Mathematics Study Guide for the TABE Test

Numbers and Operations: Fractions

Like decimals, fractions represent values between whole number values. Even outside the math classroom, fractions pop up quite a lot. Chefs, carpenters, and nurses are just a sample of the types of professionals who use fractions every day.

Fraction Basics

Fractions are all based on dividing a value into several parts and focusing on one or more of the parts. For example, see how this larger square below is divided into nine equal, smaller squares:

We could say that one of those smaller squares is one of nine parts. In math we would write that as \(1/9\) or \(\frac{1}{9}\) (which is read as “one-ninth”). The bottom number, the denominator, tells us how many total parts there are (in this case, nine). The top number, the numerator, tells us how many parts we are focusing on (one).

Looking at the shaded parts of the big square, we see that four of the nine parts are shaded. The shaded part makes up \(\frac{4}{9}\) (four-ninths) of the big square. If all nine small squares were shaded, the resulting fraction would be \(\frac{9}{9}\), which is really just another way of representing the value \(1\). Anytime the numerator and denominator are the same, the fraction equals one.

It is important to note that there are multiple ways to divide a shape into fractional parts. As long as the parts are of equal size, the fractions are the same. All three of these rectangles are divided into fourths:

As we said, fractions pop up all the time. Suppose in a classroom there are \(12\) boys and \(17\) girls. The girls make up what fraction of the class? First, we need to know the total number of parts of the class, which is \(12+17=29\). We know \(17\) of the students are girls, so the girls are \(17\) parts of the whole \(29\) parts. As a fraction, we write that as \(\frac{17}{29}\).

Types of Fractions

There are a few different types of fractions that you should know. The first is a proper fraction, like \(\frac{4}{9}\), which is when the numerator is smaller than the denominator. As you might surmise, if there are proper fractions, there are also improper fractions. These are fractions when the numerator is greater than the denominator, such as \(\frac{9}{4}\).

There is one other type of fraction to familiarize yourself with, which is known as a mixed number (or mixed fraction). This type of fraction includes both a whole number and a fraction. For instance, \(2 \frac{1}{4}\) is a mixed number. The interesting thing about a mixed number is that it can be rewritten as an improper fraction, and vice versa. You do that by multiplying the whole number portion by the denominator and then adding that result to your numerator. So, \(2 \frac{1}{4}\) becomes \(\frac{2\times4 + 1}{4} = \frac{9}{4}\).

Reducing and Equivalent Fractions

Sometimes it is necessary to reduce or simplify a fraction, which in simple terms means making it smaller. This is only possible when the numerator and denominator have the same factors. For instance, \(\frac{10}{40}\) can be reduced to \(\frac{1}{4}\) because both \(10\) and \(40\) share \(10\) as a factor. We divided the numerator and denominator in the original fraction by the same number, thereby resulting in a smaller, more manageable fraction. Here are other examples of reduced fractions:

\[\frac{6}{18} = \frac{1}{3}\] \[\frac{9}{21} = \frac{3}{7}\]

Note: On the TABE, the answer choices will often be given in the most reduced form. If you have an answer and you don’t see it among the choices, check to see if your fraction can be reduced further.

Two fractions are equivalent if they can be reduced to the same fraction. For example, \(\frac{12}{16}\) is equivalent to \(\frac{6}{8}\) because both can be reduced to \(\frac{3}{4}\). They both describe the same part of a whole.

The figure below shows a square divided into four parts:

Three of the four parts in the first square are shaded, so we can say that \(\frac{3}{4}\) of the square is shaded.

Now, this figure below shows the square divided into \(16\) parts:

Twelve of the \(16\) parts are shaded, so we say that \(\frac{12}{16}\) of the large square is shaded. Notice that the same area is shaded in both squares. That tells us that \(\frac{3}{4}\) and \(\frac{12}{16}\) mean the same thing. They are equivalent.

If you are given a fraction, you should be able to generate other fractions that are equivalent to it. You’ve already learned how to do this by reducing. Another way is to multiply the numerator and denominator by the same number. Any number will do, because, remember, when the numerator and denominator are the same number, we’re simply multiplying by \(1\).

For instance, start with \(\frac{3}{4}\) and multiply the top and bottom by \(4\):

\[\frac{3 \times 4}{4 \times 4} = \frac{12}{16}\]

Once again, we have shown that \(\frac{12}{16}\) is equivalent to \(\frac{3}{4}\).

