Keeping equations balanced

One of the coolest things about equations is that you can do almost anything you want to them as long as you remember to do the exact same thing to both sides of the equation. This rule is called keeping the equation balanced. For example, if you have the equation , you can add 3 to both sides of the equation, and it still balances out: . You can divide both sides by 3, and it still balances: .

Equation balancing becomes especially handy in algebra (see the later section “Alphabet Soup: Tackling Algebra Review”).

Obeying the order of operations

In math, you must solve equations by following steps in a proper order. If you don’t, you won’t get the right answer. Many of the most frequent math errors occur when people don’t follow the order of operations when solving mathematical problems.

Keep in mind the following order of operations:

1. Start with any calculations in brackets or parentheses.

   When you have nested parentheses or brackets (parentheses or brackets inside other parentheses or brackets), do the inner ones first and work your way outward.

   Groupings where parentheses are implied, such as numerators or denominators (like ) or the numbers under a radical (like ), are performed first, just as they would be if there were official parentheses around them.

2. Do any terms with exponents and roots.
3. Complete any multiplication and division, in order from left to right.
4. Do any addition and subtraction, in order from left to right.

An easy way to remember this order is to think of the phrase “Please Excuse My Dear Aunt Sally” (Parentheses, Exponents, Multiply and Divide, Add and Subtract). The order of operations isn’t absolute, though, because you can simplify different parts of an expression in the same step — as long as you’re simplifying and not combining terms that don’t belong together. (The more you can simplify an equation, the better. You’ll save precious time on the ASVAB and reduce the possibility of copying errors, too.)

Take the following expression out for a ride.

Solve: .

Do the calculations in the parentheses first:

Next, simplify the exponents:

Do multiplication and division from left to right:

Finally, perform addition and subtraction from left to right:

33.5

Mental math: Mixing it up with the commutative, associative, and distributive properties

Although the order of operations tells you to do steps in a certain order (see the preceding section), some mathematical properties let you choose a different path. Using the commutative, associative, and distributive properties can make numbers smaller and easier to work with — sometimes easy enough that you can do calculations in your head instead of relying on your scratch paper. Anything that saves you time and brain power on the ASVAB is great, because you’re working on a limited time budget … and you can’t use a calculator.

The commutative and associative properties: Moving numbers in your head

The commutative and associative properties let you break the rules about adding or multiplying from left to right. The commutative property of addition says you can rearrange the numbers you’re adding without changing the result:

Similarly, the associative property of addition lets you decide how to group the numbers you’re adding:

Together, these properties let you add a string of numbers in whatever order you like. For example, you can make calculations easier by pairing up numbers whose ones digits add up to 10 before adding other numbers in the list.

Because subtracting is essentially the same thing as adding a negative number, you can extend these addition properties to subtraction problems, too — just be careful to keep track of the negative signs (see the later section “Playing with Positive and Negative Numbers” for details). The following example shows how smart groupings can let you add and subtract figures faster. Notice which calculations are easier to do in your head.

Similarly, the commutative and associative properties of multiplication let you multiply numbers in any order you like. Check out how switching the numbers around can make mental math easier:

You can even use these multiplication properties with division, as long as you remember that division is the same thing as multiplying by a fraction (see the later section “Multiplying fractions”):

The distributive property: Breaking up large numbers

Have you ever envied those people who can perform calculations on large numbers in their heads? What if I told you that you can be one of those people? That’s right. All you have to do is practice the distributive property of math.

The distributive property, often referred to as the distributive law of math, lets you separate or break larger numbers into parts for simpler arithmetic. It basically says that is the same as .

Suppose you want to mentally multiply 4 by 53; is the same as . Four times 50 is easy; it’s 200. Four times 3 is also easy. It’s 12. Two hundred plus 12 is 212.

Try another one with a bit of a twist.

Mentally perform the calculation .

is equivalent to .

You can quickly mentally calculate that 12 times 20 is 240 and that 12 times 1 is 12. Subtract 12 from 240, and you have 228.

Figure 9-1 illustrates how this process works.

If is still too large for mental calculation, you can break it down as , or .

You can also use the distributive property for division, although that takes a bit more practice: is the same as . You can quickly calculate that 340 divided by 2 is 170, and 170 divided by 2 is 85.

