Question

Abbey Road Motors pays a percent commission to its sales people. They are paid

a percent of the profit the dealership makes on a car. If the profit is under $1,000,
the commission rate is 20%. If the profit is at least $1,000 and less than or equal to
$2,000, the commission rate is 20% of the first $1,000 and 24% of the remainder of
the profit. If the profit is above $2,000, the rate is 20% of the first $1,000 of profit,
24% of the next $1,000 of profit, and 29% of the amount of profit over $2,000. If p
represents the profit, express the commission c(p) algebraically as a split function.

Answer 1

c(p) = {0.2 p ⇒ p < 1000

0.24 p - 40 ⇒ 1000 ≤ p ≤ 2000

0.29 p - 140 ⇒ p > 2000}

Explanation:

* Lets explain how to solve the problem

- The profit is represented by p

1. If the profit is under $1,000, the commission rate is 20%

∵ The profit is p < 1000

∵ 20% of p = = 0.2 p

∵ c(p) is the function of the commission

∴ c(p) = 0.2 p when p < 1000

2. If the profit is at least $1,000 and less than or equal to $2,000, the

commission rate is 20% of the first $1,000 and 24% of the remainder

of the profit

- At least means greater than or equal

∵ The profit 1000 ≤ p ≤ 2000

- The commission is divided into 20% of first $1000 and 24% of

the reminder

∵ 20% of 1000 = = 200

∵ The remainder of the profit = p - 1000

∵ 24% of the remainder profit =

= 0.24(p - 1000) = 0.24 p - 240

∴ The total commission = 200 + 0.24 p - 240

∴ The total commission = 0.24 p - 40

∴ c(p) = 0.24 p - 40 when 1000 ≤ p ≤ 2000

3. If the profit is above $2,000, the rate is 20% of the first $1,000

of profit, 24% of the next $1,000 of profit, and 29% of the amount

of profit over $2,000

∵ The profit p > 2000

- The commission is divided into 20% of first $1000 and 24% of the

next $1,000 of profit, and 29% of the amount of profit over $2,000

∵ 20% of 1000 = = 200

∵ 24% of 1000 = = 240

- The amount of profit over $2,000 = p - 2000

∵ 29% of the amount of profit over $2,000 =

= 0.29(p - 2000)

= 0.29 p - 580

∴ The total commission = 200 + 240 + 0.29 p - 580

∴ The total commission = 0.29 p - 140

∴ c(p) = 0.29 p - 140 when p > 2000

* The commission function is:

c(p) = {0.2 p ⇒ p < 1000

0.24 p - 40 ⇒ 1000 ≤ p ≤ 2000

0.29 p - 140 ⇒ p > 2000}* Lets explain how to solve the problem

- The profit is represented by p

1. If the profit is under $1,000, the commission rate is 20%

∵ The profit is p < 1000

∵ 20% of p = = 0.2 p

∵ c(p) is the function of the commission

∴ c(p) = 0.2 p when p < 1000

2. If the profit is at least $1,000 and less than or equal to $2,000, the

commission rate is 20% of the first $1,000 and 24% of the remainder

of the profit

- At least means greater than or equal

∵ The profit 1000 ≤ p ≤ 2000

- The commission is divided into 20% of first $1000 and 24% of

the reminder

∵ 20% of 1000 = = 200

∵ The remainder of the profit = p - 1000

∵ 24% of the remainder profit =

= 0.24(p - 1000) = 0.24 p - 240

∴ The total commission = 200 + 0.24 p - 240

∴ The total commission = 0.24 p - 40

∴ c(p) = 0.24 p - 40 when 1000 ≤ p ≤ 2000

3. If the profit is above $2,000, the rate is 20% of the first $1,000

of profit, 24% of the next $1,000 of profit, and 29% of the amount

of profit over $2,000

∵ The profit p > 2000

- The commission is divided into 20% of first $1000 and 24% of the

next $1,000 of profit, and 29% of the amount of profit over $2,000

∵ 20% of 1000 = = 200

∵ 24% of 1000 = = 240

- The amount of profit over $2,000 = p - 2000

∵ 29% of the amount of profit over $2,000 =

= 0.29(p - 2000)

= 0.29 p - 580

∴ The total commission = 200 + 240 + 0.29 p - 580

∴ The total commission = 0.29 p - 140

∴ c(p) = 0.29 p - 140 when p > 2000

* The commission function is:

c(p) = {0.2 p ⇒ p < 1000

0.24 p - 40 ⇒ 1000 ≤ p ≤ 2000

0.29 p - 140 ⇒ p > 2000}