*Image by Shopify Partners*

Essentially everyone will have to turn words into math on some level while navigating the responsibilities of adult life. Creating and using mathematical models (through writing equations, expressions, tables, statements, etc.) isn’t just for highly-advanced programmers and scientists at Google or NASA. A model is really just a tool you can use to make predictions and decisions. The ability to turn words into math allows individuals and businesses to plan their finances, maximize their profits, solve interesting problems, and make well-informed decisions. This skill comes down to figuring out how one piece of information relates to another.

*Image by Matthew Henry*

Back to our canvas bags in lesson 1 — let’s work that problem through.

You look through the receipts for your supplies and write down the following list of what you have spent so far:

- 10 yards of canvas = $9 per yard

- 20 yards of canvas strapping = $9 for 5 yards

- 1, 100-yard spool of thread = $1

- 3 rubber blocks to carve = $1.75 each

- basic carving/printing tools = $25

- 6 containers of ink = $23

Therefore you have spent $180.25 total

You already have a sewing machine, but if you didn’t, that would be another expense. You have made a prototype of your bag and written down the amount of material you used: each bag uses \( \frac{2}{3} \) of a yard of fabric, a pair of straps (each strap is 2 feet long), \( \frac{1}{30} \) of a spool of thread, and \( \frac{1}{10} \) of a container of ink. You are using only 1 rubber block for the pattern on your first group of bags (once you’ve carved your rubber stamp, you can keep reusing it). Note: there are 3 feet in 1 yard.

**1. How many bags can you make with these materials?**

- 10 yards of canvas → \( \frac{2}{3} \) of a yard per bag

\( \frac{10 \, \mathrm{yds}}{\frac{2}{3} \, \mathrm{yds \, per \, bag}} \) = 15 bags

- 20 yards of canvas strapping → 2 feet per strap * 2 straps = 4 feet (3 feet per 1 yard)

\( 4 \, \mathrm{ft} * \frac{1 \, \mathrm{yd}}{3 \, \mathrm{ft}} = \frac{4}{3} \) yds per bag

\( \frac{20 \, \mathrm{yds}}{\frac{4}{3} \, \mathrm{yds \, per \, bag}} \) = 15 bags

- 1, 100-yard spool of thread → \( \frac{1}{30} \) of a spool per bag

\( \frac{1 \, \mathrm{spool}}{\frac{1}{30} \, \mathrm{spool \, per \, bag}} \) = 30 bags

- 6 containers of ink → \( \frac{1}{10} \) of a container per bag

\( \frac{6 \, \mathrm{containers}}{\frac{1}{10} \, \mathrm{ containers \, per \, bag}} \) = 60 bags

You therefore have canvas for 15 bags, straps for 15 bags, thread for 30 bags, and ink for 60 bags. **The limiting materials are the canvas and strapping: you can make 15 bags before you need to buy more of both.**

**2. After doing some research, you find that similar handmade bags to yours cost around $20 each. You therefore want to see if you will make any money if you charge that much.**

To do this, figure out roughly how much each bag will cost you and subtract that from $20 to find your estimated profit. You can do that most precisely with weighted averages, which we will cover in a future lesson. To get a ballpark amount for now, we will calculate how much it will cost you to make 1 bag with your materials.

Subtracting out your one-time expenses from the list of materials, you see that you spend $180.25 – $25 – $5.25 = $150 on the canvas, strapping, and ink to make 15 bags. Dividing this amount by 15 shows you that you will spend roughly $10 per bag. If you subtract $10 from the $20 you will charge per bag, you make around $10 per bag.

**3. You have practiced making the bag a few times, and it takes you about 1.5 hours to construct one. Using $10 as the rough profit per bag, how much will you make per hour on the bags?**

If you make roughly $10 per bag, and it takes 1.5 hours to make each bag, then \( \frac{$10}{1 \, \mathrm{bag}} * \frac{1 \, \mathrm{bag}}{1.5 \, \mathrm{hrs}} = \frac{$10}{1.5 \, \mathrm{hours}} = $6.67 \, \mathrm{per \, hr.} \)

**4. Using $10 as the rough cost per bag, approximately how many bags do you need to make before you earn back the money you spent on your initial list of supplies?**

If you spent $180.25 initially and make about a $10 profit on each bag, then you will recoup your initial investment on your 19th bag sold. $10 per bag x 19 bags = $190; 18 bags would be too few because you paid a bit more than \( $10 * 18 = $180 \) for your initial supplies.

Feel like you’ve got the hang of it? Try our 5-question Words to Math quiz below.