#### Quantitative and Physical History

# Basic Calculations Using Units

#### By Mark Ciotola

First published on February 27, 2019

## Basic Calculations

Most historical calculations will involve units. There are a few tricks that will working with units easier and more useful.

### Units Analysis

Physicists have a secret called “units analysis”. By assigning all quantities a unit, they can then check that the final answer results in the required unit. If it does not, then there has been a mistake. Also, a unit is required for physical units to be meaningful. A weight of “100” does not mean much, whereas a weight of 1 billion points is a bit more impactful.

Here is an example of units analysis:

Jean walks 10 miles. It takes Jean 5 hours to walk that distance. To find Jean’s mean speed, we divide distance by time:

speed = distance/time

speed = 10 miles / 5 hours

speed = 2 miles per hour.

Miles per hour is indeed a unit of speed, so the answer could be correct. Conversely, if the answer came out to be hours/mile the answer would clearly be incorrect.

### Converting and Canceling Units

Using units and calculating results will often involve converting one unit into another. An example will make this clear.

A farmer has an orchard with 10 trees. During the summer, each tree provides 2 bushels of apples per week, for 3 weeks. How many bushels of apples are produced by the orchard each summer?

bushels of apples/summer = (3 weeks/summer) x (2 bushels of apples/tree/week) x 10 trees

bushels of apples/summer = (3 ~~weeks~~/summer) x (2 bushels of apples/tree/~~week~~) x 10 trees

bushels of apples/summer = (3/summer) x (2 bushels of apples/~~tree~~) x 10 ~~trees~~

bushels of apples/summer = (3/summer) x (2 bushels of apples) x 10

Rearrange:

bushels of apples/summer = (3 x 2 x 10) (bushels of apples/summer)

Answer: 60 bushels of apples/summer

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