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# A certain car dealership sells only Full-size and Mid-size c

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Joined: 29 Nov 2012
Posts: 802
A certain car dealership sells only Full-size and Mid-size c  [#permalink]

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12 Sep 2013, 03:26
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Difficulty:

35% (medium)

Question Stats:

77% (01:35) correct 23% (01:15) wrong based on 132 sessions

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A certain car dealership sells only Full-size and Mid-size cars. One of its sales representatives receives an annual salary of $15,000. He also receives a commission of$800 for each full size car he sells and $500 for each midsize car he sells. What is the least number of cars he must sell in a year to receive total annual earnings of exactly$25,000?

A) 7
B) 12
C) 13
D) 14
E) 17

I'm getting E can someone explain?

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Re: A certain car dealership sells only Full-size and Mid-size c  [#permalink]

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12 Sep 2013, 03:36
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fozzzy wrote:
A certain car dealership sells only Full-size and Mid-size cars. One of its sales representatives receives an annual salary of $15,000. He also receives a commission of$800 for each full size car he sells and $500 for each midsize car he sells. What is the least number of cars he must sell in a year to receive total annual earnings of exactly$25,000?

A) 7
B) 12
C) 13
D) 14
E) 17

I'm getting E can someone explain?

$$15,000+800F+500M=25,000$$ --> $$8F+5M=100$$.

We need to minimize F+M. To do that we need to minimize M (or which is the same we need to maximize F) --> $$M=4$$ and $$F=10$$ --> $$F+M=14$$.

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Re: A certain car dealership sells only Full-size and Mid-size c  [#permalink]

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30 Jun 2018, 19:47
10,000=800*12+400
10,000=800*11+1200
10,000=800*10+2000

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Joined: 22 May 2016
Posts: 1979
A certain car dealership sells only Full-size and Mid-size c  [#permalink]

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30 Jun 2018, 21:34
fozzzy wrote:
A certain car dealership sells only Full-size and Mid-size cars. One of its sales representatives receives an annual salary of $15,000. He also receives a commission of$800 for each full size car he sells and $500 for each midsize car he sells. What is the least number of cars he must sell in a year to receive total annual earnings of exactly$25,000?

A) 7
B) 12
C) 13
D) 14
E) 17

He gets $15,000 in salary, so to earn$25,000, he needs $10,000 in sales. Focus on the$800 cars. In order to minimize the total number of cars, we should maximize the number of more expensive full size cars, F -- we need fewer of them to get a high total dollar amount.

Another reason to focus in the $800 cars: divisibility.$800 and $500 are awkward together. We need the$800 car number to make a round total that is divisible by $500 for medium cars, M. He needs exactly$10,000 from commissions.

To be a "round" number divisible by $500,$800 must be multiplied by 5, 10, or 15

$800 * 5F =$4,000
$6,000 left$6,000/$500 per M = 12M Total cars: 17$800 * 10F = $8,000$2,000 left
$2,000/$500 per M = 4M
Total cars: 14

The next number of $800F that is round is 15, which is a greater number of cars than the 14 we just calculated. Reject. Least number of cars: 14 10 are F 4 are M Answer D _________________ In the depths of winter, I finally learned that within me there lay an invincible summer. -- Albert Camus, "Return to Tipasa" Target Test Prep Representative Status: Founder & CEO Affiliations: Target Test Prep Joined: 14 Oct 2015 Posts: 3528 Location: United States (CA) Re: A certain car dealership sells only Full-size and Mid-size c [#permalink] ### Show Tags 04 Jul 2018, 19:24 fozzzy wrote: A certain car dealership sells only Full-size and Mid-size cars. One of its sales representatives receives an annual salary of$15,000. He also receives a commission of $800 for each full size car he sells and$500 for each midsize car he sells. What is the least number of cars he must sell in a year to receive total annual earnings of exactly $25,000? A) 7 B) 12 C) 13 D) 14 E) 17 Since the sales rep makes$15,000 per year, he needs to make exactly $10,000 in commision. Let f = the number of full-size cars and m = the number of mid-size cars he sells. We have: 800f + 500m = 10,000 8f + 5m = 100 m = (100 - 8f)/5 Since we want the least number of cars he must sell to make the exact$10,000 commision, he must sell as many full-size cars as possible. We see that the maximum value of f can be 10 where (100 - 8f) is still a multiple of 5. Therefore, he sells 10 full-size cars and (100 - 8(10))/5 = 20/5 = 4 mid-size cars, for a total of 14 cars.

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Re: A certain car dealership sells only Full-size and Mid-size c &nbs [#permalink] 04 Jul 2018, 19:24
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