Mathematics: Levels E, M, D, and A Study Guide for the TABE Test

Number and Operations: Fractions

Percentage of Test Level Specifically Assessing Number and Operations: Fractions (— = Assumed)

L	E	M	D	A
not tested	12%	20%	—	—

Even if you’re not in a math classroom, fractions pop up quite a lot. Can you imagine being a carpenter and not knowing how to read a tape measure, or a cook trying to double a recipe, without having a basic knowledge of fractions?

Fraction Basics

Fractions are all based on dividing something into several parts and focusing on just one or more of the parts. For example, see that the big square below is divided into nine equal, smaller parts. We could say that one of those parts is one of nine parts. In math, we would write it as \(1/9\) or \(\frac{1}{9}\). The bottom number, the denominator, tells us how many total parts there are. The top number, the numerator, tells us how many parts we are focusing on. In this case, 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 \(\frac{4}{9}\) of the big square.

As another example, suppose in a classroom there are \(12\) boys and \(17\) girls. The girls make up what fraction of the class? The total number of parts of the class is \(29\), and \(17\) of them are girls. The girls are \(17\) parts of the whole \(29\) parts, so the fraction we write is \(\frac{17}{29}\).

Fractions on a Number Line

Between the marked whole numbers on a number line, there are so many fractional numbers that it would be impossible to name them all, let alone fit them on the number line. Often, there aren’t specific marks for even common fractions, but they all exist.

For example, 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 and Equivalent Fractions

To compare fractions to each other, you will want to 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.

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

\(\frac{1}{4}\) of a cake means the cake has been cut into four 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 \(\frac{1}{4}\).

If the denominators are the same, the numerator tells you which is bigger. The bigger it is, the bigger the fraction: \(\frac{3}{4}\) of a cake is definitely more than \(\frac{1}{4}\) of a cake.

Using Models

Circle divided into thirds:

Circle divided into fifths:

This is one well-known way to model fractions. It’s pretty clear that \(\frac{1}{3}\) is larger than \(\frac{1}{5}\).

Reasoning

These have been mentioned earlier, but remember two things about comparing fractions:

• The bigger the top number, the bigger the fraction.
• The bigger the bottom number, the smaller the fraction.

Use these facts to tell which is larger, \(\frac{5}{9}\) or \(\frac{4}{10}\).

• \(5\) is larger than \(4\), so by itself, that would tend to make \(\frac{5}{9}\) larger.
• \(9\) is smaller than \(10\), so that too would make \(\frac{5}{9}\) larger.

Both factors point toward \(\frac{5}{9}\) being larger, so it is.

Equivalent Fractions

Two fractions are equivalent if they can be reduced to the same fraction. For example, \(\frac{3}{4}\) is equivalent to \(\frac{6}{8}\) because \(\frac{6}{8}\) 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.

This figure shows the square divided into sixteen parts. Twelve of the sixteen parts are shaded, so we can 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. Here’s one way: Start with the given fraction and multiply the top and bottom by the same number. Any number will do.

\[\text{Start with } \frac{3}{4} \text{ and multiply the top and bottom by }4\text{:}\] \[\frac{3 \times 4}{4 \times 4} = \frac{12}{16}\] \[\frac{12}{20}\text{ is equivalent to }\frac{3}{5}\]

The other way that can work is to reduce the fraction if you can. In this case, you divide the top and bottom by the same number.

\[\text{Start with } \frac{12}{20} \text{ and divide the top and bottom by }4\text{:}\] \[\frac{12 \div 4}{20 \div 4} = \frac{3}{5}\] \[\frac{12}{20}\text{ is equivalent to }\frac{3}{5}\]

Percents

Presenting a number in the percent form is a way of showing the value of a part compared to a whole, or the quantity for every hundred. Percent can be converted to a fraction or a decimal and back. For instance, \(50\%\) is a way to show that there is \(50\) for every \(100\), or \(0.50\), or \(\frac{1}{2}\) of something.

To change \(40\%\) to fraction form, write \(40\) as the numerator and \(100\) as the denominator, then reduce to the lowest equivalent fraction:

\[40\% = \frac{40}{100} = \frac{4}{10} = \frac{2}{5}\]

To change \(40\%\) to its decimal form, simply divide \(40\) by \(100\):

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

Some TABE questions will require the computation of the percentage of a given number. For example, to solve for \(35\%\) of \(60\), simply multiply \(0.35\) by \(60\):

\[0.35 \cdot 60 = 21\]

Percent questions are often simplified by this useful equation:

\[\frac{is}{of} = \frac{\%}{100}\]

Example: \(77\) is what \(\%\) of \(92\)?

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

Word Problems with Fractions

Be able to solve real-world problems using multiplication of fractions and mixed numbers (a whole number with a fraction). For example, suppose the following are the ingredients in a brownie recipe:

\(1 \frac{1}{4}\) 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 the recipe was cut in half?

