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## Ice Cream Quandary

Math brain teasers require computations to solve.

 Puzzle ID: #26432 Fun: (2.69) Difficulty: (2.22) Category: Math Submitted By: signmeup74 Corrected By: signmeup74

Zee's Ice Cream Shop is world renowned for making new and inventive ice cream flavors from scratch. They've made great profits on their Marshmallow Mustang and Cinnebago flavors, but their Purple Pinto is just not turning a profit!
Scott, Zee's President, says that customers just don't want to eat purple ice cream. He thinks they should pull the flavor and move on. However Sally, Zee's Marketing Director, disagrees. She has taste tested the flavor with a sampling poll and it got rave reviews.
Daisy, the lowly Flavor Engineer, loves the Purple Pinto and knows that it has profit potential! She thinks the problem might be stemming from the small batch sizes and the fact that Sally takes scoops for sample polls every time they run a batch of the flavor.
Daisy has collected the following information:
1. Currently, there are 30 scoops in a batch.
2. It costs Zee's \$12.90 to make each batch, right now.
3. A scoop sells for \$0.90.
4. A small profit is not enough! Scott wants \$25 per batch to make up for lost time.
5. Sally will not stop taking 17 sample scoops per batch.
First of all, is Daisy right? Or is the Purple Pinto just doomed?
If she is, can you help Daisy determine how large batches should be to ensure that Zee's makes a profit and keeps making her favorite flavor?

Daisy is right.
She should up batch size to 86 scoops to give Scott his \$25/batch profit.

Cost to Zee's = \$12.90/batch or \$0.43/scoop
Cost to Customer = \$0.90/scoop (WOW! That's more than a 100% mark-up)
BUT:
17 samples cannot be sold.
So a batch is now only 13 scoops
Cost to Customer = \$11.70/batch
Zee's loses \$1.20 per batch.
So, Daisy is right!

Now, to let's help her convince management to increase the batch size.
To break even: Our cost to make the scoops < Sales dollars from scoops
To make a \$25 profit: Our cost to make the scoops + \$25 profit < Sales dollars from scoops

Let's make s=the optimal number of scoops in a batch

Our cost to make the scoops is:
\$0.43 * s + \$0.43 * 17 (because we gave 17 away to marketing)

Sales from the scoops are:
\$0.90 *s

Therefore:
\$0.43 * s+ \$0.43*17 + \$25 < \$0.90 * s

0.43s+32.31<0.90s
0.47s>32.31
s > 68.75 scoops to sell

Adding in the samples Daisy gave away to Sally:
s = 69 + 17 = 86 scoops/batch

Daisy needs to up the batch size to 86 scoops to ensure a \$25 profit per batch.
And, most importantly, she can use her math to convince Sally and Scott to keep making the Purple Pinto!

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