Creating Rational Expressions: average cost per drink, etc

[jlaw]

Okay, okay .....I think I got this one right (I've thought that before and was wrong) but no...I really, really, for real think I may have, maybe got this one right (A and B). Could someone be a life saver by checking and sharing your honest input? I can handle the critique if I'm wrong, just can't handle a failing grade. :wink:

Problem: To save the environment, the Lion's Den Cafeteria sells an empty reusable 24-ounce plastic mug to students for $2.99. Using the mug, a student can get a 24 ounce soft drink for a reduced price of 69 cents.

a. Write a rational expression for which the value of the expression is the average cost per drink and n is the number of times the mug is used.

2.99 + .69n= cost per drink
n

b. If a student buys the mug and uses it only once, then what is the cost for that one drink?
2.99 + .69n = cost per drink
n

2.99 + .69 (1)
1

3.68 = 3.68 is the cost for one drink
1

c. If the regular price of a 24 ounce soft drink is 89 cents, then when does the student start saving money?
This one I do not know how to answer unless from trial and error on a calculator, which turned out to be anywhere from
599-600 or > depending on how the numbers are rounded off. How would it be evaluated without the n (number) already in it?

2.99 + .89 (600) = 0.894
600

The cost of a drink would equal 89 cents when a student reached their 600th drink?
Thank you much for your valuable time and much appreciated shared knowledge.

 

[soroban]

Re: Creating Rational Expressions

Hello, jlaw!

Good job on the first two . . .

The cafeteria sells an empty reusable 24-oz. plastic mug to students for $2.99.
Using the mug, a student can get a 24-oz soft drink for a reduced price of 69 cents.

c. If the regular price of a 24-oz soft drink is 89 cents, when does the student start saving money?

\(\displaystyle \text{The regular price is 0.89 per serving.}\)
\(\displaystyle \text{With the new offer, the average price per serving is: }\frac{2.99+0.69n}{n}\)

\(\displaystyle \text{They are equal when: }\:0.89\:=\:\frac{2.99 + 0.69n}{n}\)

\(\displaystyle \text{So we have: }\;0.89n\:=\:2.99 + 0.69n \quad\Rightarrow\quad 0.2n \:=\:2.99 \quad\Rightarrow\quad n \:=\:14.95\)

Therefore, the students save money for \(\displaystyle n \,\geq\,15.\)
. . (They start saving money at their 14th refill.)

 

[jlaw]

Re: Creating Rational Expressions

I'm sorry, soroban I don't understand. 89 cents per drink multiplied by 15 drinks is $13.35 for fifteen drinks plus the cost of the original cup $2.99 which is altogether 16.34 totaled. Thus, a student has spent 16.34 for 15 drinks and a mug....doesn't that equal to 1.08933 per drink so far (when divided by number of drinks purchased 15)? Am I missing something?

Sincerely,
Julia

 

[wjm11]

I'm sorry, soroban I don't understand. 89 cents per drink multiplied by 15 drinks is $13.35 for fifteen drinks plus the cost of the original cup $2.99 which is altogether 16.34 totaled. Thus, a student has spent 16.34 for 15 drinks and a mug....doesn't that equal to 1.08933 per drink so far (when divided by number of drinks purchased 15)? Am I missing something?

Hi, Julia,

You’re mixing up the costs of the drink with and without the mug. Perhaps this will help:

Student with mug:

$2.99 + .69(15) = $13.34

Student without mug:

.89(15) = $13.35

So, for 15 drinks, the student with the mug spends less money.

 

[jlaw]

Thank you Wjm11 for clearing that up! It seems that the problem is left up to the reader's interpretation. That being the case I will certainly include all possible solutions that were presented here. Thanks so much, sincerely, everyone for your expertise!

Grateful,
Julia

 

[wjm11]

Please study Soroban's method and steps very carefully. That is exactly what you need to understand and be able to repeat.