Lesson 2 · from §19 of the 1910 text
Turning Words into Algebra
Word problems are mostly a matter of translation. Pick a letter for one unknown quantity, then write every other quantity in terms of that letter.
For a rectangle twice as long as it is wide, let the width be x. Then the length is 2x, and the perimeter (two widths plus two lengths) is 2x + 2 \cdot 2x = 6x.
Consecutive integers like 7, 8, 9 each go up by 1, so if the first is n the next ones are n + 1 and n + 2. Consecutive odd (or even) numbers go up by 2: n, n + 2, n + 4.
Quick check
If n is an even number, what represents the next even number?
Sentences into equations
Phrases map onto symbols: is becomes =, more than and added to become +, less than and taken from become -, and times becomes multiplication. Watch the order in "less than": "x is 2 less than 10" means x = 10 - 2.
Typing tip: type an equation with one equals sign, like 3x - 12 = x + 50.
Quick check
Write as an equation: Five times x is 7 more than x.
From the history of algebra
From "heap" to x. Writing equations simply took thousands of years. The very first mathematical writer we know of, an Egyptian priest named Ahmes who lived nearly 2000 years before Christ, called the unknown heap. One of his problems reads: "Heap; its seventh, its whole, it makes nineteen," which we'd write \tfrac{x}{7} + x = 19. The Hindus often called the unknown color, using different colors for different unknowns, so xy might be "black times yellow." The Arabs used the word root, which is why we still call the solution of an equation its root. Early Europeans called the unknown res, Latin for thing. Only after Vieta did a letter regularly stand for the unknown, and using x for it began with Descartes in 1637.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Represent each quantity with a letter and numbers.
A rectangle is three times as long as it is wide. If x is the width, what represents the length? the perimeter?
A rectangle is 10 feet longer than it is wide. If x is the width in feet, what represents the length? the perimeter?
A rectangle is 18 feet longer than it is wide. If x is the length, what represents the width? the perimeter?
The length of a rectangle is 4 feet more than twice the width. If x is the width, represent the length and the perimeter.
The numbers 3, 4, 5, 6, … are consecutive integers. How much greater is each than the one before?
If n is an integer, what represents the next consecutive integer?
If n is an integer, represent the next two consecutive integers, and their sum.
If x is the first of four consecutive integers, represent the other three and the sum of all four.
The numbers 3, 5, 7, 9, 11, … are consecutive odd integers. How much greater is each than the one before?
If n is any odd number, what represents the next greater odd number? the odd number before n?
Represent three consecutive odd numbers of which n is the first, and their sum.
Represent four consecutive even numbers, starting with n, and find their sum.
Represent three numbers, starting with n, where the second is twice the first and the third is three times the second.
Represent three numbers, starting with n, where the second is 10 more than the first and the third is 7 less than the second.
If a man's age now is x, what represents his age 4 years ago? 6 years from now?
A's age is twice B's. Let B's age be x. Represent each age (a) now, (b) 7 years ago, (c) 12 years from now.
Express each statement as an equation. (Don't solve it.)
Write as an equation: The sum of 8 and x is 5.
Write as an equation: x is 2 less than 10.
Write as an equation: x is 3 greater than 5.
Write as an equation: x is 5 more than y.
Write as an equation: Three times x is 21.
Write as an equation: Four times x is greater than 18 by 2.
Write as an equation: Eight added to x gives the same result as x taken from 34.
Write as an equation: Nine subtracted from 2x gives the same result as 14 added to x.
Write as an equation: Twice x added to three times x is 48 more than x.
Write as an equation: Twelve taken from three times x gives the same result as x added to 50.