Or you can express as . You can mentally calculate 100 divided by 4 as 25. Forty divided by 4 is also easy — it’s 10. So . Keep practicing, and you’ll be known as the neighborhood lightning calculator.

Understanding types of factors

A factor can be either a prime number or a composite number (except 1 and 0 are neither prime nor composite). As I mention in the section “Making the Most of Math Terminology” earlier in this chapter, prime numbers have only themselves and 1 as factors, while composite numbers can be divided evenly by other numbers.

The prime numbers up to 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.

Finding prime factors

Any composite number can be written as a product of prime factors. Mathematicians call this process prime factorization. To find the prime factors of a number, you divide the number by the smallest possible prime number and work up the list of prime numbers until the result is itself a prime number.

Say you want to find the prime factors of 240. Because 240 is even, start by dividing it by the smallest prime number, which is 2: . The number 120 is also even, so it can be divided by 2: . Then and . Now, 15 isn’t even, so check to see whether you can divide it by 3 (the next highest prime number); , which itself is a prime number, so 240 is now fully factored.

Now, simply list what you divided by to write the prime factors of 240. The prime factors of 240 are .

Find the LCM of 45 and 50.

• Multiples of 45: 45, 90, 135, 180, 225, 270, 315, 360, 405, 450
• Multiples of 50: 50, 100, 150, 200, 250, 300, 350, 400, 450

The LCM of 45 and 50 is 450.

What is the least common multiple of 5, 27, and 30?

List the prime factors of each number:

• Prime factors of 5: 5
• Prime factors of 27:
• Prime factors of 30:

The number 3 occurs a maximum of three times, 5 occurs a maximum of one time, and 2 occurs a maximum of one time: . Check your answer by seeing whether 5, 27, and 30 can all divide evenly into 270.

Defining parts and types of fractions

The top number of a fraction is called the numerator. The bottom number is known as the denominator. For example, in the fraction , 7 is the numerator and 16 is the denominator.

You may also see numerators and denominators separated by a / sign rather than one on top of the other; is the same as .

If the numerator is smaller than the denominator, the fraction is less than a whole (smaller than 1). This kind of fraction is called a proper fraction. The fraction is a proper fraction, as is .

If the numerator is larger than the denominator, the fraction is larger than a whole (larger than 1), and the fraction is called an improper fraction. The fraction is an improper fraction.

Converting improper fractions to mixed numbers is customary in math, especially after all mathematical operations are complete. A mixed number is a whole number plus a fraction. The easiest way to convert an improper fraction to a mixed number is to divide the numerator by the denominator. You convert to a mixed number by dividing 17 by 16: , with a remainder of 1, so the improper fraction converts to .

Simplifying fractions

Simplifying (or reducing) fractions means to make the fraction as simple as possible. You’re usually required to simplify fractions on the ASVAB math subtests before you can select the correct answer. For example, if you worked out a problem and the answer was , the correct answer choice on the math subtest would probably be , which is the simplest equivalent to .

Many methods of simplifying fractions are available. In this section, I give you the two that I think are the easiest; you can decide which is best for you.

Multiplying fractions

Multiplying fractions is very easy. All you have to do is multiply the numerators by each other, multiply the denominators by each other, and then simplify the result, as shown in the following equation:

The fraction can be simplified to (see the earlier section “Simplifying fractions”).

If you can cross-cancel before you multiply, multiplying fractions is even easier. Here’s the same problem. The 3 in the numerator and the 15 in the denominator have a common factor: They’re both divisible by 3. If you divide both by 3 before multiplying, you don’t have to reduce the fraction at the end:

Before multiplying mixed numbers, change them to improper fractions by multiplying the denominator by the integer and adding it to the numerator. For example, is a mixed number. To convert it to an improper fraction, multiply 8 by 2 and add 5 to the result:

The end result is .

Dividing fractions

Dividing fractions is almost the same as multiplying, with one important difference: You have to convert the second fraction (the divisor) to the reciprocal and then multiply. As I explain in the earlier “Making the Most of Math Terminology” section, the reciprocal is simply a fraction flipped over.

Solve: .