Fractions on a Number Line

Between the marked whole numbers on a number line, there are an infinite number of fractional numbers. It would be impossible to name them all, let alone fit them on the number line. Still, many number lines include at least some fractional values:

Often, there aren’t specific marks for the common fractions, but even if they aren’t there, the values still exist. For example, on the number line below, the arrow at \(A\) is pointing at the number \(1\frac{1}{2}\), or \(1.5\), and the arrow at \(B\) is pointing at \(3\frac{3}{4}\) or \(3.75\):

Comparing Fractions

When comparing fractions, remember that the larger the denominator, the smaller the fraction. For example, \(\frac{1}{2}\) is larger than \(\frac{1}{4}\). Think in terms of a cake:

• One-half (\(\frac{1}{2}\)) of a cake means the cake has been cut into two equal pieces and you have one.

• One-fourth (\(\frac{1}{4}\)) of a cake means the cake has been cut into four equal pieces and you have one.

The more pieces you cut the cake into, the smaller each piece is, so \(\frac{1}{2}\) is larger than \(\frac{1}{4}\), which is larger than \(\frac{1}{8}\), and so on.

If the denominators of two fractions are the same, the numerator tells you which is bigger. The bigger it is, the bigger the fraction: three-fourths (\(\frac{3}{4}\)) of a cake is more than one-fourth (\(\frac{1}{4}\)).

Using Models

One way to make comparing fractions easier is by using a model to represent the whole and splitting that model into equal parts. For instance, using our cake example, let’s assume the cake is a circle. We could divide that model circle into thirds:

Or we could divide that circle into fifths:

This is one way to model fractions that makes comparisons simple to see. From the two models above, it’s pretty clear that \(\frac{1}{3}\) is larger than \(\frac{1}{5}\).

Reasoning

There are two rules that are generally true:

• The bigger the numerator, the bigger the fraction.
• The bigger the denominator, the smaller the fraction.

You need to be careful, though. Just because one fraction’s numerator is bigger than another fraction’s doesn’t mean it is the bigger fraction. For instance, of the fractions \(\frac{3}{9}\) and \(\frac{2}{4}\), the numerator is bigger in the first. However, \(\frac{3}{9}\) is equivalent to one-third, and \(\frac{2}{4}\) is equivalent to one-half, which is greater than one-third. You must always compare the full fraction.

Fractions as Division

In some division problems, using a fraction can help point you toward the answer. Whenever you have a number, \(a\), that you want to divide by a second number, \(b\), you can just write it as a fraction, \(\frac{a}{b}\). The reduced form of that fraction is your answer.

Think about this pretty simple problem. If three people share a six-pound ham, how much will each person get? Are you thinking \(6 \div 3 = 2\) pounds of ham per person? Or maybe you pictured it this way: \(\frac{6}{3} = 2\). Either way is correct. The point is that a fraction is a different but perfectly fine way to show division.

Now think about what would change if there were five people sharing the ham. Following the second method above, you could simply write \(\frac{6}{5}\) pounds. Most likely, you would want to write it as a mixed number: \(1\frac{1}{5}\).

Estimating Fractions

You should be able to use benchmark fractions to compare two fractions and to estimate the correctness of your answers. Benchmark fractions are ones that are fairly common and simple, such as \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\), and \(\frac{3}{4}\). They are most useful when you need to compare fractions where the answer isn’t obvious.

Here’s how they are used. Suppose you are asked which is bigger, \(\frac{3}{5}\) or \(\frac{2}{3}\). One way to find the answer is to use a pair of number lines with benchmark fractions marked on them. You could do it on one number line, but it gets a little crowded and harder to compare.

The benchmark fractions in the top line are fifths, and the bottom line uses thirds as benchmarks:

The fraction farther to the right is larger, so \(\frac{2}{3}\) is the larger of the two fractions.

Operations with Fractions

As with whole numbers, doing operations with fractions follow specific rules. As long as you know those rules, the operations are simple to complete.

Addition and Subtraction

As you will see in more detail below, the one thing you absolutely need to do when adding or subtracting fractions is ensure they have the same denominator. Once that is done, you need only to add the numerators. Sometimes, you may need to change the result to a mixed number. The one thing you never do is add or subtract the denominators.

Fractions Only

If you’re given fractions with the same denominators, simply add or subtract the numerator and copy the common denominator:

\[\frac{13}{35} + \frac{4}{35} = \frac{17}{35}\] \[\frac{8}{5} - \frac{2}{5} = \frac{6}{5}\]

More often, however, you will be given fractions with different denominators. In such cases, the first step is finding the least common multiple (LCM) of the denominators. A multiple is like the opposite of a factor. It’s what you get when you multiply any number by another number. So, \(15\) is a multiple of \(3\), because \(3 \times 5 = 15\). At the same time, the LCM of \(3\) and \(5\) is \(15\) because that’s the smallest number both numbers fit into.

