7 Formulas

You may use a calculator throughout this module if needed.

A baby drinking formula from a bottle
Photo by Rainier Ridao on Unsplash.

A formula is an equation or set of calculations that takes a number (or numbers) as input, and produces an output. The output is often a number, but it could also be a decision such as yes or no.

Each unknown number in a formula is called a variable because its value can vary. A variable is usually represented with a letter of the alphabet. To evaluate a formula, we substitute a number (or numbers) into the formula and then perform the steps using the order of operations.

Formulas with One Input

Many formulas will have just one input variable. Note: When a number is written directly next to a variable, it indicates multiplication. For example, 0.24w means 0.24\cdot{w}.

Exercises

The formula C=0.24w+1.26 gives the cost, in dollars, of mailing a large envelope weighing w ounces through the USPS.[1]

  1. Find the cost of mailing a 6-ounce envelope.
  2. Find the cost of mailing a 12-ounce envelope.

Radio Cab charges the following rates for a taxi ride: a fixed fee of \textdollar3.80 to get in the taxi, plus a rate of \textdollar2.80 per mile.[2] The total cost, in dollars, of a ride m miles long can be represented by the formula C=3.80+2.80m.

  1. Find the cost of a 5-mile ride.
  2. Find the cost of a 7.5-mile ride.
  3. Find the cost of getting in the taxi, then changing your mind and getting out without riding anywhere.

The number of members a state has in the U.S. House of Representatives can be approximated by the formula R=P\div{0.76}, where P is the population in millions.[3] The 2020 populations of three states are as follows:[4]

Oregon 4.2\text{ million}
Washington 7.7\text{ million}
California 39.6\text{ million}

Round all answers to the nearest whole number.

  1. How many U.S. Representatives does Oregon have?
  2. How many U.S. Representatives does Washington have?
  3. How many U.S. Representatives does California have?

The number of electoral votes a state has can be approximated by the formula E=P\div{0.76}+2, where P is the population in millions.

  1. How many electoral votes does Oregon have?
  2. How many electoral votes does Washington have?
  3. How many electoral votes does California have?

Temperature Conversions

You probably know that 32° Fahrenheit is the freezing point of water, and 212° Fahrenheit is the boiling point of water. The Celsius equivalents are 0°C and 100°C. The formulas shown below allow us to convert between the two temperature scales.

Temperature Formulas

F=\frac{9}{5}C+32

F=1.8C+32

C=\frac{5}{9}(F-32)

C=(F-32)\div{1.8}

Exercises

  1. The record high temperature in Portland, Oregon occurred during the “heat dome” event in June, 2021. As shown, it was 46°C at your author’s house and the squirrels were laying low. Convert this temperature to Fahrenheit.
Based on the evidence, the melting point of squirrel is right around 46°C.
  1. The traditionally accepted “normal” body temperature of a human[5] is 98.6°F. What is this temperature in Celsius?
  2. The FDA recommends that a freezer be set below -18°C. What is the Fahrenheit equivalent?
  3. A package of frozen pancakes from IKEA calls for the oven to be set to 392°F. IKEA is based in Sweden, and this temperature clearly was originally calculated in Celsius. What is the corresponding Celsius temperature?

Formulas with More than One Input

Some formulas have more than one input variable. Just pay attention to which number goes in for each variable.

Exercises

When a patient’s blood pressure is checked, they are usually told two numbers: the systolic blood pressure (SBP) and the diastolic blood pressure (DBP). The mean arterial pressure (MAP) can be estimated by the following formula: MAP=\frac{SBP+2\cdot{DBP}}{3}. (The units are mm Hg.) Calculate the mean arterial pressure for each patient.

  1. SBP = 120, DBP = 75
  2. SBP = 140, DBP = 90

a three-dimensional rectangular box with the bottom front edge labeled l, the bottom right edge labeled w, and the vertical right edge labeled hUPS uses the following formula[6] to determine the “measurement” of a package with length l, width w, and height h: m=l+2w+2h. Determine the measurement of a package with the following dimensions.

  1. length 18 inches, width 12 inches, height 14 inches
  2. length 16 inches, width 14 inches, height 15 inches

Formulas with a Yes or No Answer

Some formulas give a yes or no answer: success or failure, approved or disapproved, etc. After calculating the result from the formula, we need to compare it to a given number to see whether the result is within a specified range.

Exercises

In Australia, a chicken egg is designated “large” if its mass, in grams, satisfies the following formula: |m-54.1|\leq4.1. Determine whether each egg qualifies as large.[7]

  1. Egg 1’s mass is 57.8 grams.
  2. Egg 2’s mass is 58.3 grams.
  3. Egg 3’s mass is 49.8 grams.
  4. Egg 4’s mass is 50.0 grams.

Exercise Answers

  1. Source: https://pe.usps.com/text/dmm300/Notice123.htm#_c037 ↵
  2. Source: https://www.radiocab.net/services-radio-cab/ ↵
  3. The value 0.76 comes from dividing the total U.S. population in 2020, around 331 million people, by the 435 seats in the House of Representatives. ↵
  4. Source: https://www.census.gov/data/tables/2020/dec/2020-apportionment-data.html ↵
  5. See this article for a counterargument: https://www.nytimes.com/2023/10/12/well/live/fever-normal-body-temperature.html ↵
  6. Source: https://www.ups.com/us/en/help-center/packaging-and-supplies/prepare-overize.page ↵
  7. Source: https://en.wikipedia.org/wiki/Chicken_egg_sizes ↵
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Technical Mathematics, 2nd Edition Copyright © 2024 by Morgan Chase is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.