Words to Symbols

Before you begin

This quiz is about turning words into math. A sentence such as “a plumber charges $100 to show up and $50 for every hour” is written in English. Math can say the same thing in a few symbols, 100 + 50h, and once it is written that way you can work with it: find the bill for any number of hours, or work backward from a bill to the hours. Most of the questions on the GED math test arrive as words first, so this step, from words to symbols, comes before almost everything else in algebra.

The quiz goes in order. First, what a letter is doing in the middle of some math. Then the words that call for each of the four operations, and the few phrases that put things in the opposite order from how they are said. Then the word is, which turns an expression into an equation, and the one move that solves a simple equation: undo what was done. It ends with a honey bee, which does something very like this every day: it turns a place into a dance, and the bees that watch it turn the dance back into a place.

A letter that stands for a number

When you do not know a number yet, or when it can change, you can write a letter in its place. A letter used this way is called a variable. In 100 + 50h, the h stands for the number of hours the plumber works. It might be 1, or 3, or 4.5. The variable keeps the place for the number until you know it.

Any letter will do. People often pick the first letter of the word, h for hours or c for cost, because it helps them remember what the variable means. When a problem just says “a number,” n or x is the usual choice. The letter does not change the math: n + 4 and x + 4 say the same thing.

Two ways of writing turn up again and again, and they are worth knowing on sight.

• A number written right next to a variable means multiply. 50h means 50 × h. The times sign is left out because × looks too much like the letter x. So 3n is 3 times n, and if n is 7, then 3n is 21, not 37.
• Division is usually written as a fraction. n divided by 4 is written n4, or n/4.

If these are new, The Thing Called x goes through them slowly, one at a time, and Algebra Practice 1 has a hundred problems of this kind to practice on.

A piece of math with no equals sign, like 100 + 50h or n − 6, is called an expression. An expression is a name for a number. It does not say anything is true yet; it only describes how to get a number once you know what the variable stands for.

The words for the four operations

English has many words for each of the four operations. The GED uses all of them. Here they are in one place.

The chart is wider than your screen: slide it sideways with your finger, or turn your phone.

For adding and multiplying, the order does not matter: n + 7 and 7 + n are the same number, and so are 2n and n × 2. For subtracting and dividing, the order matters a great deal. 10 − 3 is 7, but 3 − 10 is −7. So the next part is about subtraction and division, and about the phrases that trick people there.

The phrases that run backward

Most of the time you can write the symbols in the same order as the words. “A number minus 6” is n − 6. “The difference between a number and 6” is also n − 6: the number is said first, and it is written first.

A few phrases do not work that way. The most common is less than. “6 less than a number” means: start with the number, and take 6 away. So it is n − 6, even though the 6 is said first.

A good way to be sure is to try it with real numbers. “6 less than 10” is 4, because 4 is six less than ten. And 10 − 6 is 4, while 6 − 10 is −4. So the number you start from goes first. The same reversal happens with fewer than (“6 fewer than a number” is n − 6) and with subtracted from (“6 subtracted from a number” is also n − 6). More than turns the phrase around too, but for adding the order makes no difference, so “6 more than a number” can be written n + 6 or 6 + n.

Division has the same issue. “A number divided by 4” is n4, and “4 divided by a number” is 4n, and those are different numbers. Whatever comes right after “divided by” goes on the bottom.

When to use parentheses

Some phrases describe two operations, and then the order they happen in matters. Compare these two:

• “Three times the sum of a number and 5.” The words “the sum of” tell you to add first and multiply the result by 3. Parentheses show that the adding comes first: 3(n + 5).
• “Three times a number, plus 5.” Here you multiply first and add 5 at the end: 3n + 5.

They are different numbers. If n is 2, the first is 3 × 7 = 21 and the second is 6 + 5 = 11. Parentheses are needed in the first one because, without them, the order of operations from Quiz 6 would multiply before adding. So when a phrase says the sum of or the difference of two things, and then multiplies or divides that whole result, put the sum or the difference in parentheses.

