A rehearsal room charges a booking fee of 12 dollars plus 4 dollars per hour, and the receipt says 36 dollars. How many hours was the room used? The receipt is really an equation. One block of 12 dollars, then a number of 4 dollar blocks, adding up to 36. Word problems on the SAT work the same way. Once you see which quantities are fixed, which repeat and what the total is, the algebra is usually short.
See also Variables, Parameters, and Building a Model in Foundations for Calculus.
Key terms
Unknown quantity
Definition An unknown quantity is a number the problem does not give but that you need, either because the question asks for it or because the answer depends on it. You name it with a variable and state three things about it, what it counts or measures, its unit, and which values the situation allows, for example whether it must be a whole number or must be positive.
In plain words Before writing any algebra, write one line that says what your letter means. Most wrong answers to word problems come from a letter that meant two different things at once.
Example For the rehearsal room, let be the number of hours the room was booked. Its unit is hours and it cannot be negative. Whether must be a whole number depends on the room’s rules. If the room bills only full hours, then is a whole number, and if it bills by the minute, then values like also make sense.
Relationship
Definition A relationship is a statement in the problem that connects quantities, such as a sum, a difference, a multiple, a comparison or a rate. Translating a relationship means writing the same statement in symbols, keeping track of which quantity is compared with which and in what order. Translate the whole statement, not single trigger words, because the same word can lead to different symbols depending on the sentence around it.
| Words | Symbols |
|---|---|
| 5 less than | |
| less than 5 | |
| is 5 less than | |
| 3 more than twice | |
| is 3 times as large as | |
| the sum of and is 40 |
In plain words Read the full sentence and ask what it says about which number. The phrase “less than” flips the order you read the numbers in, and “3 times as many juniors as seniors” means the juniors are the bigger group.
Example A school club has 3 times as many juniors as seniors, and it has 40 more juniors than seniors. Let be the number of seniors and the number of juniors. A trigger word reading of the first sentence gives , which is backwards. The juniors are the larger group, so the two relationships are
Both expressions equal , so , which gives and . Then . As a check, and .
Per
Definition The word “per” means “for each” and describes a rate, a quotient of two quantities with different units. A rate of units of per unit of means
and the units multiply and cancel like numbers. For example, dollars per hour times hours gives dollars. “Percent” means per hundred, so percent of a quantity is times that quantity.
In plain words A “per” number is a price tag for one unit. Multiply it by how many units there are to get the total, and check that the units come out right.
Example A car travels 32 miles per gallon of fuel. Let be the number of gallons needed for a 200 mile trip. Then , so gallons. The units check, since miles per gallon times gallons is miles.
A percent works the same way. A jacket has a price of dollars before an 8 percent sales tax, and the total with tax is 27 dollars. The tax adds , so
which gives dollars.
Equation model
Definition An equation model of a situation is an equation in which every variable is defined with its unit, every term on both sides has the same unit, and the solutions of the equation correspond to the values the question asks about. A common shape is a fixed amount plus a rate times a count, set equal to a total,
In plain words An equation model says the same thing twice, once in words and once in symbols. If one side is in dollars, every term on both sides must be in dollars.
Example For the rehearsal room the receipt becomes
where is in dollars, is dollars per hour times hours, and is in dollars. Undo the fixed fee first and then split the rest into hourly blocks,
The room was booked for 6 hours. Check in the original story. Six hours cost dollars, and with the 12 dollar fee the total is 36 dollars.
At least and at most
Definition These phrases describe inequalities, not equations.
| Words | Symbols |
|---|---|
| is at least | |
| is at most | |
| is no more than | |
| is no less than | |
| is more than | |
| is less than |
“At least” and “at most” include the boundary value , while “more than” and “less than” exclude it.
In plain words “At least 10” means 10 is fine and anything bigger is fine too. “At most 10” means 10 is fine and anything smaller is fine too.
Example The band can spend at most 60 dollars on the rehearsal room. The cost must satisfy
Subtract 12 to get , then divide by the positive number 4, which keeps the direction, to get . Hours cannot be negative, so the budget allows any booking of up to 12 hours, and 12 hours is the longest booking the band can afford. Dividing by a negative number would reverse the sign, which is covered in Solving Linear Inequalities.
Constraint
Definition A constraint is a condition that the unknowns must satisfy because of the situation. Some constraints are stated in the problem, such as a budget of at most 60 dollars. Others come from the meaning of the variable, such as a count being a whole number or a length being positive. A solution of the equation model is an answer to the problem only if it satisfies every constraint. A negative or fractional algebraic solution may be impossible in the context, and then either the problem has no valid answer or you must choose the nearest allowed value in the direction the situation requires.
In plain words The algebra does not know that you cannot buy half a ticket. You have to check that yourself, every time.
Example A school needs buses for 210 students, and each bus holds 48. Let be the number of buses, a whole number. Every student needs a seat, so
The smallest whole number that works is . Rounding to the nearest whole number would give 4 buses, which seat only 192 students.
A fractional answer can also show that a situation is impossible. Suppose 14 tickets cost 100 dollars in total, adult tickets cost 9 dollars and child tickets cost 5 dollars. With adult tickets there are child tickets, so
which gives . A ticket count must be a whole number, so no purchase matches these numbers.
Check yourself
- A plumber charges a 75 dollar visit fee plus 40 dollars per hour of work. A customer’s bill was 215 dollars. Which equation can be used to find , the number of hours of work? (A) (B) (C) (D)
Show the answer
The correct choice is (B). The fee is paid once and the 40 dollars repeats for every hour, so the bill is . Choice (A) swaps the fixed and the hourly amounts. Choice (C) treats the 75 dollar fee as if it were 75 more hours, and (D) adds the fee to the hourly rate as if it were paid every hour. Solving (B) gives , so hours.
- Pens are sold only in packs of 12. A teacher needs at least 150 pens. What is the minimum number of packs the teacher must buy?
Show the answer
The answer is . With packs the teacher has pens, and the constraint is , so . The number of packs is a whole number, so the smallest choice is . The tempting answer gives only 144 pens.
- Phone plan A costs 30 dollars per month plus 0.10 dollar per minute of calls. Plan B costs 0.25 dollar per minute with no monthly charge. For how many minutes of calls in a month do the two plans cost the same? (A) (B) (C) (D)
Show the answer
The correct choice is (C). With minutes the costs are equal when
so and . Both plans then cost 50 dollars. Choice (B) is and ignores plan A’s per minute charge, (D) is and ignores plan B, and (A) is the common cost in dollars rather than the number of minutes.