Translations and Word Problems
Some problems simply ask you to 'translate' English into math. For these problems, you need to simply translate word for word from left to right. Seems obvious, right? So many students get these questions wrong because they don't do this! They try to skip steps or lump stuff together, and that's NOT what you should do. A few examples:
The sum of three times a number and 4
The square root of the quantity 4x + 6
Nine less than the product of 2 and a number squared
To know what to replace them with, we need to know what those words mean. Memorize this list:
The word/phrase |
Means: |
of/times/multiplied |
• (multiply) |
decreased by /difference/ less than |
- |
sum/more than |
+ |
what/a number |
x, n, y (a variable) |
Percent |
/100 |
is/was/has |
= |
for/per |
÷ |
the quantity |
( ) |
squared/times itself |
^{2} |
-Word-for-Word Rule: For every translation problem, you should read left to right and translate word for word. Don't skip any words and don't try to do it all at once. They are testing your ability to follow a series of steps on these questions, so it's important that you do so.
Let's talk first about some of the tricky ones.
"Less Than/Fewer than:" 'Less than' does mean subtract, but why is this one tricky? Because it means the first number in the equation comes second in the word problem. Think about it for a second. If you are 17 now, and someone were to ask you "What's four less than your age?" you would quickly be able to tell them 13. But what did you really do in terms of math? You took the second number in the word problem (your age) and put it first to do the math (17-4).
If a translation problem says "less than," leave a blank spot in front of your minus sign and put the number less than after the minus sign. Let's take a look at the example from above:
Nine less than the product of 2 and a number squared: Keeping the above in mind, we need to leave a blank spot in front of the minus sign and put 9 after it. So your paper should like this:
_ - 9
with a blank spot in front of that subtraction symbol for whatever they're about to tell you.
Continuing on in this example, we run into another tricky situation:
"Minus the product/Times the sum/etc:" If someone asks you for 4 times 2, you know enough to write 4•2. But what about if they ask for 4 times the sum of something? You can't write 4•sum, though if you're following the Word-for-Word Rule, you should. So what does that mean? It means you need to multiplying 4 by the entire sum of whatever's to follow, which in short, means you need parentheses. Let's continue on with the example from above:
'Nine less than the product of' means you are subtracting 9 from that entire product, not just a number. So your paper should now look like this:
(__) - 9
'2': Easy.
(2_) - 9
'and': Another tricky one!
-SAT Math Hint: The word 'and' in a translation problem does NOT mean add! It means 'do the mathematical operation here.' So a question can say 'the product of 4 and 2' or 'the quotient of 3x and y' – neither of those questions have any addition in them.
So in this case, what does 'and' mean? Well, we were asked for the product, so 'and' means multiply.
(2•_) - 9
'a number:' From the chart, we know this means to put in a variable:
(2•n) - 9
'squared:' Because this word directly follows 'a number' we know that it's just the number that's squared, not the whole thing. So your final expression should look like this:
(2•n^{2}) - 9
Where to put the parentheses can sometimes be confusing, so let's look at it again. What's the difference between these two expressions?
The sum of 4 times x and y
Four times the sum of x and y
Remember, the word 'and' means 'do the math here'. So the first expression, word for word, should read as:
4x + y
The second expression, however, should read:
4(x + y)
Why? Because it's "Four times the sum" NOT "four times x". When you see "four times the sum" or "the quotient of the difference" or "the sum of the product," you need parentheses. Also, be careful that it makes sense. The first example above must be 4(x + y) instead of4+ x + y or 4+ xy because we must go word for word. The first instruction you are given says "The sum of" which means we are looking for where to place that addition symbol right from the beginning. You can't have 4+x if the problem says 'times.'
Let's go through the other two examples from above:
The sum of three times a number and 4:
'The sum of:' So we're looking to add.
'Three times:' Write down "3•"
'a number': Put in a variable. So your paper now reads: "3•n"
'and': Here's the math operation. Which one is it? Sum. Put in a plus sign: "3•n +"
'4:' Put in a 4: 3n + 4
Let's look at the next one.
