# Page 3 Arithmetic Study Guide for the ACCUPLACER® test

#### Fraction and Percent Equivalencies

All fractions can be written as percentages. Likewise, all percentages can be written as fractions.

$\frac{1}{2}$, for example, can be read as “one divided by two,” “one half,” “one over two,” or “the ratio of one to two.”

Actually performing the operation “one divided by two,” will yield the result of $0.50$, which can then be converted to a percentage by multiplying by $100$ and adding a percent sign:

This is read as “fifty percent.” So, $\frac{1}{2}$ can also be thought of, or described as, “fifty percent.” This is the idea underlying fraction and percent equivalencies.

In order to convert a fraction to a percent, perform the division indicated by the fraction. Divide the numerator by the denominator to find the decimal equivalent of the fraction, then multiply this decimal by $100$ (which is the same as moving the decimal two places to the right) and add the percent sign.

To convert a percent to a fraction, divide the percentage by $100$ and simplify. For example:

What is 76% written as a fraction?

Place $76$ over $100$ and reduce: $\frac{76}{100} = \frac{38}{50} = \frac{19}{25}$.

It is helpful to become familiar with a few different percentage and fraction equivalences.

$\frac{1}{1} = 100\%$, $\frac{2}{1} = 200\%$, etc.

$\frac{1}{10} = 10\%$, $\frac{2}{10} = 20\%$, etc.

$\frac{1}{5} = 20\%$, $\frac{2}{5} = 40\%$, etc.

$\frac{1}{4} = 25\%$, $\frac{2}{4} = 50\%$, etc.

$\frac{1}{3} = 33.333…\%$, $\frac{2}{3} = 66.666…\%$

#### Estimation with Percents and Decimals

Percentages and decimals are different ways of expressing the same value.

$50\%$ is equivalent to $0.50$

To convert a percentage to a decimal, drop the percentage sign and divide by $100$. This is the same as moving the decimal point two places to the left.

To convert a decimal to a percentage, multiply by $100$ and add the percent sign. This is the same as moving the decimal point two places to the right.

### Applications and Problem Solving

This topic is a vast one and encompasses operations such as identifying rate, percentages, and measurements, as well as geometry and distribution. When studying for this portion of the test, study rates and ratios and how they are discovered and used. Rates are most easily found in prices: a simple trip to the grocery store will reveal a plethora of rates, such as \$1.50 for one bag of red onions. Ratios simply compare two different numbers in order to create a relationship between them. Practice identifying and developing both rates and ratios to study for this test. Study the different formulas used to achieve percentages, and practice turning different numbers into percentages. A basic knowledge of measurements can assist you in completing this portion of the test that requires some knowledge in measurement. Learn to identify different systems of measurement (pounds, ounces, pints, quarts, etc.) and how they relate to one another. Also study the basic principles of geometry, ranging from how different shapes function to the different types of angles found in geometric operations. Practice identifying angles and the formulas attached to different shapes (area, diameter, etc.).

The test will include items that require you to find the answer to problems by doing the correct procedure for:

#### Finding the Rate

A rate is a relationship between two different types of values. “Miles per hour” and “cost per unit” are some common rates. Consider the expression, “she drove 89 miles in 3 hours.” This expression can also be written as:

$\frac{89\, mi.}{3\, hr}$, which can also be read as “89 miles for every 3 hours,” or “89 miles per 3 hours.”

After generating this expression, we can find the unit rate, or number of miles per hour, by setting up and solving a proportion:

Cross multiply to solve:

When setting up a proportion to solve a rate problem, it is crucial to place the units in corresponding places. Notice that miles is in the numerator on both sides of the equation, and that hours is in the denominator on both sides of the equation.

Let the wording of the question dictate the position of the values in your expression. The word ‘per’ is often used to express the relationship between the values. “Miles per hour,” for example, can be rewritten as a fraction: $\frac{miles}{hour}$.