A starting amount and a rate

Many GED problems have the same shape: an amount you pay once, plus an amount that is paid again and again. The plumber from Quiz 25 charges $100 to show up, then $50 for every hour. In symbols, with h for the number of hours:

cost = 100 + 50h

The 100 is paid once, however long the job takes. The 50 is paid once for every hour, so it is multiplied by the hours. To find the cost of a 3-hour job, put 3 in place of h: 100 + 50 × 3 = 100 + 150 = $250. Putting a number in place of the variable like this is called evaluating the expression, and it is how you check that an expression says what you meant. Try h = 0: the cost is $100, the show-up fee, which is right, because the plumber charges that even for no work at all.

That is also why Quiz 25 said this plumber is not proportional: a $100 head start does not double when the hours double. A relationship that is proportional has no head start. Pay of $22 an hour, with nothing paid up front, is just 22h.

The word “is” becomes an equals sign

An expression names a number. An equation says that two things are equal, and it has an equals sign. In a word problem, the equals sign is usually the word is, or one of its cousins: equals, is the same as, gives, results in, comes to, was.

“A number increased by 12 is 30” becomes n + 12 = 30. “The plumber’s bill came to $325” becomes 100 + 50h = 325.

Solving an equation means finding the number that makes it true. For n + 12 = 30, the number is 18, because 18 + 12 really is 30. Any other number makes the equation false. So you can always check a solution by putting it back in and seeing whether both sides come out the same.

Solving in one step: undo it

An equation like n + 12 = 30 tells you what was done to the number: 12 was added to it. To get the number back, undo that. Each operation has an opposite that undoes it:

The rule that keeps this honest: whatever you do to one side of the equation, do to the other side too. An equation is a balance. The two sides weigh the same. If you take 12 off one side and not the other, it tips, and the equation stops being true. So you subtract 12 from both sides:

n + 12 − 12 = 30 − 12,  so  n = 18.

On the left, adding 12 and taking 12 away cancel, and the variable is left by itself. On the right, the arithmetic gives the answer. Then check: 18 + 12 = 30. It works. The Balance shows this with weights on a scale, if you want to see it happen.

The most common mistake is to do the same operation instead of the opposite one: to see n − 9 = 14 and subtract 9, getting 5. The check catches it at once: 5 − 9 is −4, not 14. The subtraction was already done to the number; you undo it by adding.

Two steps: undo in reverse order

Back to the plumber. The bill came to $325. How many hours did the job take? The equation is

100 + 50h = 325

Two things were done to h: first it was multiplied by 50, then 100 was added. To get h back, undo them in the reverse order: take away the 100 first, then divide by 50.

Step by step, doing each thing to both sides: subtract 100 from both sides, and 100 + 50h = 325 becomes 50h = 225. Divide both sides by 50, and h = 4.5. The job took four and a half hours. Check: 100 + 50 × 4.5 = 100 + 225 = 325. It works.

Reverse order is the same thing you do with shoes and socks: socks go on first and shoes second, so shoes come off first and socks second. The last thing done to the number is the first thing you undo.

Cross-multiplying, as algebra writes it

Quiz 24 solved proportions like 712 = x30 by cross-multiplying, and promised that algebra had its own way of saying the same thing. Here it is, and it is only the undoing move you have just learned.

In 712 = x30, the x has been divided by 30. Undo that by multiplying both sides by 30:

30 × 712 = x,  so  x = 210 ÷ 12 = 17.5.

That is the same answer Quiz 24 got. When the variable is on the bottom of a fraction instead, as in 712 = 35x, multiply both sides by both bottoms, 12 and x. The 12 cancels on the left and the x cancels on the right, leaving 7x = 12 × 35 = 420, so x = 60. Look at what was left: 7 times x on one side, 12 times 35 on the other. Those are the two diagonal products. Cross-multiplying is this move, with the canceling done in your head.