The square root of the quantity4x + 6: Write your square root, and then put the whole quantity underneath.
Examples & Solutions
Try these:
The product of 3 and the difference between x and 2y
Three less than the sum of x raised to the third power and 7
6 more than the quotient of x and 2 + y
What did you get? You should have:
The product of 3 and the difference between x and 2y: 3•(x - 2y)
Three less than the sum of x raised to the third power and 7: (x^{3} + 7) - 3
6 more than the quotient of x and 2 + y:
The same translation chart works for regular word problems. Try these:
Bill ran 10 miles farther than Mark
Liz ate 2/3 of the candy
Tom has 5 fewer marbles than twice the number Jane has. If Tom has 12 marbles, how many does Jane have?
You should have:
Bill ran 10 miles farther than Mark: Bill and Mark are unknowns, so put in variables. "Ran" is a verb. Like 'is' or 'was,' it defines Bill for the purpose of this question, so it's equivalent to an equals sign. So your equation should be: B = 10 + M
Liz ate 2/3 of the candy: Again, Liz and the candy are both unknowns. 'Ate' defines her for our purposes, and 'of' means 'times':
Tom has 5 fewer marbles than twice the number Jane has. If Tom has 12 marbles, how many does Jane have? T = 2J - 5 Then substitute in 12 for T to solve.
Before we go back to percents, there's another strategy you should know for Word Problems:
-Make It Real Strategy: Whenever you're given a word problem, take a moment and literally sit back and think about the actual situation presented. What are they talking about? Does it make sense? Before you do any work, throw in a few numbers to check your hunch. If a problem says, "Mary can eat up to 6 slices of pizza" and you feel like the equation should be m ≤ 6, check with numbers. Is it okay if she eats 5? 4? 8? Ah, no she can't eat eight. So check your equation. 5 and 4 are both less than six, eight isn't. So your hunch was right.
If an unknown is presented, make up your own numbers to see what the heck they're talking about. You can even make up your own situation if that helps. Let's say a question discussed someone shopping for fabric. "Bernard needed to make a certain number of pants for the dance competition. If fabric is sold in yards and he needs f feet worth of fabric to make his costumes, how many y yards of fabric should he buy?" There are a lot of unknowns in this question, and in fact, the answers all contain variables only, so this would be a good question to use ITOS on. But even before you do that, make it real for you. If you don't care about costumes, make it about football, or cooking. How do you do that? By putting it into terms you care about. Let's make it about football to figure out what's going on. So basically, in that case, we could say that he needs to accrue a certain number of feet for his stats, but they're measuring in yards. So how many yards is it he needs? Oh, okay. That makes sense. We're basically just converting feet to yards. Now go back and put in real numbers, using ITOS, to get a nice real number that you can take to the answer choices.
Lastly, if there are numbers, but they're ugly, change them until you get the problem itself figured out in your head. Let's say the question is: "If someone is buying 423 shares of stock at 4 5/8 per share and the stock rises to 9 1/8 before falling to 3/4, how much did they lose from their initial investment?" Those are some ugly numbers with a lot of fractions. For now, use different numbers simply to get the situation clear in your head. Let's use 100 for the first number, and 4, 10, and 2 for the next ones. That would mean they bought 100 shares of stock at $4 each. Okay. How much is that? What math did you have to do to get that? Oh, multiply, right. So you're going to need to multiply the first number by the second. Next, with the easier numbers, the price rose to $10. Do we care? No. We only care about the final amount. So there's actually no work to do here. Then it fell to $2 a share. You have 100 shares. So you multiply again. Okay. Oh, and now you subtract 200 from 400. Pretty easy. So those are the steps you need to follow with the more complicated numbers - multiply the number of shares by the final price, and subtract the difference. Got it.
SAT Word problems very much go back to our Most Important Rule: Let Yourself Feel Like an Idiot. Get rid of your pride or your desire to do everything really quickly and brilliantly without even thinking about it. It won't work. Take it slow, figure it out, and then the work will almost be done for you.
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