Arithmetic Reasoning Study Guide for the ASVAB

Page 3

Fractions

Fractions represent part(s) of a whole and are written in the form:

\[\frac{\text{numerator}}{\text{denominator}}\]

where the numerator is the number of parts of the whole, and the denominator represents the whole number of parts. For example: If Jimmy ate three out of five apples in a bag, he ate \({3\over 5}\) of the apples.

Types of Fractions

Mixed Number: A fraction that contains a whole number and fractional part. If Johnny ate one whole bag containing five apples and also ate three out of five apples in a second bag, the fraction this represents would be mixed:

\[1\frac{3}{5}\]

Improper Fraction: With an improper fraction, the numerator is bigger than the denominator. To convert a mixed fraction to an improper fraction, multiply the denominator by the whole integer part, then add this to the numerator. This is the new numerator of the converted improper fraction. The original denominator is retained. For example, with \(1\frac{3}{5}\), multiply 5 by 1, and add this product to 3 to get:

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

To convert an improper fraction to a mixed fraction, divide the denominator into the numerator to find the whole number part. The remainder from this division becomes the numerator:

\[\frac{8}{5} = 1 \frac{3}{5}\]

Reciprocal Fraction: To take the reciprocal of a fraction is to invert it so that the denominator becomes the numerator and the numerator becomes the denominator.

The reciprocal of \(\frac{3}{7}\) is \(\frac{7}{3}\).

Operations with Fractions

You will find that you’ll need to manipulate fractions in a number of ways to answer some of the questions on this test. Here are the basics.

Equivalent Fractions

Equivalent fractions look different at first glance, but are actually the same in value. For instance, \(\frac{1}{2}\) is equivalent to:

\[\frac{7}{14}, \frac{4}{8}, \frac{2}{4}, \frac{3}{6}, \frac{5}{10}\]

They are all equivalent fractions.

In a question, you may need to find a number that is equivalent to \(\frac{1}{2}\) but has a denominator of 10.

You will get the missing numerator by using this relationship:

\[\frac{1}{2} = \frac{\text{numerator}}{10}\]

You will find the numerator by cross-multiplying 10 and 1, which gives 10, and dividing 10 by 2, which gives 5. The numerator is 5.

This gives you the equivalent fraction of \(\frac{5}{10}\).

Simplifying and Reducing to Lowest Possible Terms

During or after performing other operations with fractions, you may have to “reduce” or “simplify” fractions. Here’s what that means and how to do it.

Many questions require simplifying fractions. It means reducing a fraction to its simplest form.

Take the following fractions:

\[\frac{3}{6}, \;\frac{5}{10},\; \frac{7}{14},\; \frac{4}{8},\; \frac{2}{4}\]

They all actually reduce to \(\frac{1}{2}\). Did you find it easy simplifying those fractions to \(\frac{1}{2}\)? If not, here’s how:

Let’s take the fraction: \(\frac{7}{14}\)

Factor the numerator and denominator:

\[\frac{(7 \cdot 1)}{(7 \cdot 2)}\]

The 7s in the numerator and denominator cancel out, leaving the reduced fraction:

\[\frac{1}{2}\]

Addition and Subtraction of Fractions

When adding or subtracting two fractions having different denominators, the fractions may have to be converted into equivalent fractions so that they have the same denominator. Then, the numerators can be added or subtracted.

The conversion is done by finding the lowest common denominator (the least common multiple [LCM]), then dividing the original denominator into the new denominator, and finally multiplying this quotient by the original numerator to get a new numerator.

For example, with \(\frac{1}{4} + \frac{3}{20}\), the LCM is 20, so \(\frac{1}{4}\) must be converted into an equivalent fraction with 20 as its denominator.

Divide 4 into 20 to get 5, and then multiply 5 by 1 to get the new numerator in our equivalent fraction:

\[\frac{5}{20}\]

Multiplication of Fractions

To multiply two fractions, multiply the numerator by the numerator to get a new numerator, then multiply the denominator by the denominator to get a new denominator. Reduce the result to lowest possible terms if it is not in this form already.

Example:

\[\frac{3}{8}\cdot \frac{4}{5} =\frac{12}{40}\]

Reduce this to get:

\[\frac{3}{10}\]

Division of Fractions

To divide two fractions, take the reciprocal of the divisor (second) fraction and multiply it by the dividend (first) fraction.

