A farmer keeps hens and rabbits on his farm. One day, he counted a total of 70 heads and 196 legs. How many more hens than rabbit does he have? Explain your working please. Your help is greatly appreciated.
number of hens --- x
number of rabbits --- y
x + y = 70 , using heads info
2x + 4y = 196 , using legs info
or
x + 2y = 98
subtract the two equations ....
y = 28
then x = 42
so ... 42 hens and 28 rabbits.
Let's solve this problem step-by-step:
Step 1: Let's assume the number of hens is H and the number of rabbits is R.
Step 2: The number of heads can be calculated by adding the number of hens and rabbits: H + R = 70.
Step 3: The number of legs can be calculated by multiplying the number of hens by 2 (as each hen has 2 legs) and multiplying the number of rabbits by 4 (as each rabbit has 4 legs) and then adding them: 2H + 4R = 196.
Step 4: Now, we have a system of two equations:
H + R = 70 ---(1)
2H + 4R = 196 ---(2)
Step 5: We can use the method of substitution or elimination to solve this system of equations.
Using the method of substitution:
From equation (1), we can rearrange it to get H = 70 - R.
Substituting this value of H into equation (2):
2(70 - R) + 4R = 196
140 - 2R + 4R = 196
2R = 196 - 140
2R = 56
R = 56 / 2
R = 28
Step 6: We substitute this value back into equation (1) to find the value of H:
H + 28 = 70
H = 70 - 28
H = 42
Step 7: Now we can calculate the difference between the number of hens and rabbits:
Difference = H - R
Difference = 42 - 28
Difference = 14
Therefore, the farmer has 14 more hens than rabbits.
To solve this problem, we need to use a system of linear equations. Let's denote the number of hens as 'h' and the number of rabbits as 'r'.
We know that h + r = 70, as the total number of heads (hens + rabbits) is 70.
We also know that the total number of legs for hens and rabbits combined is 196. Since hens have 2 legs and rabbits have 4 legs, we can express this equation as:
2h + 4r = 196.
Now, we can solve the system of equations to find the values of 'h' and 'r'.
We have the equations:
h + r = 70 --> Equation 1
2h + 4r = 196 --> Equation 2
To make Equation 1 easier to work with, we can isolate 'h' in terms of 'r' by subtracting r from both sides of the equation:
h = 70 - r
Substituting this expression for 'h' into Equation 2, we get:
2(70 - r) + 4r = 196
Now, we can simplify and solve for 'r':
140 - 2r + 4r = 196
2r = 196 - 140
2r = 56
r = 56/2
r = 28
Once we have the value of 'r', we can substitute it back into Equation 1 to find 'h':
h + 28 = 70
h = 70 - 28
h = 42
So, the farmer has 42 hens and 28 rabbits on his farm.
To find out how many more hens than rabbits the farmer has, we can simply subtract the number of rabbits from the number of hens:
42 - 28 = 14
Therefore, the farmer has 14 more hens than rabbits.