Take the reciprocal of the second fraction and multiply it by the other fraction:

The fraction is an improper fraction, which you can convert to (see the earlier section “Defining parts and types of fractions”). Then you can simplify to (see the earlier section “Simplifying fractions”).

Adding and subtracting fractions

Adding and subtracting fractions can be as simple as multiplying and dividing them, or it can be more difficult. As the following sections show, it all depends on whether the fractions have the same denominator.

Adding and subtracting fractions with like denominators

To add or subtract two fractions with the same denominator, add (or subtract) the numerators and place that sum (or difference) over the common denominator:

Adding and subtracting fractions with different denominators

You can’t add or subtract fractions with different denominators. You have to convert the fractions so they all have the same denominator, and then you perform addition or subtraction as I explain in the preceding section.

Converting fractions so they share the same denominator involves finding a common denominator. A common denominator is nothing more than a common multiple of all the denominators, as I describe in the earlier section “Looking at Least Common Multiples.”

Find a common denominator for the fractions and .

The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, and 40.

The multiples of 8 are 8, 16, 24, 32, and 40.

A common denominator for the fractions and is 40.

The next step in the addition/subtraction process is to convert the fractions so they share the common denominator. To do this, divide the original denominator into the new common denominator and then multiply the result by the original numerator.

Start with . Divide the original denominator (5) into the new common denominator (40): . Next, multiply the result (8) by the original numerator (3): . The equivalent fraction is .

Perform the same operation with the second fraction, . Divide the original denominator (8) into the new common denominator (40): . Next, multiply this (5) by the original numerator (1): . The equivalent fraction is .

Now that the fractions have the same denominator, you can add or subtract them as shown in the preceding section:

Performing multiple operations

Sometimes you have to work through more than one operation on a set of fractions. Give this one a try:

On the surface, this problem looks complicated. But if you remember the order of operations (see the earlier section “Obeying the order of operations”) and take the problem one step at a time, it’s really easy.

Under the order of operations, you do the work in the implied sets of parentheses (the top and bottom of the big fraction) first:

and

If you write the big fraction bar as a division sign, the problem now reads .

Continue by performing the next operation in the parentheses:

The problem is now much simpler: .

Converting fractions to decimals

Some math problems require you to perform operations on both decimal numbers (see the later section “Dealing with Decimals”) and fractions. To properly perform such calculations, you must either convert the fraction to a decimal number or convert the decimal to a fraction.

Converting a fraction to a decimal number is easy. You simply divide the numerator by the denominator. For example is

Try the following:

Solve: .

Convert the fraction to a decimal by dividing the numerator by the denominator:

Now you can easily perform the operation: .

Comparing fractions

The two math subtests of the ASVAB often ask you to compare fractions to determine which one is the largest or smallest. If the fractions all have the same denominator, it’s easy. The fraction with the largest numerator is the largest, and the one with the smallest numerator is the smallest.

But how do you compare fractions that have different denominators? I’ll leave it up to you to determine which of the following proven methods you like the best.

Getting rational about ratios

Ratios represent how one quantity is related to another quantity. A ratio may be written as A:B or or using the phrase “A to B.”

A ratio of 1:3 says that the second quantity is three times as large as the first. A ratio of 2:3 means that the second quantity is three times as large as one-half of the first quantity. A ratio of 5:4 means the second quantity is four times as large as one-fifth of the first quantity.

A ratio is actually a fraction. For example, the fraction is also a ratio of 3 to 4. Solve problems including ratios the same way you solve problems that include fractions.

Converting decimals to fractions

To convert a decimal to a fraction, write all the digits following the decimal point in the numerator. If you see zeros before any nonzero digits, you can ignore them.

The denominator is always a one followed by zeros. The number of zeros in the denominator is determined by the total number of digits to the right of the decimal point (including the leading zeros):

• One digit: Denominator = 10. Example:
• Two digits: Denominator = 100. Example:
• Three digits: Denominator = 1,000. Example:
• Four digits: Denominator = 10,000. Example:

If necessary, reduce the fraction (see the earlier section “Simplifying fractions”).

Of course, you can also convert fractions to decimals (see the “Converting fractions to decimals” section earlier in this chapter).

