Math and Logic Study Guide for the CCAT

General Information

One of the three types of questions on the Criteria Cognitive Aptitude Test (CCAT) assesses your skills in mathematics and logic. No higher-level math is necessary to answer these questions, and a calculator is not allowed. This is fine, though, because one should not be needed. The emphasis is on utilizing basic numeration (number and operations) skills as well as some very basic algebra skills to solve word problems. The key is to do so very quickly, since you will only have an average of about 18 seconds to answer each question.

In this study guide, we have outlined some of the most important math and logic skills you may need to use on the test. If you have trouble with any of them, you should seek additional practice if you have time before taking the test.

Remember, though, that there are 50 questions on this test, and you are only given 15 minutes to try to answer all of them. Since the average number of questions completed is around 24, you most likely will not answer them all. You’ll be trying to answer the easier ones first, and the more of the following skills you are familiar with, the more questions will seem easy to you.

Numeration

Numeration is a fundamental concept in mathematics. It provides a structured framework for the representation and organization of numbers. In essence, it encompasses the methods of naming, ordering, and categorizing numbers, laying the groundwork for various mathematical operations and problem-solving strategies. Numeration is the bedrock for advanced mathematical concepts and applications.

Decimals

Decimals are a fundamental aspect of our number system, providing a way to express parts of a whole in a more precise manner than whole numbers alone. A decimal number includes both a whole number part and a fractional part, separated by a decimal point. For example, in the decimal $123.456$, “$123$” is the whole number part, and “$456$” is the fractional part. The position of each digit after the decimal point determines its place value.

1 Place Value Chart (3).png

Above is a place value chart that shows the place values of three digits both before and after the decimal point.

The first digit after the decimal point is in the tenths place, the second digit after the decimal point is in the hundredths place, and the third digit after the decimal point is in the thousandths place.
The first digit to the left of the decimal point is in the ones place, the second digit to the left of the decimal point is in the tens place, and the third digit to the left of the decimal point is in the hundreds place.

Note: Place values continue infinitely in both directions from the decimal point, but for the purposes of the CCAT, you should only need to know the first few.

Let’s look at an example problem that involves identifying place value in a decimal number.

In the number $871.209$, what are the place values of the digits $7$ and $9$?

Solution

The digit $7$ is the second digit to the left of the decimal point. From the place value chart, we see that it has a place value of ten. The digit $9$ is the third digit to the right of the decimal point. So, per the chart, its place value is thousandths.

Percentages

In mathematics, percentages are a way to express a fraction of a whole in terms of $100$. The word percent means “per $100$,” and the symbol $\%$ represents percent. To understand percentages better, let’s consider some simple examples.

If you have a pizza divided into $10$ slices and you ate two slices, you would have consumed $20\%$ of the pizza.
Similarly, if your mom has $\$100$ and tells you to take $40\%$ of that amount, you will take $\$40$.

Working with Percentages

Percentage problems involve various scenarios where you will need to either find what percentage one number is of another number or calculate a certain percentage of a given number.

Finding What Percentage One Number Is of Another Number

The problems you will encounter on this test will generally have this form:

What percentage is $x$ of $y$?

Mathematically, this type of question can be represented as $\frac{x}{y} \times 100\%$.

Let’s try an actual problem.

If you were given $20$ apples out of a total of $50$, what percentage of the total apples do you have?

To find the percentage of apples you have out of the total, you can use the following formula:

\[\text{percentage}=\frac{\text{number of apples you have}}{\text{total number of apples}}\times 100\]

In this case:

\[\%=\frac{20}{50}\times100\] \[= 0.4\times 100\] \[=40\%\]

So, you have $40\%$ of the total apples.

Calculating a Certain Percentage of a Given Number

This type of problem will be expressed like this:

What is $x\%$ of $y$?

Mathematically, it can be expressed as $\frac{x}{100} \times y$.

Let’s try an actual problem.

What is $15\%$ of $80$?

To find $15\%$ of $80$, you can use the following formula:

\[\text{percentage of a number}= \frac{\text{percentage}}{100}\times \text{number}\]

In this case, it would be:

\[15\% \,\text{of}\, 80 =\frac{15}{100}\times 80\] \[= 0.15 \times 80\] \[=12\]

So, $12$ is $15\%$ of $80$.

Converting Percentages

In some questions you will be given a bar chart or a circle graph from which to extract information to convert to percentage form. Consider this graph of students’ favorite colors:

2 Bar Graph (fixed).png

This is a two step problem. If you were asked to find the percentage of students from the class who like green best, you would first need to find the total number of students.