The bee and its dance

A honey bee that finds good flowers flies home and tells the other bees where they are. She does it in the dark, on the comb, which stands on its edge inside the hive. She runs a short straight line, shaking her body from side to side as she goes, then loops back around and runs the line again, again and again. This is called the waggle dance, and the straight part is the waggle run. The bees around her follow her, and then leave the hive and fly to the flowers.

The dance holds two numbers. The direction of the waggle run, measured from straight up, is the direction of the flowers measured from the sun. The time the waggle run lasts gives the distance: the farther the flowers, the longer she waggles. Researchers at North Carolina State University give the example of a waggle run that lasts 2.5 seconds, which points to flowers about 2,625 meters away. That is 1,050 meters for each second, so for these bees the rule can be written with a variable:

d = 1,050t

where t is the time of the waggle run in seconds and d is the distance in meters. It is a rough rule, and the exact number differs from one population of bees to another. But it is a rule, and it works both ways. A 2-second waggle means flowers about 2,100 meters away. And flowers 1,575 meters away call for a waggle of t seconds, where 1,050t = 1,575: divide both sides by 1,050, and t = 1.5 seconds. A bee turns a distance into a time; the bees that follow her turn the time back into a distance. That is the whole of this quiz, done by an insect.

Three to study before you start

Now you

Work without a calculator, on paper. Questions 3 and 8 are multiple choice: choose the one best answer. For every expression you write, put in a real number to check that it says what you meant; for every equation you solve, put your answer back in.

1. Write each phrase in symbols, using n for the number. (a) 9 more than a number. (b) A number divided by 4. (c) 6 less than a number. (d) The product of 7 and a number.
2. Write each phrase in symbols. (a) Three times the sum of a number and 5. (b) Three times a number, plus 5. Then find the value of each when the number is 2.
3. Marcus wrote “8 less than a number” as 8 − n. Which statement names what he did?
   • A) Nothing: 8 − n and n − 8 are the same, because the order does not matter in subtraction.
   • B) He wrote the words in the order he heard them, but “less than” runs backward. Start with the number and take 8 away: n − 8.
   • C) He should have multiplied, because “less than” means times: 8n.
   • D) He should have added, because a number that is 8 less must first have 8 added: n + 8.
4. A plumber charges $85 to show up, plus $60 for every hour. Write an expression for the cost of a job that takes h hours. Then find the cost of a 3-hour job.
5. Solve each equation, and check each answer by putting it back in. (a) x + 17 = 42. (b) x − 9 = 14. (c) 6x = 54. (d) x5 = 8.
6. Write each sentence as an equation, then solve it. (a) A number decreased by 13 is 29. (b) Four times a number is 92.

ReadingA reading rest stop, for the stubborn ones.

Read Honey Bee Dance Language, a short page from North Carolina State University’s extension service, by David Tarpy and Jennifer Keller. Land on the paragraph about how long the waggle run lasts: the 2.5 seconds and the 2,625 meters are there, and they are where question 10’s rule comes from.

The photograph above was taken at Malibu Lagoon, California, by the National Park Service’s Santa Monica Mountains National Recreation Area; it is not from NC State’s page.

7. After a raise of $1.75 an hour, Rosa earns $19.50 an hour. Let w be her wage before the raise. Write an equation, and solve it to find her old wage.
8. The moving company from Example 3 charges $120 for the truck plus $45 for every hour. Another bill came to $390. Which equation could be solved to find the number of hours, h?
   • A) 120h + 45 = 390
   • B) 45h = 390
   • C) 390 + 120 = 45h
   • D) 120 + 45h = 390
9. Solve the proportion 915 = x40 the algebra way: say what was done to x, and undo it on both sides.
10. Use the bees’ rule d = 1,050t, where t is the time of the waggle run in seconds and d is the distance to the flowers in meters. (a) How far away are the flowers when a waggle run lasts 2 seconds? (b) Flowers are 3,150 meters from the hive. Write an equation for the time of the waggle run, and solve it.

Check your work