Adding and subtracting decimals

You add and subtract decimals just as you do regular numbers (integers), except that before you perform your operation, you arrange the numbers in a column with the decimal points lined up one over the other.

Add the numbers 3.147, 148.392, and 0.074.

Put the numbers in an addition column with the decimal points lined up and perform the addition:

Multiplying decimals

Multiplying decimals requires three steps:

1. Convert the decimals to whole numbers by moving the decimal points to the right, remembering to count how many spaces you move each decimal point.
2. Multiply the whole numbers just as you’d perform any other multiplication.
3. Place the decimal point in the product by moving the decimal point to the left the same number of total spaces you moved the decimal points to the right at the beginning.

Multiply: .

First, convert the decimals to whole numbers by moving the decimal points to the right (remember to count).

• 3.724 becomes 3,724 (decimal moved three spaces).
• 0.0004 becomes 4 (decimal moved four spaces).
• 9.42 becomes 942 (decimal moved two spaces).

Next, perform the multiplication on the whole numbers:

Finally, put the decimal point in the correct position by moving it to the left the same number of places you moved the points to the right. You moved the decimal points a total of nine spaces to the right at the beginning, so now place the decimal point nine spaces to the left:

14,032,032 becomes 0.014032032

If you run out of numbers before you’re finished counting spaces to the left, add zeros (as shown in the example) until you’ve finished counting.

Dividing decimals

Dividing decimals can be a challenge. You have to use both subtraction and multiplication. You also need to be pretty good at rounding (see the later “Rounding” section) and estimating numbers.

You’re not allowed to use a calculator on the ASVAB math subtests.

You can divide decimals in two ways: long division and conversion.

Method 1: Long division

To do long division with decimals, follow these steps:

1. If the divisor isn’t a whole number, move the decimal point in the divisor all the way to the right (to make it a whole number), and move the decimal point in the dividend the same number of places to the right.
2. Position the decimal point in the result directly above the decimal point in the dividend.

3. Divide as usual.

   If the divisor doesn’t go into the dividend evenly, add zeros to the right of the last digit in the dividend and keep dividing until it comes out evenly or a repeating pattern shows up.

Try the following division problem:

Write the problem on your scratch paper in long-division form:

Now move the decimal point one place to the right, which makes the divisor a whole number. Also move the decimal point in the dividend one place to the right:

Position the decimal point in the result directly above the decimal point in the dividend:

Divide as usual. Seven goes into 70 ten times with 4 left over; then drop the 2 down:

Seven goes into 42 six times:

When you’re finished dividing decimals, you’re finished. You don’t have to move the decimal point around like you do after you’ve multiplied decimals.

Method 2: Conversion

The other way to divide decimals is to convert the decimals to fractions and then divide the fractions (see the earlier sections “Converting decimals to fractions” and “Dividing fractions,” respectively).

Try the problem from the preceding section, using the conversion method.

First, convert the decimals to fractions:

Take the reciprocal of the divisor (flip the second fraction upside down) and then multiply:

The fraction can be simplified (see “Simplifying fractions”) to . Convert to a decimal (see “Converting fractions to decimals”), and the answer is 10.6.

Rounding

Rounding a number means limiting a number to a few (or no) decimal places. For example, if you have a $1.97 in change in your pocket, you may say, “I have about $2.” The rounding process simplifies mathematical operations.

To round a number, you first determine what place you’re rounding to. For example, the math subtests that make up the AFQT may ask you to round to the nearest tenth. Then look at the number immediately to the right of that place. If the number is 5 or greater, round the digit to the left up; for any number under 5, round the digit to the left down. Thus, you’d round 1.55 up to 1.6 and 1.34 down to 1.3.

You can also round other numbers, such as whole numbers. For example, 1,427 becomes 1,400 when you round to the nearest 100. However, most of the rounding operations you encounter on the Mathematics Knowledge subtest involve rounding decimals to the nearest tenth or nearest hundredth.

Perusing percents

Percent literally means “part of 100.” That means, for example, that 25 percent (25%) is equal to , which is equal to 0.25.

If a problem asks you to find 25 percent of 250, it’s asking you to multiply 250 by 0.25.