The chart shows that green is the favorite color of five students, blue is the favorite of $13$ students, purple is the favorite of eight students, yellow is the favorite of four students, orange is the favorite of $10$ students, and red is the favorite of seven students. So, the total number of students is:

\[5 + 13 + 8 + 4 + 10 + 7 = 47\]

Now, let’s input the two numbers into the formula from earlier to find the percentage of students who like green:

\[\frac{5}{47} \times 100 = 10.63 \approx 11\%\]

So, green is the favorite color of approximately $11\%$ of the students.

In other situations, we will need to convert percentages given on a graph to numbers.

Let’s use the chart below to do an example problem.

3 Circle Graph (fixed).png

The results of a survey of $200$ people show their favorite foods in this chart. How many people prefer hot dogs?

From the circle graph, we see that $30\%$ of all the people surveyed prefer hot dogs over any other food. So, we need to find $30\%$ of $200$:

\[30\% \text{ of } 200 =\frac{30}{100}\times 200\] \[= 0.3 \times 200\] \[= 60\]

So, hot dogs are the favorite food of $60$ of the $200$ surveyed people.

Using the same chart, we could similarly calculate the number of people who prefer each food option.

Fractions

Fractions represent a part-to-whole relationship and consist of two main parts: the numerator (top number) and the denominator (bottom number). The numerator indicates how many parts of the whole we have, while the denominator represents the total number of equal parts the whole is divided into.

For example, in the fraction $\frac{2}{5}$, $2$ is the numerator, indicating we have two parts of the whole, and $5$ is the denominator, representing the total parts in the whole. The top and bottom numbers are separated by a small horizontal line, which is known as the fraction bar.

4 Fraction Parts.png

Fractional Drawings

Graphical representations, or fractional drawings, are tools for visually understanding and solving problems related to fractions. In these drawings, the parts that are either shaded or unshaded in a figure correspond to the numerator and denominator of a fraction.

Let’s consider the example below. A circle is divided into eight equal parts, and five of those parts are shaded:

5 Circle Divided into Fractions.png

By observing the fractional drawing, you can see there are five shaded parts (numerator) and eight total parts (denominator). Therefore, the fraction representing the shaded area is $\frac{5}{8}$. Moreover, the fraction representing the unshaded part is $\frac{3}{8}$, since three out of the eight sections are unshaded.

Sometimes, it requires a bit of mental math to extract the information you need from a fractional drawing. Suppose you are given the image below, and a question asks, “What fraction of the figure is shaded?”

6 Box with Fractional Parts (fixed).jpg

You can see that the entire figure is first divided into fourths, but only one-half of two boxes are shaded.

So, to figure out the shaded part of the whole, mentally divide the entire box into eighths, like this:

7 Box Divided into Fractions (fixed).jpg

There are now eight parts, and it’s easy to see that two of the eight parts are shaded in the original figure, which equals:

\[\frac{2}{8}\]

Generally, for your answers, you will want to reduce the fractions if possible. That means lowering the numerator and denominator to their simplest forms by dividing them by their shared greatest common factor (GCF). In this case, reducing the above fraction yields:

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

Comparing Fractions

Comparing fractions involves determining which fraction is larger or smaller. When fractions have the same denominator, comparing them is straightforward. The larger fraction is simply the one with the larger numerator. For example:

\[\frac{14}{23} > \frac{8}{23}\]

However, when fractions have different denominators, additional steps are needed.

Let’s look at an example.

Which of the fractions $\frac{4}{15}$ and $\frac{5}{16}$ is larger?

First, we need to give both fractions the same denominator. We do this by finding the least common multiple (LCM) of $15$ and $16$. In this case, that is simply the product of the two numbers: $15 \times 16 = 240$.

Let’s make each fraction have a denominator of $240$:

\[\frac{4}{15} \times \frac{16}{16} = \frac{64}{240}\] \[\frac{5}{16} \times \frac{15}{15} = \frac{75}{240}\]

Now that both fractions have the same denominator, we can simply compare the numerators to find out which original fraction is larger:

Since $75 > 64$, we know $\frac{75}{240} > \frac{64}{240}$.

Consequently, $\frac{5}{16} > \frac{4}{15}$.

Using Tables

Tables are a graphical representation of data organized in rows and columns. Being able to interpret and use information presented in tabular form is a helpful skill in various problem-solving scenarios, especially on standardized tests like the CCAT. Let’s look at a couple examples.

