Everyday Chemistry

Dimensional Analysis and Money

Introduction

Every measurement or value you come across has an associated, important unit that describes what the measurement describes. If you’re at the store with your friend and she asks you to lend her fifty, it makes a difference to your wallet whether she’s asking for 50 cents or 50 dollars. 20 ounces of water is a refreshing drink, but 20 gallons of water is enough for a long shower. Some measurements are given in derived units, like meters per second or kilowatt-hours, which are combinations of more than one unit.

Units determine how we can compare and combine measurements. Is 1 kilometer more or less than 2 pounds? Even though 1 is less than 2, the question can’t be answered because kilometers are a measurement of length while pounds are a measurement of weight. It’s important to use consistent units to avoid making comparisons that don’t make sense.

Sometimes, you’ll want to convert a measurement given in one unit to another unit. Some conversions are easy – like “to convert feet to inches, divide by 12” – but how do you handle more complicated conversions, or ones where we don’t know the direct conversion factor? We use the skill of dimensional analysis to break difficult conversion problems into easy ones.

Examples

Your friend tells you that he is 1.88 meters tall. You find out from your math book that 1 meter = 39.4 inches. How tall is he in inches?

We can treat the conversion factor as an equation. From the rules of algebra,

Remember that anything multiplied by 1 is itself. This means we can multiply the value we want to convert by the conversion factor fraction, and cancel out units that appear both above and below the fraction bar. Units can be multiplied and divided like numbers can.

We can arrange these fractions in a table to make it easier to read:

1.88 meters / 39.4 in / = 74.1 inches
1 meter

This is an appropriate answer because the answer’s unit is the unit we wanted.

You get a part-time job at the supermarket as a cashier. Your starting salary is $8.00/hour, and you predict that you will work 3 days a week, for 4 hours a day. If you want to make $2,000, how many weeks should you work?

In this problem, we want to convert $2,000 into weeks. Start by setting up the table:

$2,000 / = ? weeks

From the problem, we see that 1 week = 3 days of work, 1 day of work = 4 hours of work, and 1 hour of work = $8. Add these to the table, arranging the fractions to make sure that the units will cancel out:

$2,000 / 1 hour / 1 day / 1 week / = ? weeks
$8 / 4 hours / 3 days

Now we simplify by canceling and multiplying:

$2,000 / 1 hour / 1 day / 1 week / = 20.83 weeks
$8 / 4 hours / 3 days

CO-OP ACTIVITY

Objective

Research a career of your choice, and use dimensional analysis to answer financial questions

Roles

  1. Chief financial officer(coordinator) – keeps the group on task and makes sure all members get a chance to participate and be heard
  2. Financial analyst (researcher) – responsible for using the Internet to answer research questions
  3. Accountant (recorder) – keeps track of calculations and results
  4. Manager (liaison) – the only member allowed to communicate with other groups or ask for help

Sounds like / Looks like
Let’s get on task. / Smile
Let’s do that later. / Nod
Come on! / Hand gestures to get attention
Let’s get going. / Beckoning
We need you. / Look at group members
Let’s get with this. / Eye contact
We can talk about that at lunch. / Tap on shoulder to get attention
We’re running out of time. / Point to work

Materials

  • Computer with internet access
  • Calculator
  • This worksheet

Procedure

  1. Using the internet, look at the Occupational Outlook Handbook (
  2. Look for an occupation whose median earnings are reported as an annual salary.
  3. Record which occupation your group chose.
  4. Record the median annual wage listed in the Handbook.
  5. Use dimensional analysis to answer these questions about the job you chose for question 2. Be sure to show your work.
  6. If you worked at this job and made the median annual wage, how much money would you make each month?
  7. Assume there are 26 pay periods per year and you work 10 days per pay period. How much would you make per work day?
  8. Choose a model of car and look up its price on the Internet. How many days of work is this car worth?
  9. Using the converter at find how many Japanese yen (JPY, ¥) one U.S. dollar is worth. At the current exchange rate, how many days of work would it take you to make one million yen?
  10. Now, look in the Handbook for a job whose median earnings are reported as an hourly wage.
  11. Record which occupation your group chose.
  12. Record the median hourly wage of this occupation.
  13. Use dimensional analysis to answer these questions about the job you chose for question 4. Be sure to show your work.
  14. If you worked at this job and made the median hourly wage, assuming you worked 8 hours a day, 5 days a week, how much money would you make in a year?
  15. Look up the price of another model of car. Assuming you work eight hours a day, how many days of work at this job is this car worth?
  16. Using look up how many Brazilian reales (BRL, R$) one U.S. dollar is worth. At the current exchange rate, assuming you work 8 hours a day, how many days would it take you to make R$100,000?
  17. Check your answers. Be sure that the unit of your answers all make sense.