To convert a percent to a decimal number, remove the percent sign and move the decimal point two places to the left: 15 percent is 0.15, and 15.32 percent is 0.1532. Conversely, to change a decimal number to percent, add the percent sign and move the decimal point two places to the right: 4.321 is equal to 432.1 percent.

Advice about exponents

Exponents are an easy way to show that a number is to be multiplied by itself a certain number of times. For example, is the same as , and is the same as . The number or variable that is to be multiplied by itself is called the base, and the number or variable showing how many times it’s to be multiplied by itself is called the exponent.

Here are important rules when working with exponents:

• Any base raised to the power of 1 equals itself. For example, .

• Any base raised to the zero power (except 0) equals 1. For example, .

  In case you were wondering, according to most calculus textbooks, is an “indeterminate form.” What mathematicians mean by “indeterminate form” is that in some cases it has one value, and in other cases it has another. This stuff is advanced calculus, however, and you don’t have to worry about it on the ASVAB math subtests.

• To multiply terms with the same base, you add the exponents. For example, .
• To divide terms with the same base, you subtract the exponents. For example, .
• If a base has a negative exponent, it’s equal to its reciprocal with a positive exponent. For example, .
• When a product has an exponent, each factor is raised to that power. For example, .

Roots

A root is the opposite of a power or an exponent. There are infinite kinds of roots. You have the square root, which means “undoing” a base to the second power; the cube root, which means “undoing” a base raised to the third power; a fourth root, for numbers raised to the fourth power; and so on. However, on the ASVAB math subtests, the only questions you’re likely to see will involve square roots and possibly a couple of cube roots.

Scientific notation

Scientific notation is a compact format for writing very large or very small numbers. Although it’s most often used in scientific fields, you may find a question or two on the Mathematics Knowledge subtest of the ASVAB asking you to convert a number to scientific notation, or vice versa.

Scientific notation separates a number into two parts: a characteristic, which is always greater than or equal to 1 and less than 10, and a power of ten. Thus, means 1.25 times 10 to the fourth power, or 12,500; means 5.79 times 10 to the negative eighth power. (Remember that a negative exponent is equal to its reciprocal with a positive exponent, so means ). In this case, the scientific notation comes out to 0.0000000579.

Visiting variables

Most algebraic equations involve using one or more variables. A variable is a symbol that represents a number. Usually, algebra problems use letters such as n, t, or x for variables. In most algebra problems, your goal is to find the value of the variable. For example, in the equation , you’d try to find the value of x by using the rules of algebra.

Following the rules of algebra

Algebra has several rules or properties that — when combined — allow you to simplify equations. Some (but not all) equations can be simplified to a complete solution:

• You may combine like terms. This rule means adding or subtracting terms with variables of the same kind. The expression simplifies to 8x. is equal to 3y. The expression simplifies to 9.
• You may use the distributive property to remove parentheses around unlike terms (see “The distributive property: Breaking up large numbers” earlier in this chapter).
• You may add or subtract any value as long as you do it to both sides of the equation.
• You may multiply or divide by any number (except 0) as long as you do it to both sides of the equation.

Using the distributive property

I know what you’re thinking: Combining like terms is pretty cool, but what if you have unlike terms contained within parentheses? Doesn’t the order of operations require you to deal with terms in parentheses first? Indeed, it does, and that’s where the distributive property comes in.

As I explain earlier in the chapter, . For example, is mathematically the same as .

Applying the same principle to algebra, the distributive property can be very useful in getting rid of those pesky parentheses:

Using addition and subtraction

You can use addition and subtraction to get all the terms with variables on one side of an equation and all the numeric terms on the other. That’s an important step in finding the value of the variable.

The equation has only the variable term on one side and only a number on the other. The equation doesn’t.

You can add and subtract any number as long as you do it to both sides of the equation. In this case, you want to get rid of the number 4 on the left side of the equation. How do you make the 4 disappear? Simply subtract 4 from it:

The equation simplifies to .

Using multiplication and division

The rules of algebra also allow you to multiply and divide both sides of an equation by any number except zero. Say you have an equation that reads , or 3 times x equals 21. You want to find the value of x, not three times x.

What happens if you divide a number by itself? The result is 1. Therefore, to change 3x to 1x (or x), divide both sides of the equation by 3:

But what if the equation were ? What would you do then?

