Calculating Income vs. Sales

Date: 3/26/96 at 0:15:23
From: Anonymous
Subject: problem solving

  Dr Math:

  Can you help?

A factory owner knows that some of her expenses are going to occur
regardless of how many items she makes.  These are fixed costs and total 
$3000 per week.  This means that if she makes no sales she still has to 
pay the $3000 per week.  The remaining expenses are variable costs.  The 
most obvious variable cost is the stock she buys, but there are others 
such as labour.  She has found the variable costs are almost directly 
related to sales.  In a week in which she made 500 items, variable costs 
were $7500. The items are all made to order and so are all sold.  They 
sell at $20 each.

1. Draw up a table to show some values for income to the shop, i, for
different numbers of items sold, n.

number of items sold n    0    100     500    1000    15000

income in dollars    

2. draw up a table for the total cost for different numbers of items 
sold.  These are the fixed costs added to the variable costs.  Remember 
that the variable costs are directly related to the number sold.

number of items sold     0    100       500     1000   1500

expenses in dollars

3. The difference between expenses and income is the factory owner's 
profit.  If she made no sales she would still have to pay the fixed 
costs, so she would have a very large loss.  If she made many sales, she 
would have easily covered her fixed costs and made a good profit.
In a week in which she sold 500 items did she make a profit or a loss?

In a week in which she sold 1000 items did she make a profit or a loss?

On the same pair of axis draw the two graphs to show income related to 
the number of items sold and expenses related to the number of items 
sold.  Use a large scale as you will need to read values from your 
graph.

From the graph, estimate the number of items she needs to sell each week 
to break even.  Calculate the expenses and income for that number of 
items to check your estimate.

Use your graph to estimate the number of items she needs to sell each 
week to make a profit of $1000 per week.  Calculate the expenses and 
income for that number of items to check your estimate.

Write an equation for income which shows how the number of dollars 
taken each week, m, is related to the number of items sold, n.

Write another equation for expenses which shows how the number of 
dollars spent each week, m, is related to the number of items sold, n. 

I know it's a long problem, but my parents can't help me.  Thanks. 

Date: 3/28/96 at 19:13:20
From: Doctor Jodi
Subject: Re: problem solving

Hi Marta!  Let me go through and give you some hints to get you started. 
Please write us back if you have specific questions or need more help. 
Okay?

>1. draw up a table to show some values for income to the shop i, for
>different numbers of items sold,n.
>
>number of items sold n    0    100     500    1000    15000

If everything sells for $20, then to get the income, multiply n* 20.  
For example, above, where there are 500 items, the income is 

500 * 20 = 10000

Does this make sense?

>income in dollars    
>
>2. draw up a table for the total cost for different numbers of items 
>sold. These are the fixed costs added to the variable costs. remember 
>that the variable costs are directly related to the number sold.
>
>number of items sold     0    100       500     1000   1500

If above, the variable costs are 7500 and there are 500 items, the 
variable costs are $7500/500 items.  How much is that per item?  Now can 
you figure out what the total variable costs are, depending on the items 
sold?

>expenses in dollars
>
>3. the difference between expenses and income is the factory owner's 
>profit.  If she made no sales, she would still have to pay the fixed 
>costs, so she would make a very large loss.  If she made many sales she 
>would have easily covered her fixed costs and made a good profit.
>In a week in which she sold 500 items did she make a profit or a loss?

To find the profit/loss, here is what you do.

Take income and subtract fixed cost ($3000/week) and variable cost 
(depends on number of items sold).

Does this make sense?  If you get a positive number, that is the profit. 
If it's negative profit, it's also called loss.

>in a week in which she sold 1000 items  Did she make a profit or a 
>loss?
>
>On the same pair of axis draw the two graphs to show income related to 
>the number of items sold and expenses related to the number of items 
>sold.  Use a large scale as you will need to read values from your 
>graph.

You need $ on one axis and items sold on the other axis.

Now you can plot two different sets of points:

One is items sold versus variable expenses.
For example, one point would be (500, 10500) since 7500 + 3000 = 10500

These are the numbers in your first table with 3000 added to each of the 
costs.

The other is items sold versus income - 
for example, one point would be (500, 10000)

Basically, these are just the numbers in your second table.

>from the graph estimate the number of items she needs to sell each week 
>to break even.  Calculate the expenses and income for that number of 
>items to check your estimate.

Where do incomes equal expenses?  This is the "break-even point."

I hope this helps.  Please write us back with other questions.  Good 
luck!

-Doctor Jodi,  The Math Forum