Product	Quantity Sold	Unit Price (in $)
A	50	10
B	30	15
C	20	20
D	40	8

From the table above, which product generated the highest revenue?

To find the revenue of each product, we take the quantity sold of each product and multiply it by the unit price:

product A: $50 \times 10 = \$500$
product B: $30 \times 15 = \$450$
product C: $20 \times 20 = \$400$
product D: $40 \times 8 = \$320$

Thus, product A generated the most revenue.

Sometimes, the questions will require you to alter the information given in a table. Let’s try another example that also references the table above.

If the company offers a $10\%$ discount on product C, how much revenue will be generated if the same number of units are sold?

To answer this question, we need to first find the discounted price of product C. This requires converting the percentage into a decimal, which is $0.90$. We can drop the trailing $0$, so:

\[0.9 \times 20 = \$18\]

If $20$ units are sold at $\$18$ per unit, the revenue will be:

\[20 \times 18 = \$360\]

Number Series

In mathematics, working with number series (or sequences) involves identifying patterns and rules used in those sequences of numbers. This skill is used to predict or find the next number in a given series. Understanding the various patterns that can occur in number series is key. Let’s explore some common patterns and rules:

addition series—In an addition series, each number is obtained by adding a constant value to the previous one. Here is an example:

\[2, \,5, \,8, \,11, \,14…\]

A common type of problem may ask you to determine the next number in this series. As with any number series question, look for clues in the numbers that are provided. You can see that each number in this sequence is $3$ added to the previous number. So, the next number in the series would be $14 + 3 = 17$.

subtraction series—The numbers in this type of series are obtained by subtracting a constant value from the previous one. A series of this type might be:

\[25, \, 20, \,15, \,10 …\]

To get the next number in the series, we subtract $5$ from the previous number: $10 - 5 = 5$. So, $5$ would be the next number in this series.

multiplication series—A multiplication series involves multiplying each number by a constant value to get the next one. An example of this type of series is:

\[2, \, 4, \, 8, \,16 …\]

You can see that each successive number is the previous number multiplied by $2$, so $16 \times 2 = 32$ will be the next number in the series.

division series—In this type of series, each number is obtained by dividing the previous one by a constant value, such as this series:

\[243, \, 81, 27, 9…\]

Based on the given numbers, you should see that each successive number is found by dividing the previous number by $3$, which means the next number in the series will be $9 \div 3 = 3$.

In number series, patterns can extend beyond basic arithmetic operations, incorporating more intricate sequences such as squares, cubes, and roots. Here are a couple other patterns:

square series—In a series like $1, \, 4, \, 9, \,16, \,25...$, each number is the square of its position in the series. For instance, $9$ is in the third position of the sequence, and $3^2=9$.

root series—This type of series is exemplified by $1, \sqrt{2}, \sqrt{3}, 2, \sqrt{5}...$ In it, each number is the square root of its position. So, if you were asked for the ninth number in the series, you would know that it would be $\sqrt{9}$, which is $3$.

Determining the series rule requires a keen eye for the underlying pattern. Whether it’s addition, subtraction, multiplication, division, squares, cubes, or roots, once the pattern is discerned, you can leverage it to predict any number in the sequence.

Averages

An average or a mean provides a composite central value for a set of numbers. Calculating the average involves adding all the values in a data set and then dividing by the number of values. This is expressed by the formula:

\[\text{average} = \frac{\text{sum of values}}{\text{number of values}}\]

Let’s see how this concept can be employed in a multi-step word problem.

Ronald’s farm produced $20$ liters of milk on Wednesday, $25$ liters on Thursday, and $35$ liters on Friday. Betty’s farm, on the other hand, produced $15$ liters of milk on Wednesday, $20$ liters on Thursday, and $65$ liters on Friday. Whose farm produced the higher average liters of milk per day?

Solution

First, let’s find the average amount of milk produced per day for Ronald’s farm by adding the values for the three days:

\[\text{sum} = 20 + 25 + 35 = 80\]

Now, we divide that sum by $3$ (the number of values) to find the average:

\[\text{average} = 80 \div 3 \approx 26.67\]

On average, Ronald’s farm produces approximately $26.67$ liters of milk a day.

Let’s do the same for Betty’s farm. First, the sum:

\[\text{sum} = 15 + 20 + 65 = 100\]

Now, we divide that sum by $3$ to find the average:

\[\text{average} = 100 \div 3 = 33.33\]

On average, Betty’s farm produces $33.33$ liters of milk a day. So, Betty’s farm produces more milk per day than Ronald’s farm.