Question

A repairman purchased several furnace-blower motors for a total cost of $450. If his cost per motor had been $5 less, he could have purchased 1 additional motor. How many motors did he buy at the regular rate?

Answer 1

The repairman bought 9 motors at the regular rate, which was found by setting up an algebraic equation based on the given conditions and solving for the unknown number of motors.

Step-by-step explanation:

The question involves solving a word problem to find out the number of motors a repairman bought at the regular rate.

Let's let x represent the number of motors the repairman bought. Then the cost per motor would be $450/x. According to the problem, if the cost per motor had been $5 less, i.e., $(450/x - 5), he could have bought one additional motor, which would be x + 1 motors. Therefore, we can set up the following equation to represent the situation:

450/x = 450/(x + 1) + 5

Now we solve for x:

1. Multiply both sides by x(x + 1) to get rid of the denominators: 450(x + 1) = 450x + 5x(x + 1).
2. Simplify the equation: 450x + 450 = 450x + 5x² + 5x.
3. Cancel out the 450x from both sides and solve for x: 450 = 5x² + 5x.
4. Divide through by 5: 90 = x² + x.
5. Rewrite the equation as x² + x - 90 = 0 and solve for x using the quadratic formula or factoring.
6. By factoring, we find (x - 9)(x + 10) = 0, which gives us x = 9 or x = -10. Since the number of motors cannot be negative, the number of motors he bought is 9.

Therefore, the repairman bought 9 motors at the regular rate.

Answer 2

Given:

A repairman purchased several furnace-blower motors for a total cost of $450.

Let the number of motors = x

And the cost of one motor = y

So,

`xy=450\rightarrow(1)`

If his cost per motor had been $5 less, he could have purchased 1 additional motor

So,

`(x+1)(y-5)=450\rightarrow(2)`

Divide equation (2) by equation (1)

`\begin{gathered} ((x+1)(y-5))/(xy)=1 \\ (x+1)(y-5)=xy \\ xy-5x+y-5=xy \\ -5x+y-5=0 \\ y=5x+5\rightarrow(3) \end{gathered}`

substitute with (y) from equation (3) into equation (1) then solve for x

`\begin{gathered} x(5x+5)=450 \\ 5x^2+5x=450\rightarrow(/5) \\ x^2+x=90 \\ x^2+x-90=0 \\ (x+10)(x-9)=0 \\ x+10=0\rightarrow x=-10 \\ x-9=0\rightarrow x=9 \end{gathered}`

The negative result will be rejected

So, x = 9

So, the answer will be:

The number of motors at the regular rate = 9 motors.

And the regular price = $50