Math Strategies for Armed Services Vocational Aptitude Battery (ASVAB) Study Guide (page 2)
- Don't work in your head! Use your test book or scratch paper to take notes, draw pictures, and calculate. Although you might think that you can solve math questions more quickly in your head, that's a good way to make mistakes. Write out each step.
- Read a math question in chunks rather than straight through from beginning to end. As you read each chunk, stop to think about what it means and make notes or draw a picture to represent that chunk.
- When you get to the actual question, circle it. This will keep you more focused as you solve the problem.
- Glance at the answer choices for clues. If they are fractions, you probably should do your work in fractions; if they are decimals, you should probably work in decimals; etc.
- Make a plan of attack to help you solve the problem.
- If a question stumps you, try one of the backdoor approaches explained in the next section. These are particularly useful for solving word problems.
- When you get your answer, reread the circled question to make sure you have answered it. This helps avoid the careless mistake of answering the wrong question.
- Check your work after you get an answer. Test takers get a false sense of security when they get an answer that matches one of the multiple-choice answers. Here are some good ways to check your work if you have time:
- Ask yourself if your answer is reasonable, if it makes sense.
- Plug your answer back into the problem to make sure the problem holds together.
- Do the question a second time, but use a different method.
- Approximate when appropriate. For example:
- $5.98 + $8.97 is a little less than $15. (Add: $6 + $9)
- .9876 × 5.0342 is close to 5. (Multiply: 1 × 5)
- Skip hard questions and come back to them later. Mark them in your test book so you can find them quickly.
Backdoor Approaches for Answering Tough Questions
Many word problems are actually easier to solve by backdoor approaches. The two techniques that follow are timesaving ways to solve multiple-choice word problems that you don't know how to solve with a straightforward approach. The first technique, nice numbers, is useful when there are unknowns (like x) in the text of the word problem, making the problem too abstract for you. The second technique, working backward, presents a quick way to substitute numeric answer choices back into the problem to see which one works.
- When a question contains unknowns, like x, plug nice numbers in for the unknowns. A nice number is easy to calculate with and makes sense in the problem.
- Read the question with the nice numbers in place. Then solve it.
- If the answer choices are all numbers, the choice that matches your answer is the right one.
- If the answer choices contain unknowns, substitute the same nice numbers into all the answer choices. The choice that matches your answer is the right one. If more than one answer matches, do the problem again with different nice numbers. You will only have to check the answer choices that have already matched.
Example: Judi went shopping with p dollars in her pocket. If the price of shirts was s shirts for d dollars, what is the maximum number of shirts Judi could buy with the money in her pocket?
To solve this problem, let's try these nice numbers: p = $100, s = 2; d = $25.Now reread it with the numbers in place:
Judi went shopping with $100 in her pocket. If the price of shirts was 2 shirts for $25, what is the maximum number of shirts Judi could buy with the money in her pocket?
Since 2 shirts cost $25, that means that 4 shirts cost $50, and 8 shirts cost $100. So our answer is 8. Let's substitute the nice numbers into all 4 answers:
The answer is b because it is the only one that matches our answer of 8.
You can frequently solve a word problem by plugging the answer choices back into the text of the problem to see which one fits all the facts stated in the problem. The process is faster than you think because you will probably only have to substitute one or two answers to find the right one.
This approach works only when:
- All of the answer choices are numbers.
- You are asked to find a simple number, not a sum, product, difference, or ratio.
Here's what to do:
- Look at all the answer choices and begin with the one in the middle of the range. For example, if the answers are 14, 8, 2, 20, and 25, begin by plugging 14 into the problem.
- If your choice doesn't work, eliminate it. Determine if you need a larger or smaller answer.
- Plug in one of the remaining choices.
- If none of the answers works, you may have made a careless error. Begin again or look for your mistake.
Example: Juan ate of the jellybeans. Maria then ate of the remaining jellybeans, which left 10 jellybeans. How many jellybeans were there to begin with?
Starting with the middle answer, let's assume there were 90 jellybeans to begin with:
Since Juan ate of the jellybeans, that means he ate 30 ( × 90 = 30), leaving 60 of them (90 – 30 = 60). Maria then ate of the 60 jellybeans, or 45 of them ( × 60 = 45). That leaves 15 jellybeans (60 – 45 = 15).
The problem states that there were 10 jellybeans left, and we wound up with 15 of them. That indicates that we started with too big a number. Thus, 90, 120, and 140 are all incorrect! With only two choices left, let's use common sense to decide which one to try. The next lower answer is only a little smaller than 90 and may not be small enough. So, let's try 60:
Since Juan ate of the jellybeans, that means he ate 20 ( × 60 = 20), leaving 40 of them (60 – 20 = 40). Maria then ate of the 40 jellybeans, or 30 of them ( × 40 = 30). That leaves 10 jellybeans (40 – 30 = 10).
Because this result of 10 jellybeans remaining agrees with the problem, the right answer is a.
Denominator The bottom number in a fraction. Example: 2 is the denominator in
Difference Subtract. The difference of two numbers means subtract one number from the other.
Divisible by A number is divisible by a second number if that second number divides evenly into the original number. Example: 10 is divisible by 5 (10 ÷ 5 = 2, with no remainder). However, 10 is not divisible by 3. (See Multiple of .)
Even Integer Integers that are divisible by 2, like…–4, –2, 0, 2, 4.…(See Integer.)
Integer Numbers along the number line, like…–3, –2, –1, 0, 1, 2, 3.…Integers include the whole numbers and their opposites. (See Whole Number.)
Multiple of A number is a multiple of a second number if that second number can be multiplied by an integer to get the original number. Example: 10 is a multiple of 5 (10 = 5 × 2); however, 10 is not a multiple of 3. (See Divisible by.)
Negative Number A number that is less than zero, like…–1, –18.6, – .…
Numerator The top part of a fraction. Example: 1 is the numerator of .
Odd Integer Integers that aren't divisible by 2, like…–5, –3, –1, 1, 3.…
Positive Number A number that is greater than zero, like…2, 42, , 4.63.
Prime Number Integers that are divisible only by 1 and themselves, like…2, 3, 5, 7, 11.… All prime numbers are odd, except for number 2. The number 1 is not considered prime.
Product Multiply. The product of two numbers means the numbers are multiplied together.
Quotient The answer you get when you divide. Example: 10 divided by 5 is 2; the quotient is 2.
Real Number All the numbers you can think of, like… 17, –5, , –23.6, 3.4329, 0.… Real numbers include the integers, fractions, and decimals. (See Integer.)
Remainder The number left over after division. Example: 11 divided by 2 is 5, with a remainder of 1.
Sum Add. The sum of 2 numbers means the numbers are added together.
Whole Number Numbers you can count on your fingers, like…1, 2, 3.…All whole numbers are positive.
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