a messenger charges $10 to make a delivery to an address. in addition each letter delivered cost $3, and each package cost $8.if there are 15 more letters than

Question

a messenger charges $10 to make a delivery to an address. in addition each letter delivered cost $3, and each package cost $8.if there are 15 more letters than packages, what is the maximum number of items that can be delivered for $95?

Answer 1

We can immediately strike $10 from our pool of cash since that is the base cost of any delivery, leaving us with:

 

95 - 10 = 85                                                                                                                (1)

 

Next, we need to deal with the most important part of this problem--there are 15 more letters than packages. Mathematically, this can be written as:

 

L = P + 15                                                                                                                   (2)

 

Where "L" represents the number of letters, and "P" represents the number of packages. We also know that each letter costs $3 to deliver and each package costs $8 to deliver. Recalling that we have a total of 85$ that we can spend (see equation (1)), this gives the following:

 

3L + 8P = 85                                                                                                                (3)

 

Lastly, since it's very likely that this will not yield whole number answers, we need to make sure we choose an answer that will yield the most number of items delivered.

 

Now that the preliminaries are out of the way, we can move on to actually solving the problem. We have two equations each containing two unknown variables. This means we can solve the problem by either substitution or elimination, depending on your preference. I find substitution to be far easier in this case, so that is what I'll use here. We apply substitution by replacing "L" in equation (3) with our definition for "L" from equation (2). This gives:

 

3(P + 15) + 8P = 85

3P + 45 + 8P = 85

11P + 45 = 85

11P = 40

P = 40/11                                                                                                                     (4)

 

Here I've applied some simple algebra techniques: combining like terms and solving for a variable. We can then substitute this answer back into (2) to find L:

 

L = P + 15

L = 40/11 + 15

L = 205/11                                                                                                                    (5)

 

Now it should be clear that (4) and (5) don't provide our final answer since one cannot send fractional letters or packages! There's one extra step needed, and you'll need to keep the condition in equation (2) in mind! Since "P" is between 3 and 4, and "L" is between 18 and 19, we know that choosing L = 19 and P = 4 would give an answer that is larger than our allotted money. This means the answer must be L = 18 and P = 3. If this is unclear, think about it this way: if we choose any allowed combination with fewer letters than 18 and 3, let's say the combination L = 17 and P = 2 as an example, we would be within our allowed dollar amount but would have fewer total items delivered; if we chose anything over 18 letters, let's say the combination L = 19 and P = 4 as an example, we would be spending more money than we are allowed. Thus, L = 18 and P = 3 is the "sweet spot."

 

Our final answer:

 

The maximum number of items that can be delivered for $95 dollars is 21. You can deliver 18 letters and 3 packages and only spend $88. 

 

Note: There are several other ways to approach this problem, I just thought this would be the most descriptive way.