Page 3 Arithmetic Reasoning Study Guide for the ASVAB

Fractions

Fractions represent part of a whole and are written in the form: $\frac{numerator}{denominator}$ where numerator is the number of parts of the whole, and the denominator represents the whole number of parts:

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 (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: $\frac{5}{20}$

To multiply two fractions, multiply the numerator times 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 $\frac{3}{10}$

To divide two fractions, take the reciprocal of the fraction in the denominator and multiply it by the fraction in the numerator. Reciprocal is the inverse of something. To take the reciprocal of a fraction is to invert it so that the denominator becomes the numerator and the numerator becomes the denominator.

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 ten in the (unseen) denominator (the whole).

A decimal like 0.045 has an unseen denominator equal to a power ten, 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: $\frac{45}{1000}$

Scientific Notation

A number written in scientific notation is expressed as some integer whose absolute value is greater than 1 and less than 10, times some power of 10. Writing numbers in this form is useful in describing and talking about very large or very small numbers. A number like 13,549,000,000 is hard to say and harder to understand, but scientific notation makes it easy by putting the magnitude of that number in the power of ten. $1.3549\cdot 10^{10}$ is easier to grasp than the long string of numbers.

Exponents

An exponent tells us how many times to multiply a number by itself: $2^3$ tells us to multiply two by itself three times: $2\cdot 2\cdot 2 =8$ There are three important rules to remember when simplifying and evaluating expressions having exponents:

For a number or variable multiplied by itself, the exponents can be added.

General rule: $a^m\cdot a^n = a^{m+n}$

Example: $2^2\cdot 2^3 = 2^{2+3} = 2^5$

For a number or variable divided by itself, the exponents can be subtracted.

General rule: $a^m\div a^n = a^{m-n}$

Example: $3^5\div 3^2 = 3^{5-2} = 3^3$

For a number or variable raised to a power, then to a power again, the exponents can be multiplied.

General rule: ${(a^m)}^n = a^{m\cdot n}$

Example: ${(2^2)}^3 = 2^{2\cdot 3} = 2^6$

A radical is similar to an exponent but in the opposite direction; it tells us how many times some number is multiplied by itself to get the given number. A radical is written with this symbol $\sqrt{}$ and a number in its small corner (if greater than 2), which is called the index. The most common radical is the square root of some number. This tells us we want to know what number multiplied by itself equals the given number.

A radical can also be written using a fraction as the power the base number is raised to. The denominator, like the index, tells us we want to know the number, which when multiplied by itself that number of times , equals the base number.

The square root of a number does not show an index because it is implicit: $\sqrt{4}$. This is the same as using $\frac{1}{2}$ as the power the base number is raised to.

Factorials

A factorial is made using the exclamation point symbol (!). It tells us to multiply a counting number by decreasing increments from itself in whole units, starting with one less than the original number, until the number one is reached.

For example, 6!, which is spoken and read as “six factorial”, is:

$6\cdot 5\cdot 4\cdot 3\cdot 2 \cdot 1$
$30\cdot 4\cdot 3\cdot 2\cdot 1$
$120\cdot 3\cdot 2\cdot 1$
$360\cdot 2\cdot 1$
$720$

Six is multiplied by descending increments of whole counting numbers, starting with one less than itself.