I’ll give you a hint: If you multiply any fraction by its reciprocal, the result is 1. Remember, a reciprocal is a fraction flipped upside down.

Remember to multiply both sides of the equation by .

Solve: .

The inequality says that 3x plus 4 is less than 25. You solve it in the same way as you would the equation :

Method 1: The square-root method

Simple quadratic equations (those that consist of just one squared term and a number) can be solved by using the square-root rule:

If , then , as long as k isn’t a negative number.

Remember to include the sign, which indicates that the answer is a positive or negative number. Take the following simple quadratic equation:

Method 2: The factoring method

Most quadratic equations you encounter on the ASVAB math subtests can be solved by putting the equation into the quadratic form and then factoring.

The quadratic form is , where a, b, and c are just numbers. All quadratic equations can be expressed in this form. Want to see some examples?

• : This equation can be expressed in the quadratic form as . In this case, , , and .
• : You can express this equation as . So , , and .
•  : Expressed in quadratic form, this equation reads . So , , and .

Ready to factor? How about trying the following equation?

Solve: .

Because I like you, I’ve already expressed the equation in quadratic form (the expression on the left is equal to zero), saving you a little time.

You can use the factoring method for most quadratic equations where and c is a positive number.

The first step in factoring a quadratic equation is to draw two sets of parentheses on your scratch paper, and then place an x at the front of each, leaving some extra space after it. As with the original quadratic, the equation should equal zero:

The next step is to find two numbers that equal c when multiplied together and equal b when added together. In the example equation, and , so you need to hunt for two numbers that multiply to 6 and add up to 5. For example, and . In this case, the two numbers you’re seeking are positive 2 and positive 3.

Finally, put these two numbers into your set of parentheses:

Any number multiplied by zero equals zero, which means that and/or . The solution to this quadratic equation is and/or .

When choosing your factors, remember that they can be either positive or negative numbers. You can use clues from the signs of b and c to help you find the numbers (factors) you need:

• If c is positive, then the factors you’re looking for are either both positive or both negative:
  • If b is positive, then the factors are positive.
  • If b is negative, then the factors are negative.
  • b is the sum of the two factors that give you c.
• If c is negative, then the factors you’re looking for are of alternating signs; that is, one is negative and one is positive:
  • If b is positive, then the larger factor is positive.
  • If b is negative, then the larger factor is negative.
  • b is the difference between the two factors that gives you c.

Try another one, just for giggles:

Solve: .

Start by writing your parentheses:

In this equation, and . Because b is negative and c is positive, both factors will be negative.

You’re looking for two negative numbers that multiply to 6 and add to –7. Those numbers are –1 and –6. Plugging the numbers into your parentheses, you get . That means and/or .

Method 3: The quadratic formula

The square-root method can be used for simple quadratics, and the factoring method can easily be used for many other quadratics, as long as . (See the two preceding sections.) But what if a doesn’t equal 1, or you can’t easily find two numbers that multiply to c and add up to b?

You can use the quadratic formula to solve any quadratic equation. So why not just use the quadratic formula and forget about the square-root and factoring methods? Because the quadratic formula is kind of complex:

The quadratic formula uses the a, b, and c from , just like the factoring method.

Armed with this knowledge, you can apply your skills to a complex quadratic equation:

Solve: .

In this equation, , , and . Plug the known values into the quadratic formula and simplify:

Rounded to the nearest tenth, and .

Knowing all the angles

Angles are formed when two lines intersect at a point. Many geometric shapes are formed by intersecting lines, which form angles. Angles can be measured in degrees. The greater the number of degrees, the wider the angle is:

• A straight line is exactly 180°.
• A right angle is exactly 90°.
• An acute angle is more than 0° and less than 90°.
• An obtuse angle is more than 90° but less than 180°.
• Complementary angles are two angles that equal 90° when added together.
• Supplementary angles are two angles that equal 180° when added together.

Take a look at the different types of angles in Figure 9-3.

Common geometric shapes

I’m not going to explain all the possible geometric shapes for two reasons: Doing so would take this entire book, and you don’t need to know them all to solve the math problems you find on the ASVAB. However, you should recognize the most common shapes associated with geometry.