Once you have the LCM, you determine the equivalent of the fractions with the common denominator, then proceed to addition or subtraction. Let’s try an example.

Solve: \(\frac{1}{7} + \frac{2}{3}\)

Solution

First, let’s find the LCM of \(7\) and \(3\). If you don’t know it off the top of your head, you can write out the first few multiples of each:

• \(\boldsymbol{3}\)\(\text{: } 3, 6, 9, 12, 15, 18, 21, 24…\)
• \(\boldsymbol{7}\)\(\text{: } 7, 14, 21, 28, 35, 42, 49…\)

The LCM of \(3\) and \(7\) is \(21\). To find the fraction equivalent to \(\frac{1}{7}\) with the denominator of \(21\), multiply the fraction by \(\frac{3}{3}\):

\[\frac{1}{7} \cdot \frac{3}{3} = \frac{3}{21}\]

Now, multiply the fraction \(\frac{2}{3}\) by \(\frac{7}{7}\):

\[\frac{14}{21}\]

We may now proceed to adding the fractions with common denominators:

\[\frac{1}{7} + \frac{2}{3} = \frac{3}{21} + \frac{14}{21} = \frac{17}{21}\]

The same method is applied when subtracting fractions with unlike denominators, except that numerators are subtracted in the last step.

Fractions and Whole Numbers

Adding fractions and whole numbers can be done by writing the whole number to the left of the fraction, making a mixed number. For example:

\[2 + \frac{2}{3} = 2 \frac{2}{3}\]

Subtraction with whole numbers and fractions can be done by converting the whole number into a fraction. Just remember that \(1 = \frac{4}{4} = \frac{5}{5} =\frac{12}{12}\) and so on. Likewise, \(2 = \frac{8}{4} = \frac{10}{5}\) and \(3= \frac{12}{4} =\frac{15}{5}\).

Suppose you want to subtract \(\frac{5}{8}\) from \(3\). The easiest way to do this is to convert \(3\) into an improper fraction with the same denominator as \(\frac{5}{8}\). Rewrite \(3\) as \(\frac{24}{8}\) and subtract the numerators:

\[\frac{24}{8}-\frac{5}{8} = \frac{24-5}{8} = \frac{19}{8}=2\frac{3}{8}\]

Note: When doing any operation with fractions, if one of your numbers is a mixed number, it’ll be much easier if you first convert that to an improper fraction. Then you simply do the normal process for two fractions.

Fractions and Mixed Numbers

The addition or subtraction of mixed numbers involves a couple extra steps, but it really isn’t more complex than adding proper fractions. There are two different ways of doing the process. The first way involves converting mixed numbers into improper fractions, as you learned to do earlier.

Take this addition problem for example:

\[2\frac{1}{2} + 3\frac{1}{4}\]

First, let’s convert both to improper fractions:

\[2\frac{1}{2} = \frac{2\times2+1}{2}=\frac{5}{2}\] \[3\frac{1}{4} = \frac{3\times4+1}{4}=\frac{13}{4}\]

Now, we’ll get the same denominator and add:

\[\frac{5}{2} +\frac{13}{4} = \frac{10}{4}+\frac{13}{4}=\frac{23}{4}\]

Converting our improper fraction into a mixed number, we get:

\[23 \div 4 = 5 \text{ R } 3 = 5\frac{3}{4}\]

Now, there is another way to add mixed numbers. You can line up the whole numbers and the fractions like you would with basic addition:

\[\begin{array}{r} 2\frac{1}{2}\\ +3\frac{1}{4}\\ \hline \end{array}\]

Now, make the fractions have the same denominator and add them, and then add the whole numbers:

\[\begin{array}{r} 2\frac{2}{4}\\ +3\frac{1}{4}\\ \hline 5\frac{3}{4}\\ \end{array}\]

As you see, both methods gave us the same answer. We won’t go over a subtraction example here, but just know the process is essentially the same.

Multiplication

You know that when you multiply whole numbers together, you get a result that is bigger than either of the numbers you started with. With fractions, though, you always get a result smaller than either number. For example:

\[\frac{1}{10} \cdot \frac{3}{4} = \frac{3}{40}\]

This makes sense because you’re multiplying by a value less than \(1\). If multiplying by \(1\) results in the same number, and any number greater than \(1\) makes the number bigger, then any number less than \(1\) will logically make the number smaller.

