Question

Ryan has his own business selling watches, and he wants to monitor his profit per watch sold. The expression
`(200x-300)/(x)` models the average profit per watch sold, where x is the number of watches sold.

Part A: What does the numerator of this rational expression represent?

Part B: What does the denominator of this rational expression represent?

Part C: Rewrite the expression
`(200x-300)/(x)` as a sum of two fractions, and simplify.

Part D: What does each part in the expression from part C represent?

Part E: Ryan also earns money by selling colored watch bands with the watches he sells. The linear expression 100x – 50 models Ryan’s additional income. What is the expression that models the new average profit, including the bands? Note: To obtain the new average profit expression, add the linear expression to the original rational expression. Write the new profit expression as one fraction.

Answer 1

Explanation:

Given that Ryan has his own business selling watches, and he wants to monitor his profit per watch sold. The expression \frac{200x-300}{x} models the average profit per watch sold, where x is the number of watches sold.

Numerator = total profit for all x watches

Denominator = No of watches sold

C)
`(200x-300)/(x) =200-(300)/(x)`

D) 200 is the fixed profit irrespective of no of watches sold but second expression is 300/watches sold thus increases when x increases.

E)
`Additional Income = 100x-50`

New profit =
`(200x-300)/(x) +100x-50\\=(100x^2+150x-300)/(x)`

Answer 2

Part A: What does the numerator of this rational expression represent?

ANSWER:

Numerator (200x-300) gives profit when x watches are sold.

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Part B: What does the denominator of this rational expression represent?

ANSWER:

Denominator (x) represents the number of watches sold.

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Part C: Rewrite the expression
`(200x-300)/(x)`

as a sum of two fractions, and simplify.

ANSWER:

`(200x-300)/(x)`

`=(200x)/(x)-(300)/(x)`

`=200-(300)/(x)`

Hence simplified fraction form is
`200-(300)/(x)`

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Part D: What does each part in the expression from part C represent?

ANSWER:

First part is (200) which means maximum average prfit can be 200.

Second part
`-(300)/(x)`

is negative and number of watches (x) is in denominator so as the number of sold watches increases, then
`(300)/(x)`

decreases and due to negative sign, decrease in average profit value becomes less.

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Part E: Ryan also earns money by selling colored watch bands with the watches he sells. The linear expression 100x – 50 models Ryan’s additional income. What is the expression that models the new average profit, including the bands? Note: To obtain the new average profit expression, add the linear expression to the original rational expression. Write the new profit expression as one fraction.

ANSWER:

We just need to add both profit expressions:

`(200x-300)/(x)+100x – 50`

`=(200x-300)/(x)+\left(100x-50\right)\cdot(x)/(x)`

`=(200x-300)/(x)+(\left(100x^2-50x\right))/(x)`

`=(200x-300+100x^2-50x)/(x)`

`=(100x^2+150x-300)/(x)`

Hence final profit expression is
`(100x^2+150x-300)/(x)`