Getting square with quadrilaterals

A quadrilateral is a geometric shape with four sides. All quadrilaterals contain interior angles totaling 360°. Here are the five most common types of quadrilaterals:

• Squares have four sides of equal length, and all the angles are right angles.
• Rectangles have all right angles.
• Rhombuses have four sides of equal length, but the angles don’t have to be right angles.
• Trapezoids have at least two sides that are parallel.
• Parallelograms have opposite sides that are parallel, and their opposite sides and angles are equal.

Figure 9-4 gives you an idea of what these five quadrilaterals look like.

Trying out triangles

A triangle consists of three straight lines whose three interior angles always add up to 180°. The sides of a triangle are called legs. Triangles can be classified according to the relationship among their angles, the relationship among their sides, or some combination of these relationships. You should know the three most common types of triangles:

• Isosceles triangle: Has two equal sides, and the angles opposite the equal sides are also equal.
• Equilateral triangle: Has three equal sides, and all the angles measure 60°.
• Right triangle: Has one right angle (90°); therefore, the remaining two angles are complementary (add up to 90°). The side opposite the right angle is called the hypotenuse, which is the longest side of a right triangle.

Check out Figure 9-5 to see what these triangles look like.

Settling on circles

A circle is formed when the points of a closed line are all located equal distances from a point called the center of the circle. A circle always has 360°. The closed line of a circle is called its circumference. The radius of a circle is the measurement from the center of the circle to any point on the circumference of the circle. The diameter of the circle is measured as a line passing through the center of the circle, from a point on one side of the circle to a point on the other side of the circle. The diameter of a circle is always twice as long as the radius. Figure 9-6 shows these relationships.

Famous geometry formulas

The math subtests of the ASVAB often ask you to use basic geometry formulas to calculate geometric measurements. You should commit these simple formulas to memory.

Math terminology practice

Remember when your mother said, “It’s not what you say; it’s how you say it”? She wasn’t talking about math. In mathematics, it’s exactly what you say that matters. Use Activity 1 to match the signal words to the correct mathematical operations (addition, subtraction, multiplication, or division), and then check the “Answers and Explanations” section at the end of this chapter to make sure you’re right.

ACTIVITY 1 Math Terminology

Order of operations practice

Math is a lot like the military: There are right ways and wrong ways to do everything. In mathematics, it’s all about the order of operations. Remember that Aunt Sally — although we always have to apologize for her (Please Excuse My Dear Aunt Sally) — can show you how to solve any math problem in the right order (refer back to “Obeying the order of operations,” earlier in this chapter). In Activity 2, look at the problem and determine which operations you’ll do first, then which you’ll do second. You don’t need to perform the calculations. When you’re done, see whether you’re right in the “Answers and Explanations” section at the end of this chapter. When you’re only simplifying things, it’s often okay to switch the first and second operations you perform.

ACTIVITY 2 Order of Operations

Converting fractions and decimals

When you run into an ASVAB question that contains fractions and decimals, you’ll most likely need to convert one or the other to solve the problem (see “Dealing with Decimals” earlier in this chapter). Use Activity 3 to practice switching between fractions and decimals, and then check the “Answers and Explanations” section at the end of this chapter to see how well you did.

ACTIVITY 3 Fractions and Decimals

Solving equations and inequalities

When you take the ASVAB, you’ll do plenty of mental math — but you’ll also use some of the unlimited supply of scratch paper the test proctor gives you to write equations. Simple errors in an equation can throw off your entire game, though, so use Activity 4 to practice solving each equation or inequality. When you’re finished, check your work in the “Answers and Explanations” section at the end of this chapter. (If you need to brush up on balancing equations, see the “Following the rules of algebra” section earlier in this chapter.)

ACTIVITY 4 Solving Equations and Inequalities

Identifying formulas

Many of the algebra and geometry questions on the ASVAB require you to plumb the depths of your memory for a specific mathematical formula. If you can’t remember it, you’re going to have a tough time coming up with the right answer. Use Activity 5 to match each type of problem with the appropriate formula to solve it. (Note: Not all of these formulas appear in this chapter.) Make sure you connected the right pairs by checking your answers in the “Answers and Explanations” section at the end of this chapter.

ACTIVITY 5 Identifying Formulas