To take half of something, you multiply it by \(\frac{1}{2}\), so do that to each ingredient.

But first, since \(1\frac{1}{4}\) is a mixed number, change it to an improper fraction: \(1\frac{1}{4} = 1 + \frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4}\).

Now, use that \(\frac{5}{4}\) and multiply 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: \(\frac{1}{2} \times 2 = 1\)

Key Words

Be sure that you can add and subtract fractions. The main thing to do first is to make sure that the fractions have the same denominator (bottom number). Then, just add or subtract the numerators (top numbers). Here is an example.

James fills up his car with gas and drives until he’s used a quarter tank of gas. Antoine takes over and drives, using another \(\frac{3}{8}\) of a tank. In total, how much of the tank did they use up?

The key word here is total. It tells us to add.

\[\frac{1}{4} + \frac{3}{8}\]

We need both fractions to have the same bottom number. We could turn \(\frac{1}{4}\) into eighths by multiplying the top and bottom by \(2\):

\[\frac{2}{2} \times \frac{1}{4} = \frac{2}{8}\]

Now we can do our addition by adding the numerators:

\[\frac{2}{8} + \frac{3}{8} = \frac{5}{8}\]

So, James and Antoine used up \(\frac{5}{8}\) of the tank.

The next question is how much of the gas is left in the tank?

You should be able to reason this way: a full tank is \(\frac{8}{8}\), so subtract \(\frac{5}{8}\):

\[\frac{8}{8} - \frac{5}{8} = \frac{3}{8}\]

Fractions as Division

In some division problems, using a fraction can help point you toward the answer. First, think about this pretty simple problem. If three people share a six-pound ham, how much will each person get? Are you thinking \(6 \div 3 = 2\) lb of ham per person? Or maybe you pictured it this way: \(\frac{6}{3} = 2\). Either way is good. 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 \(\frac{6}{5}\) lb, and then, most likely, you would want to write it as a mixed number: \(1\frac{1}{5}\) lb of ham.

The point of this is to show that any time you have a number, \(a\), that you want to divide by a second number, \(b\), you can just write \(\frac{a}{b}\) and simplify it.

Estimating Fractions

You should be able to use benchmark fractions to compare two fractions and to estimate the correctness of problem 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, such as comparing \(\frac{3}{5}\) to \(\frac{2}{3}\).

Here’s how they are used. Suppose, as above, you are asked which is bigger, \(\frac{3}{5}\) or \(\frac{2}{3}\). You can use a pair of number lines with benchmark fractions marked on them. You could do it one 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

Some math problems may require that you perform operations with fractions. There are rules for doing this to find the correct answer that you must remember to follow.

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 to make them have the same denominator. Once that is done, you need only to add the numerators and often change the result to a mixed number. Never add the denominators.

Fractions Only

Fractions with the same denominators:

Add or subtract the numerator of fractions with the same denominators as usual, then copy the common denominator.

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

Fractions with different denominators:

To add or subtract fractions with unlike denominators, first find the least common multiple (LCM) of the denominators. Find the equivalent of the fractions with the common denominator, then proceed to addition or subtraction.

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

The LCM of \(7\) and \(3\) 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 \(\frac{7}{7}\) to get the fraction of \(\frac{14}{21}\).

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

Combining fractions with whole numbers using addition 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 can be done by remembering that \(1 = \frac{4}{4} = \frac{5}{5} =\frac{12}{12}\) and so on. Likewise, \(2 = \frac{8}{4} = \frac{10}{5} \text{ and } 3= \frac{12}{4} =\frac{15}{5}\).

Suppose you want to subtract \(\frac{5}{8}\) from \(3\):

\[3-\frac{5}{8}\]

Rewrite \(3\) as \(\frac{24}{8}\) and subtract the numerators:

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

Multiplication

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

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

Fractions Only

Multiplying fractions is quite straightforward: multiply the numerators and multiply the denominators to get the product of the fractions. Here is another 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 or mixed number, you will always get a result smaller than the whole or mixed number. Multiplying something by \(\frac{1}{2}\), for example, is the same as taking \(\frac{1}{2}\) of that something.

Here is how to handle multiplying a whole number by a fraction.

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

Change the whole number to itself over \(1\) and multiply the tops and the bottoms, then simplify:

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

Fractions as Scaling

Multiplication of fractions can be used in “scaling down” an object. To picture this, think about a plastic scale model car. Very often, a model car will be made that is \(\frac{1}{25}\) the size of a real car. 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?”

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

The model car would be \(\frac{3}{5} \text{ft}\) long. This is scaling down.

We could do the reverse. If the model is \(3\) 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 \(75 \text{ in }\) tall. This is scaling up.

Division

You should be able to divide fractions by other fractions. Also, divide fractions by whole numbers and divide whole numbers by fractions.

Fractions Only

Division of fractions involves two steps. First, get the reciprocal of the divisor. Then, proceed to multiply the two fractions:

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

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:

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

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

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