Word Problems. Two simple words that evoke more fear and loathing than most other math concepts and question types combined.
When the subject of word problems arises, you might envision the following nightmare:
Two trains are loaded with equal amounts of rock salt and ball bearings. Train A leaves Frogboro at 10:00 a.m. carrying 62 passengers. Train B leaves Toadville at 11:30 a.m. carrying 104 passengers. If Train A is traveling at a speed of 85 mph and makes four stops, and Train B is traveling at an average speed of 86 mph and makes three stops, and the trains both arrive at Lizard Hollow at 4:30 p.m., what is the average weight of the passengers on Train B?
The good news is that you won’t see anything this ugly. SSAT word problems are pretty straightforward. Generally, all you have to do is translate the prose to math and solve.
The bad news is that you can expect to see a lot of word problems on your test. Keep in mind that, while word problems are generally algebra problems, they can contain other math concepts.
Many word problems seem tricky because it’s hard to figure out what they’re asking. It can be difficult to translate English into math. The following table lists some common words and phrases that turn up in word problems, along with their mathematical translations.
|When you see:||Think:|
|sum, plus, more than, added to, combined total||+|
|minus, less than, difference between, decreased by||–|
|is, was, equals, is equivalent to, is the same as, adds up to||=|
|times, product, multiplied by, of, twice, double, triple||x|
|divided by, over, quotient, per, out of, into||÷|
|what, how much, how many, a number||x, n, etc.|
Symbolism Word Problems
Word problems, by definition, require you to translate English to math. But some word problems contain an extra level of translation. Symbolism word problems are like any other word problem; just translate the English and the symbols into math and then solve.
Assume that the notation ◽ (w, x, y, z) means “Divide the sum of w and x by y and multiply the result by z.” What is the value of: ◽ (10, 4, 7, 8) + ◽ (2, 6, 4, 5)?
First, translate the English/symbols into math.
◽ (w, x, y, z) means ((w+x) / y) × z
Next, substitute the given values into the expression.
◽ (10, 4, 7, 8) + ◽ (2, 6, 4, 5) = ((10 + 4) / 7) × 8 + ((2 + 6) / 4) × 5 = 16 + 10 = 26
Word Problems with Formulas
Some of the more difficult word problems may involve translations into mathematical formulas. For example, you might see questions dealing with averages, rates, or areas of geometric figures. Since the SSAT and ISEE does not provide formulas for you, you’ll have to know these going in.
If a truck travels at 50 miles per hour for 6 1/2 hours, how far will the truck travel?
(A) 600 miles
(B) 425 miles
(C) 325 miles
(D) 300 miles
(E) 500 miles
To answer this question, you need to remember that Distance = Rate × Time. Once you note the formula, you can just plug in the numbers.
D = 50 × 6.5
D = 325 miles, (C)
Word problems are extraordinarily susceptible to backdoor strategies. Check out Kaplan’s Picking Numbers and Backsolving strategies.
Here are a few things to remember:
- You can Pick Numbers when the answer choices contain variables.
- Pick easy numbers rather than realistic ones. Keep the numbers small and manageable.
- You have to try all the answer choices. If more than one works, pick another set of numbers.
- Don’t pick the same number for more than one variable.
- When picking a number for a remainder problem, add the remainder to the number you’re dividing by.
- Always pick 100 for percent questions.
The average of four numbers is n. If three of the numbers are n + 3, n + 5, and n − 2, what is the value of the fourth number?
(A) n − 6
(B) n − 4
(D) n + 2
(E) n + 4
Pick an easy number for n, such as 10. If the average of four numbers is 10, the sum of the four numbers is 40 (4 × 10 = 40). If three of the numbers are n + 3, n + 5, and n − 2, then those three numbers are 10 + 3, 10 + 5, and 10 − 2—13, 15, and 8. Then 13 + 15 + 8 = 36. The sum of the four numbers must equal 40, so the remaining number is 4. If you plug 10 in for n in each of the answer choices, only (A) gives you 4.
- You can Backsolve when the answer choices are only numbers.
- Always start with the middle answer choice, (C).
- If the middle answer choice is not correct, you can usually eliminate two more choices simply by determining whether the value you’re looking for must be higher or lower.
Mike has n Hawaiian shirts, and Adam has three times as many Hawaiian shirts. If Adam gives Mike six Hawaiian shirts, both boys would have an equal number of Hawaiian shirts. How many Hawaiian shirts does Mike have?
Start with the middle answer choice, 9. If Mike has 9 shirts, then Adam has three times as many, or 27. If Adam gives Mike 6 shirts, Adam now has 21 and Mike has 15. This is not equal, so (C) is not correct. Since Adam was left with too many shirts when Mike had 9, Mike must have fewer than 9. Try (B). If Mike has 6 shirts, then Adam has 18. If Adam gives Mike 6, then they both have 12 shirts. Bingo, (B) is correct.
Roman Numeral Word Problems
You might see a Roman numeral problem on your test. If you do, keep a few things in mind. In keeping with the problem style, let’s lay them out in Roman numerals . . .
I. You don’t have to work with the statements in the order they are given. Deal with them in whatever order is easiest for you.
II. If you find a statement that is true, eliminate all of the choices that don’t include it.
III. If you find a statement that is false, eliminate all of the choices that do include it.
If the product of the positive numbers x and y is 20 and x is less than 4, which of the following must be true?
I. y is greater than 5.
II. The sum of x and y is greater than 10.
III. Twice the product of x and y is equal to 40.
(A) I only
(B) II only
(C) I and III only
(D) II and III only
(E) I, II, and III
We’re told that xy = 20 and x < 4. Now let’s look at the statements. Statement I says that y > 5. Since xy = 20, y = 20/x . When x = 4, y = 5. If we replace x with a smaller number than 4 in 20/x , then 20/x , which is y, will be greater than 5. Statement I must be true. Statement I must be part of the correct answer. Eliminate choices (B) and (D). Statement II says that x + y > 10. Try picking some values such that xy = 20 and x < 4. If x = 3, then y = 20/x = 20/3 = 6 2/3 . The sum of x and y is not greater than 10. Statement II does not have to be true. It will not be part of the correct answer. Eliminate (E). Statement III says that 2(xy) = 40, or 2xy = 40. The question stem says that xy = 20. Multiplying both sides of the equation xy = 20 by 2, we have that 2(xy) = 2(20), or 2xy = 40. Statement III must be true. (C) is correct.
Now it’s time to put all of your skills into play with a practice question. Remember to translate the English to math, don’t get intimidated, and keep your cool. Good luck!
Word Problem Practice Question
Sheila charges $5 per haircut during the week. On Saturday, she charges $7.50. If Sheila has six customers each day of the week except Sunday, how much money does she earn in five weekdays and Saturday?
D: Each weekday, Sheila earns $5 × 6 haircuts = $30. Each Saturday, Sheila earns $7.50 × 6 haircuts = $45. In five weekdays, she earns 5 × $30 = $150. In one Saturday, she earns $45. So in five weekdays plus one Saturday, she earns $150 + $45, or $195.
Check out more SSAT word problem practice questions here!