Example: \(\frac{2}{5}\div \frac{3}{7} = \frac{2}{5} \cdot \frac{7}{3}\)

Decimals

Decimals, like fractions, also represent parts of a whole. But decimals, unlike fractions, always have some power of 10 in the (unseen) denominator (the whole).

A decimal like 0.045 has an unseen denominator equal to a power 10, which is based on the place value of the last digit furthest to the right.

One place to the right of the decimal, the denominator is \(10^1\) or \(10\). Two places to the right, the denominator is \(10^2\) or \(100\). Three places to the right, the denominator is \(10^3\) or \(1,000\), and so on.

The decimal 0.045 can be written in fraction form like so: \(\frac{45}{1000}\)

Here are the place values of commonly-used decimal values.

Ones Tenths Hundredths Thousandths Ten-Thousandths
1 0.1 0.01 0.001 0.000 1

Operations with Decimals

Performing basic operations with numbers having decimal places is much the same as doing so with whole numbers. You just have to be very aware of the placement of the decimal point in the answer. Here’s how to accomplish that.

Addition and Subtraction

To add or subtract decimals, the thing to remember is to write the numbers in such a way as to have the decimal points line up vertically. Also, it’s good to add zeros as needed to the right of the given numbers to make both numbers have the same amount of decimal places. The decimal point in the result goes directly under the other two decimal points. For example, add \(16.33\) and \(5.2\).

Add a zero to \(5.2\), line up the decimal points, do column addition as you normally would, and put the decimal point directly under the given two decimal points.

\[\begin{array}{r} &16.33\\ &+5.20\\ \hline &21.53\\ \end{array}\]

Now, try subtracting \(1.53\) from \(19.7\).

Add a zero to \(19.7\), line up the decimal points, do subtraction as you normally would and put the decimal point directly under the given two decimal points.

\[\begin{array}{r} &19.70\\ &-1.53\\ \hline &18.17\\ \end{array}\]

Multiplication

Multiplying decimals uses only two steps.

  1. Ignore the decimal points and multiply normally.

  2. Count the total number of decimal places in the original numbers and place a decimal point in the result so that it has that total number of decimal places.

For example, multiply \(0.07 \times 1.8\). There are three decimal places (don’t count the leading zero in \(0.07\)).

\[\begin{array}{r} &1.8\\ &\times 0.07\\ \hline &0.126\\ \end{array}\]

Now try multiplying \(1.73\) and \(0.042\).

\[\begin{array}{r} &1.73 \\ &\times 0.042\\ \hline &346\\ &692 \phantom{6}\\ \hline &0.07266\\ \end{array}\]

Notice that, ignoring decimals, we get \(7266\) for an answer, but we need five decimal places. To get five, we needed to add a zero to the left of the \(7\).

Division

The game plan for decimal division is threefold.

First, move the decimal points. In the divisor (what you are dividing by) move the decimal point to the right as many places as you need to make the divisor a whole number. Then move the decimal point in the dividend (what you’re dividing into) that same number of places. Add zeros if you need to.

Second, write the decimal point directly above where you wrote it in the dividend.

Third divide as usual, being careful to keep your numbers nicely in columns.

For example, divide \(1.7\) by \(0.2\):

\[\require{enclose} \begin {array}{r} 0.2 \enclose{longdiv}{1.7}\\ \end{array}\]

Move the decimal points one place to the right.

\[\require{enclose} \begin {array}{r} 2 \enclose{longdiv}{17}\\ \end{array}\]

Divide as usual, adding zeros to the dividend if needed.

\[\require{enclose} \begin {array}{r} 8.5\\ 2 \enclose{longdiv}{17.0}\\ \underline{16} \phantom{0}\\ 10\\ \ \underline{10}\\ \end{array}\]

Here’s another example:

\[\require{enclose} \begin {array}{r} 0.035 \enclose{longdiv}{1.54}\\ \end{array}\]

Move the decimal points to the right three places and divide normally.

\[\require{enclose} \begin {array}{r} 44\\ 35 \enclose{longdiv}{1540}\\ \underline{140} \phantom{0}\\ 140\\ \underline{140}\\ \end{array}\]

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