Mr. Abernathy purchased a selection of wrenches for his shop. The total cost was $78. He bought the same number of $1.50 and $2.50 wrenches, and half that many $4 wrenches. The number of $3 wrenches was one more than the number of $4 wrenches. How many of each did he purchase?
10, $1.50 wrenches
10, 2.50 wrenches
5, $4.00 wrenches
6, 3.00 wrenches
I got the same as emily
To solve this problem, we can use a system of equations. Let's define the variables:
Let x be the number of $1.50 and $2.50 wrenches.
Let y be the number of $4 wrenches.
Let z be the number of $3 wrenches.
From the information provided, we can write these equations:
1. The total cost of the wrenches is $78:
(1.50x + 2.50x) + (4y) + (3z) = 78
2. He bought the same number of $1.50 and $2.50 wrenches:
x = x
3. He bought half as many $4 wrenches as the $1.50 and $2.50 wrenches:
y = 0.5x
4. The number of $3 wrenches was one more than the number of $4 wrenches:
z = y + 1
We now have a system of equations that we can solve simultaneously to find the values of x, y, and z.
Let's substitute the value of y from equation 3 into equations 4 and 1:
Substituting y = 0.5x into equation 4:
z = 0.5x + 1
Substituting y = 0.5x into equation 1:
(1.50x + 2.50x) + (4 * 0.5x) + (3z) = 78
Simplifying equation 1:
(4x) + (2x) + (2x) + (3z) = 78
8x + 3z = 78
Now, substitute the value of z from equation 4 into equation 2:
Substituting z = 0.5x + 1 into equation 2:
x = x
We can now solve this system of equations using the substitution method or elimination method.
Let's solve it using the substitution method:
From equation 2, we know that x = x, so we can substitute this into equation 4:
z = 0.5x + 1
Substituting x = x into equation 4:
z = 0.5(x) + 1
Simplifying equation 4:
z = 0.5x + 1
Now, substitute the values of x and z into equation 1:
(1.50x + 2.50x) + (4y) + (3z) = 78
Substituting x for x and z for z:
(1.50(x) + 2.50(x)) + (4y) + (3(0.5x + 1)) = 78
Simplifying equation 1:
(4x) + (2x) + (2x) + (1.5x) + (3) = 78
9.5x + 3 = 78
Subtracting 3 from both sides:
9.5x = 75
Dividing both sides by 9.5:
x = 7.89
Since we can't have a fraction of a wrench, we'll round x to the nearest whole number:
x = 8
Now, let's use the values of x and y to find the value of z using equation 4:
z = 0.5x + 1
z = 0.5(8) + 1
z = 4 + 1
z = 5
Finally, we can find the value of y using equation 3:
y = 0.5x
y = 0.5(8)
y = 4
Therefore, Mr. Abernathy purchased 8 $1.50 and $2.50 wrenches, 4 $4 wrenches, and 5 $3 wrenches.