Formulas and word problems
Solve F = (9/5)C + 32 for C and you get C = (5/9)(F − 32), which turns 212 °F into 100 °C.
A formula is an equation with several letters
A formula is an equation that links several quantities, such as d = rt for distance, rate and time. You can solve a formula for any one of its letters, treating the others as if they were numbers. To solve d = rt for t, notice that t is multiplied by r, and undo that by dividing both sides by r: t = d/r. Now the formula answers a different question. If a train covers 360 kilometres at 120 kilometres per hour, t = 360/120 = 3 hours. OpenStax gives a whole section to this skill, because it saves rearranging the same formula over and over.
The same steps work with more letters. The perimeter of a rectangle is P = 2L + 2W. To solve for W, subtract 2L from both sides to get P − 2L = 2W, then divide by 2: W = (P − 2L)/2. A rectangle with perimeter 30 metres and length 9 metres therefore has width (30 − 18)/2 = 6 metres. The temperature formula from OpenStax, F = (9/5)C + 32, converts Celsius to Fahrenheit. Its slope 9/5 says that Fahrenheit rises 9 degrees for every 5 Celsius degrees. Subtract 32 and multiply by 5/9 to get C = (5/9)(F − 32), so 212 °F is (5/9) × 180 = 100 °C.
From words to an equation
A word problem is an equation in disguise. Read it once for the story, then decide what the unknown is and give it a letter, saying in words what it stands for. Translate the sentence piece by piece: "is" becomes =, "more than" becomes +, "twice" becomes 2 ×. For example, "twice a number, plus 3, is the number plus 8" becomes 2x + 3 = x + 8. Solve it, x = 5, and then answer the question in a sentence with units. Finally check the answer against the words, not only against your equation, because the equation itself might be the mistake.
Some problems have two unknowns that depend on each other. "The sum of two consecutive integers is 47" has a smaller number n and the next one, n + 1. The equation is n + (n + 1) = 47, which simplifies to 2n + 1 = 47, so n = 23 and the integers are 23 and 24. When two quantities are linked like this, write the second one in terms of the first so that only one letter remains. In the graphs course, problems with two truly separate unknowns are solved with two lines instead.
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| five more than x | x + 5 |
| twice a number | 2x |
| seven less than y | y − 7 |
| the quotient of n and 4 | n/4 |
| two consecutive integers | n, n + 1 |
| "is", "equals", "gives" | = |
- Solve d = rt for t.
- t = d/r
- t = dr
- t = r/d
- t = d − r
- A train covers 360 km at 120 km per hour. How many hours does it take?
- Solve P = 2L + 2W for W.
- W = (P − 2L)/2
- W = P − 2L
- W = P/2 − 2L
- W = 2(P − L)
- A rectangle has perimeter 30 m and length 9 m. What is its width?
- Use C = (5/9)(F − 32). What is 212 °F in Celsius?
- Use F = (9/5)C + 32. What is 20 °C in Fahrenheit?
- Twice a number, plus 3, is the number plus 8. What is the number?
- The sum of two consecutive integers is 47. What is the smaller one?
- A square has perimeter 52 cm. How long is each side?
- Which equation matches "seven less than three times a number is 11"?
- 3x − 7 = 11
- 7 − 3x = 11
- 3(x − 7) = 11
- 3x + 7 = 11
- Solve a formula for one letter by treating the others as numbers.
- d = rt gives t = d/r; P = 2L + 2W gives W = (P − 2L)/2.
- C = (5/9)(F − 32) undoes F = (9/5)C + 32.
- For word problems: name the unknown, translate, solve, answer with units, check against the words.