Fractions Only

The process of multiplying fractions is quite straightforward. You multiply the numerators and multiply the denominators to get the product of the fractions. Here is an example:

\[\frac{5}{9} \cdot \frac{2}{7} = \frac{5 \cdot 2}{9 \cdot 7} = \frac{10}{63}\]

Fractions and Whole Numbers

If you multiply a fraction by a whole number, you will always get a result smaller than the original whole number. Multiplying a number by \(\frac{1}{2}\), for example, literally means taking half of that number (or dividing it by \(2\)).

When multiplying a fraction by a whole number, the first step is converting the whole number into a fraction, as you did with adding and subtracting. However, you do not need to worry about the denominators being the same with multiplication. So, for instance, let’s say you have this problem:

\[\frac{4}{5} \cdot 25\]

You’ll change the whole number to \(\frac{25}{1}\), because putting any number over \(1\) doesn’t change its value. Now, multiply the tops and the bottoms and simplify:

\[\frac{4}{5} \cdot \frac{25}{1}\] \[\frac{4 \cdot 25}{5 \cdot 1} = \frac{100}{5} = 20\]

Scaling with Fractions

Multiplication of fractions can be used in scaling down or up a real-world value. To picture this, think about a plastic scale model car. That model car may be made at one twenty-fifth (\(\frac{1}{25}\)) the size of a real car. With what we know about fractions, we can use multiplication to answer questions such as, “If the length of the real car is \(15\) feet, how long will the model be?” This is how that works:

\[\frac{1}{25} \times 15 \text{ ft} =\frac{1}{25} \times \frac{15}{1} = \frac{15}{25} = \frac{3}{5} \text{ ft}\]

The model car would be \(\frac{3}{5}\) foot (three-fifths of a foot) long. This is scaling down.

We can also do the reverse. If that same model is three inches tall, how tall is the real car? In this case, we are going the opposite way so we invert the \(\frac{1}{25}\) and make it \(\frac{25}{1}\):

\[\frac{25}{1} \times 3 \text{ in} =\frac{25}{1} \times \frac{3}{1} = \frac{75}{1} =75 \text{ in}\]

The real car would be \(75\) inches tall. This is scaling up.

Division

As we discussed earlier, division is the inverse operation of multiplication. Doing division with fractions makes that fact very apparent.

Fractions Only

Division of fractions involves two steps. First, you must get the reciprocal of the divisor. That means, you flip (or invert) that fraction. After that step, you do fraction multiplication like normal. So, for instance, say you have this problem:

\[\frac{4}{9} \div \frac{1}{2}\]

You take the inverse of \(\frac{1}{2}\), which is \(\frac{2}{1}\) (or simply \(2\)), and multiply:

\[\frac{4}{9} \cdot \frac{2}{1} = \frac{4 \cdot 2}{9 \cdot 1} = \frac{8}{9}\]

Just as multiplying by a fraction makes the number smaller, dividing by a fraction makes the number larger.

Fractions and Whole Numbers

To divide a fraction by a whole number, picture the whole number written over \(1\), then invert it and multiply. For example:

\[\frac{6}{7} \div 3\] \[\frac{6}{7} \div \frac{3}{1}\] \[\frac{6}{7} \cdot \frac{1}{3} =\] \[\frac{6}{21} = \frac{2}{7}\]

To divide a whole number by a fraction, invert the fraction and multiply as above:

\[4 \div \frac{3}{4}\] \[\frac{4}{1} \cdot \frac{4}{3} =\] \[\frac{16}{3} = 5 \frac{1}{3}\]

Multiplication and Division of Mixed Numbers

As with addition and subtraction, multiplying and dividing mixed numbers is generally easiest if you first convert them into improper fractions. After that, the steps are exactly the same as multiplying or dividing with normal fractions. Consider this problem:

\[3\frac{2}{3} \times 2\frac{1}{4}\]

We will convert both into improper fractions:

\[3\frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{11}{3}\] \[2\frac{1}{4}= \frac{2 \times 4 + 1}{4} = \frac{9}{4}\]

Since we’re multiplying, we don’t need common denominators, we just multiply the top by the top and the bottom by the bottom:

\[\frac{11}{3} \times \frac{9}{4} = \frac{99}{12} = \frac{33}{4}\]

That fraction can be converted back to a mixed number:

\[33 \div 4 = 8\text{ R } 1 = 8\frac{1}{4}\]

Let’s try an example problem.

\[4\frac{1}{2} \div 1\frac{1}{8}\]

Solution

Let’s change both numbers into improper fractions:

\[\frac{2 \times 4+1}{2} \div \frac{8 \times 1+1}{8}\] \[\frac{9}{2} \div \frac{9}{8}\]

Now, because this is division, we’ll invert the second fraction and multiply:

\[\frac{9}{2} \times \frac{8}{9} = \frac{72}{18} = 4\]

Word Problems with Fractions

You will need to be able to solve real-world problems using multiplication of fractions and mixed numbers. Word problems with fractions may give you the numerals (e.g., \(\frac{1}{3}\)) or write the number out (e.g., one-thirds). You should be able to recognize both forms.

