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Let’s say that you want to start a small side business to earn spending money. How do you make sure that you turn a profit?

Here’s a scenario. You are going to make hand-printed canvas bags that you will sew yourself and decorate using stamps you will carve from rubber blocks. For now, you will be selling the bags through word of mouth, so you won’t need a marketing budget. How do you figure out how much to charge per bag? There are a few things you need to know. How much do your materials cost per bag? How long does it take you to make each bag?

This is just one example of a situation in which you need to turn various pieces of real-world information into math to solve a problem. You will use similar skills when you are figuring out how to budget for your expenses on a salary. 

Before we find the answer, let’s go over some terminology that will allow us to turn words into mathematical operations. 

Per, every → division

5 miles per 3 hours → \( \frac{5  \, \mathrm{miles}}{3 \, \mathrm{hours}} \) 

$50 per month → \( \frac{$50}{1 \, \mathrm{month}} \)

Flat fee, base rate, one-time expense → addition

A gym membership costs $10 per month with a $15 sign up fee. → $15 is a flat fee, so you will add it to your overall expenses one time.

Recurring expense → multiply the expense by the number of times it needs to be paid

A gym membership costs $10 per month with a $15 sign up fee. → $10 is a recurring fee. To find out how much the membership would cost per year, for example, you would multiply $10 by 12 months to get $120, then add the $15 sign up fee to get $135.

Cost of multiple items → cost for each item times number of items

If shirts cost $20 each and you buy 4 of them, multiply $20 by 4 to find your total expense of $80.

More or less than → add or subtract

5 more than 3 = 3 + 5 = 8

7 less than 20 = 20 – 7 = 13

Difference → subtract

The difference between 30 and 5 = 30 – 5 = 25

The difference between 24 and -2 = 24 – (-2) = 26

Times → multiply

5 times 7 = 5 * 7

A fraction of → multiply the number you are starting with by that fraction

\( \frac{1}{3} \)of 5 yards = \( \frac{1}{3} * 5 \)  = \( \frac{5}{3} \) yards

Increase/decrease by an amount → add or subtract

45 increased by 7 = 45 + 7 = 52

66 decreased by 5 = 66 – 5 = 61

Increase or decrease by a factor of → multiply or divide

10 increased by a factor of 3 = 10 * 3 = 30

100 decreased by a factor of 2 = \( \frac{100}{2} \)  = 50

Percent of a number → percent written as a decimal (ex: 5% = \( \frac{5}{100} \) = 0.05) times the number you want to find the percentage of

25% of 150 = .25 * 150 = 37.5

2% of 150 =  .02 * 150 = 3

125% of 150 = 1.25 * 150 = 187.5

Profit → total money made – total expenses

If you sold a painting of yours for $100, but the materials to make it cost $30, then you made $100 – $30 = $70 in profit.