Once you've learned the basic keywords for translating word problems from English into mathematical expressions and equations, you'll be presented with various English expressions, and be told to perform the translation.
Don't view the lists of keywords as holy writ, handed down from on high. Instead, use these lists as helpful hints. But always use your head! If you're not sure of the proper translation, use numbers. If you're not certain of the construction, apply it to what you might encounter "in real life". Above all, make sure that you understand what you're doing, and why. If you can clearly explain your work to a fellow student, then you're well on your way to mastery.
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The keyword "sum" tells me that they're wanting me to add the two listed quantities. This translates to:
8 + y
The order of the quantities doesn't matter here, since they're being added. But it's still a good idea to get in the habit of writing things in the specified order, because it'll matter in some contexts. While "y + 8" is technically okay, it's better to use the order "8 + y", because that's the order that they used in the English.
If you're careful now, then you'll be well-trained by the time you get to the test.
This is the "less than" construction, which is backwards in the math from the English. They've given me some unknown quantity, x, and they're telling me that they want the expression which stands for the quantity that is four units smaller than x. To find this quantity, I'll need to subtract the four from the unknown. This translates to:
x − 4
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To be clear, "four less than (the unknown)" in English means "(the unknown), less four" in algebra. If you're not sure of this, plug numbers in. If you get four dollars less an hour than (an unknown worker), you wouldn't subtract that worker's pay from 4; instead, you'd subtract 4 from that worker's pay: p − 4. Use this same order in your algebraic expression.
The keyword here is very obvious; "multiplied by" means that I'll be multiplying (the unknown) by the given value. The order of the terms here is (the unknown), followed by the value that is being multiplied onto (the unknown). However, in algebra, we put the constant (in this case, the 13) in front of (the unknown). Since order doesn't matter for multiplication, (x)(13) = (13)(x). So the English expression translates to:
13x
The keyword here is "quotient", which tells me that one of the items is divided by the other. The order of the items is important here, because order matters in division. Since (the unknown) comes first in the English expression, this tells me that it's on top in the fraction. Then this translates to:
The keyword here is "difference", telling me that one of the items is subtracted from the other. Since order matters in subtraction, I'll need to be careful with the order of the items. Since the number comes first in the English expression, it will need to come first in the math expression. Then this translates to:
5 − y
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English expressions can be more complicated than a simple relation between two items. When faced with these more-complex expressions, take your time and work carefully. Let the keywords and logic help you find the proper corresponding math expressions.
Okay; "the ratio of (this) to (that)" means "(this) divided by (that)", so I know I'll be ending up with a division. But the items being divided aren't simple. In particular, the (this) part is "9 more than x", which translates as "x + 9" (being "the variable, plus another nine more"). So this expression will be what goes in the top of the fraction which will be my ratio expression. The (that) part is just x, so this variable will be the bottom of my ratio expression. Then this translates to:
"The total of" indicates that things are being added. The things, in this case, are (a number) and 2. I'll need to pick a variable for (a number); I'll pick:
a number: n
(By explicitly saying what the variable is and what it stands for, I'll be much less likely to forget what it means; it also puts me in the good habit of naming things clearly, which always makes graders happy, and may get me partial credit if my math goes wrong at some point.)
The sum is of (a number) and the number 2. This sum is written as:
n + 2
Then I have to translate "nine less than" this sum into math. The "less than" construction is backwards in the English from the math. In this case, that means that the "nine less than", which is first in the English, actually needs to be last in the math. Then this translates to:
(n + 2) − 9
The instructions specify that I'm to "simplify" this expression, if possible, and it is indeed possible. Then my final answer is:
n − 7
Whatever the width w is, the length is 30 more than this. Remembering that "more than" means "plus that much", I'll be adding 30 to the w.
The expression they're looking for is:
w + 30
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This next one is very important; it crops up in many different word-problem contexts, but isn't usually pointed out to students. It's kinda hoped that you'll somehow figure it out on your own. It's the "how much is left" construction, and you'll usually need it when you're working with two things, like two legs of a journey, or two ingredients in one mix. You'll quickly note that you need at least one variable, but students are often at a loss as to how to handle whatever remains, after that one variable. Here's how it works:
The expression they're looking for is found by this reasoning:
I have two containers, and one tank that's pouring into them. They've given me a variable for the amount being poured into the larger container. I have to figure out an expression for the amount poured into the smaller container. There are twenty gallons total, and I've already poured g gallons of it. "The rest" is how much will be poured into the smaller container. But how many gallons is that?
I figure this out by noting that what goes into the smaller container is whatever is left over, after the larger container is taken care of. So how many gallons are left? There are the total, less whatever has already been taken care of. The amount taken care of already is the g gallons. Then the amount left over is the total, less g, or 20 − g gallons left. This is the answer they're wanting.
20 − g
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Whenever they expect you to use this "how much is left" construction, you will be given some total amount. Smaller amounts, of unspecified sizes, are added (combined, mixed, etc) to create this total amount. You will pick a variable to stand for one of these unknown amounts. After having thus accounted for one of the amounts, the remaining amount is whatever is left after deducting this named amount from whatever is the total. For example:
I'm making a big deal about this "how much is left" construction because it comes up a lot and tends to cause a lot of confusion. Make sure you understand this one!
Once you've learned to translate phrases into expressions and sentences into equations, you are ready to dive into word problems. Of course, there are infinitely-many possible word problems (physics is all word problems; business math is all word problems; "real life" can feel like an essay question...). The following links lead to explanations and examples of some basic types of word problems that you can expect to see in your classes:
"Age" problems, involving figuring out how old people are, were, or will be
"Area/volume/perimeter" problems, involving very basic geometric formulas
"Coin" problems, involving figuring out how many of each type of coin you have
"Distance" problems, involving speeds ("uniform rates"), distance, time, and the formula "d = rt".
"Investment" problems, involving investments, interest rates, and the formula "I = Prt".
"Mixture" problems, involving combining elements and find prices (of the mixure) or percentages (of, say, acid or salt).
"Number" problems, involving "Three more than two times the smaller number..."
"Percent of" problems, involving finding percents, increase/decrease, discounts, etc.
Quadratic word problems, such as projectile motion and max/min questions.
"Work" problems, involving two or more people or things working together to complete a task, and finding how long they took.
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