Suppose the following are the ingredients in a brownie recipe that makes \(20\) brownies:

• \(1 \frac{1}{4}\) cup of sugar
• \(\frac{3}{4}\) cup of flour
• \(\frac{2}{3}\) cup of cocoa powder
• \(2\) eggs

How much of each ingredient would be needed if you only wanted to make \(10\) brownies? Well, \(10\) is half of \(20\), and to take half of something you multiply it by \(\frac{1}{2}\), so we’ll do that to each ingredient. However, since \(1\frac{1}{4}\) is a mixed number, we need to first change it to an improper fraction:

\[1\frac{1}{4} = 1 + \frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4}\]

Now you can multiply \(\frac{5}{4}\) by \(\frac{1}{2}\). Remember, multiply the top and put the answer on top, then multiply the bottom and put that answer on the bottom. Reduce your answer if possible:

Sugar: \(\frac{1}{2} \times \frac{5}{4} = \frac{5}{8}\)

Flour: \(\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}\)

Cocoa: \(\frac{1}{2} \times \frac{2}{3} = \frac{2}{6} = \frac{1}{3}\)

Eggs: \(\frac{1}{2} \times 2 = 1\)

Those are the new amounts you will need to make half the original recipe.

Let’s try a different type of real-world problem.

Some small engines, like chainsaw engines, require you to mix oil with gasoline. If a gallon of gasoline needs one thirty-second of a gallon of oil, how many gallons can you treat with one half gallon of oil?

Solution

The basic question here is how many \(\frac{1}{32}\) gallons of oil are in \(\frac{1}{2}\) gallon of oil. Because you are dividing something into smaller parts, this is a job for division. Divide \(\frac{1}{2}\) by \(\frac{1}{32}\):

\[\frac{1}{2} \div \frac{1}{32}\] \[\frac{1}{2} \times\frac{32}{1}\] \[\frac{32}{2}\] \[16\]

So, the answer is \(16\) gallons.

Key Words

When doing word problems involving fractions, you should be alert for those key words you learned about earlier. These words will direct you to use certain operations to find the answer.

There are also key words that will let you know that the problem involves fractions. Besides common terms like denominator, numerator, and reciprocal, you’ll want to remember the way fractions are written out (e.g., “one-third,” “two-fifths”) as well as common phrases like half of or a quarter of.

Percentages

Percentages are a way of showing the value of a part compared to a whole, similar to fractions and decimals. However, with a percentage, we are stating the quantity of something for every hundred. “Percent” means “per hundred.”

For instance, \(50\%\) (read “fifty percent”) means there is \(50\) for every \(100\) of whatever we’re discussing. As a decimal it would be \(0.5\), and as a fraction it would be \(\frac{50}{100}=\frac{1}{2}\).

Converting Percentages

You will be expected to be able to convert between percentages, fractions, and decimals. Converting a percentage to a fraction is quite easy. For instance, to change \(40\%\) to fraction form, write \(40\) as the numerator and \(100\) as the denominator, then reduce to the lowest equivalent fraction:

\[40\% = \frac{40}{100} = \frac{2}{5}\]

To convert a fraction to a percentage, you create an equivalent fraction with \(100\) as the denominator. The new numerator is your percentage, and you just have to add the percent sign, \(\%\).

Converting a percentage to a decimal is even easier. You just divide by \(100\) and drop the percent sign:

\[40\% = 40 \div 100 = 0.4\]

To go from a decimal to a percentage, reverse that process by multiplying the decimal by \(100\) and adding the percent sign:

\[0.53 \cdot 100 = 53\%\]

Calculating Percentages

Some TABE questions will require you to compute the percentage of a given number. Remember earlier when we said the word “of” is a multiplication key word? That tells you what you need to do. You convert the first number to a decimal (by dividing by \(100\)) and multiply.

For example, to solve for \(35\%\) of \(60\), multiply \(0.35\) by \(60\):

\[0.35 \cdot 60 = 21\]

Other questions will ask you to determine what percentage a number is of another number. This is essentially the inverse of that previous technique. You multiply the first number by \(100\) and divide by the second. For example, \(77\) is what percentage of \(92\)? It’s simple:

\[\frac{77}{92} = \frac{\%}{100}\] \[\frac{77 \cdot 100}{92} = 83.70\%\]

Note: This is actually a process known as cross-multiplication, which you will learn about later in this study guide.