
Next, multiply both sides by 2/3, which means (3/2) * (2/3)y = (14) * (2/3).
The 2/3 on the left cancels out the 3/2 on the right, leaving us with y = (14) * (2/3), which is y = 28/3. Now, we don't want any fractions on our farm, do we? So, let's simplify further.
Multiplying 14 by 2 gives us 28, and multiplying 3 by 3 gives us 9. So y = 28/3 is the same as y = 9 and a third.
But hold on a second! We can't have a third of a chicken running around, can we? So, let's see how many thirds we need to make 9 whole chickens.
Since there are 3 thirds in a whole, we can divide 9 by 3 to find out how many chickens that is. The answer is 3.
Now, let's go back to x + y = 14, and substitute 3 for y. This gives us x + 3 = 14. Subtract 3 from both sides to find x. We get x = 11.
So, after all that math-juggling, we've discovered that there are 11 dogs and 3 chickens on the farm. Now we know the whole farmyard crew!
Let's solve this problem step by step:
Step 1: Assign variables.
Let's assign "d" to represent the number of dogs and "c" to represent the number of chickens on the farm.
Step 2: Translate the problem into equations.
From the problem statement, we know that the sum of dogs and chickens is 14 and the total number of legs is 36. Also, we know that there are at least twice as many chickens as dogs. This can be represented by the equation c = 2d.
Step 3: Set up the equations.
Based on the information above, we can set up the following equations:
d + c = 14 (Equation 1)
4d + 2c = 36 (Equation 2)
(Note: We multiply the number of legs by the number of animals, as each dog has 4 legs and each chicken has 2 legs.)
Step 4: Solve the equations.
Let's solve the equations using the substitution method.
From Equation 1, we can express c in terms of d:
c = 14 - d
Substitute this value of c into Equation 2:
4d + 2(14 - d) = 36
4d + 28 - 2d = 36
2d + 28 = 36
2d = 8
d = 4
Substitute the value of d back into c = 14 - d:
c = 14 - 4
c = 10
Step 5: Answer the question.
According to the calculations, there are 4 dogs and 10 chickens on the farm that satisfy the given conditions.
To solve this problem, we can use a system of linear equations. Let's represent the number of dogs as "d" and the number of chickens as "c".
From the given information, we have two equations:
1) The number of dogs and chickens on the farm add up to 14:
d + c = 14
2) The number of legs between them is 36:
4d + 2c = 36 (since a dog has 4 legs and a chicken has 2)
Now, since we know that there are at least twice as many chickens as dogs, we can modify equation 1 as follows:
d + 2d = 14
3d = 14
d = 14 / 3
d ≈ 4.67
Since the number of dogs cannot be a decimal or fraction, we can conclude that there are no dogs on the farm. Therefore, the number of chickens can be found by substituting the value of d in equation 1:
c = 14 - d
c = 14 - 4.67
c ≈ 9.33
Again, since the number of chickens cannot be a decimal or fraction, we deduce that there are no chickens either. Therefore, there are no dogs or chickens on the farm that satisfy all the given conditions.
d+c=14
4d+2c=36
Multiply the top equation by -2.
-2d-2c=-28
4d+2c=36
2d=8
d=4
c=10