Math and Logic Study Guide for the CCAT

Word Problems

There are a variety of word problems on the CCAT, and they involve a number of different math skills. Here are some examples of the types of word problems you are likely to see.

Problems Involving Money

One of the common types of word problems on this test is problems involving money. This section will equip you with the tools to navigate financial scenarios and make informed calculations.

Adding and Subtracting Money

The only difference between adding and subtracting money and doing regular addition and subtraction is the decimal point, which must be kept aligned. This is simple if all the numbers are in the same format, such as \(\$3.98, \$42.75,\) and \(\$123.16\). You simply write them in column form, being sure the decimal points are aligned.

\[\begin{array}{r} 3.98 \\ 42.75 \\ 123.16 \\ \end{array}\]

Occasionally, however, you may be presented with money amounts that are written in different formats, such as \(\$11.37\) and \(\$180\). In this case, to align the decimal points in the column, you’ll need to write \(\$180\) as \(\$180.00\).

Multiplying Money

Multiplying money involves understanding the significance of decimal points and using precision in your financial calculations. Let’s look at an example.

You are shopping for a party. One of the items you buy is cheese dip at \(\$2.95\) per container. If you buy three containers, how much will it cost?

Solution

When we buy three of an item priced at \(\$2.95\) each, the cost is \(\$2.95 \times 3\). When multiplying decimal numbers, you do the multiplication as usual, ignoring the decimal point at first:

\[\begin{array}{cr} &295\\ \times\!\!\!\!\!\!&3\\ \hline & 885\\ \end{array}\]

Now, count the number of decimal places in both of the numbers that were multiplied. There are two in \(2.95\) but none in \(3\), so move the decimal point two places to the left, resulting in \(\$8.85\).

If you didn’t pay attention to the decimal point when multiplying, you could get an answer that is inaccurate, such as \(\$885.00\) or \(\$88.50\).

Multi-Step Money Problems

Multi-step money problems often mirror real-life financial scenarios, requiring a systematic approach. Let’s tackle a common multi-step problem involving expenses and savings.

Imagine you have a monthly budget of \(\$1\text{,}500\). After allocating \(\$600\) for rent and \(\$250\) for groceries, you plan to spend \(20\%\) of the remaining budget on entertainment. You want to determine how much you have left to put in savings after your expenses.

You can calculate the savings amount in three steps.

First, subtract rent and groceries from the monthly budget:

\[1\text{,}500 - 600 - 250 = \$650\]

This is the remaining amount. Now, find \(20\%\) of this amount:

\[0.2 \times 650 = \$130\]

This is the entertainment expense.

Deduct the entertainment expense from the remaining budget to determine the savings amount:

\[650 - 130 = \$520\]

This example showcases the application of multiple mathematical steps in practical financial decision-making. By mastering these skills, you can be confident in your ability to tackle money-related challenges with precision and accuracy.

Problems Using Percentages

Word problems involving percentages not only test your mathematical skills but also require practical reasoning. Successfully solving percentage problems necessitates a solid understanding of percentages, along with proficiency in arithmetic operations and the ability to interpret and solve multi-step problems. Developing these skills can help you make informed decisions in various real-world scenarios, from financial planning to understanding trends in data.

Price Increase Problems

Price increase problems are common applications of percentages that involve calculating the new price of an item after a given percentage increase. These problems are often set in retail and business scenarios where changes in prices directly impact financial outcomes.

For example, if an item originally priced at \(\$50\) experiences a \(20\%\) increase, we need to calculate the new price to make a buying decision. The formula for calculating the new price is straightforward: multiply the original price by one plus the percentage increase (expressed as a decimal):

\[\text{new price} = \text{original price} \times (1 + \text{percentage increase})\]

In the above case, we will do the following:

\[\text{new price} = 50 \times (1 + 20\%)\] \[\text{new price} = 50 \times (1 + 0.2)\] \[\text{new price} = 50 \times 1.2\] \[\text{new price} = \$60\]

Alternatively, you could have found \(20\%\) of \(\$50\) (\(.2 \times 50 = 10\)) first and then added it to \(\$50\). It would have given you the same answer.

Percentage in a Multi-Step Problem

Percentages also play a role in multi-step problems. Whether it’s determining discounts, tax amounts, or final prices after multiple changes, percentages are an important component. In a multi-step problem, you might encounter a situation where you need to calculate successive percentage changes. This involves applying one percentage change to a value and then using the updated value to calculate another percentage change. Let’s look at an example.

A shirt is originally priced at \(\$60\). The store offers a \(25\%\) discount on the shirt, and you also have a coupon for an additional \(15\%\) off the discounted price. Calculate the final price of the shirt after both discounts.

Solution

Let’s calculate the discount on the shirt:

\[0.25 \times 60 = \$15\]

So, after the discount, the shirt is worth \(\$60 - \$15 = \$45\).

Now, there is an additional discount of \(15\%\) on top of this because of your coupon. Let’s find the results of this discount:

\[0.15 \times 45 = \$6.75\]

Thus, after the coupon is applied, the final price of the shirt becomes:

\[\$45 - \$6.75 = \$38.25\]

Note: Always read the problems carefully. If this problem had said the coupon gave an additional \(15\%\) off the original price, the answer would have been different.

Multi-Step Problems

Navigating multi-step problems requires a strategic approach to problem-solving, combining different mathematical skills and operations. These questions often test one’s ability to integrate various concepts seamlessly. If you encounter a multi-step problem that appears challenging at first glance, it can be beneficial to skip it initially and return later if time allows. This test-taking strategy ensures that you use your time effectively.

Time Elapse and Clock Terms

Time-related questions on the CCAT will require you to understand time elapse and related clock terms, particularly as they pertain to reading a clock and calculating intervals. For these problems, you should be familiar with terms like elapsed time, time interval, and time duration. All three terms refer to an amount of time between two clock readings.

You may be given different types of information to solve this type of problem, though. For example, the two clock readings may be stated, and you will be asked to determine how much time has elapsed. Alternatively, you may be given just one clock reading and the amount of time that has elapsed and be asked to determine the second clock reading. Consider a problem like this:

Karen starts watching a movie at \(8\text{:}45\) p.m. that lasts for two hours and \(15\) minutes. When will the movie finish?

To find the movie’s ending time, we will need to add \(2\) hours and \(15\) minutes to the starting time, \(8:45\). This is done in two steps:

First, we add \(2\) hours to the original \(8\) hours, which gives us \(8 + 2 = 10\) hours.

Now, adding \(15\) minutes to \(45\) minutes, we get \(45 + 15 =60\) minutes, which is equal to \(1\) hour.

So, the final time will be \(10 + 1 = 11\) hours, which is \(11\text{:}00\) p.m.

Capacity and Rate

Capacity and rate problems are two common types of multi-step problems on this test Capacity refers to the amount that a container can hold, often measured in units like liters or gallons. Rate, on the other hand, involves how quickly a quantity changes over time. Let’s try a practice problem that illustrates these concepts.

A tank is being filled with water at a rate of \(5\) liters per minute. If the tank can hold a maximum of \(120\) liters, how long will it take to fill the tank to its capacity?

Solution

To solve this, we can use this formula:

\[\text{time} = \frac{\text{amount}}{\text{rate}}\]

Here, the amount is \(120\) liters, and the rate is \(5\) liters per minute. So, let’s find the time:

\[\text{time} = \frac{120}{5}\] \[\text{time} = 24\]

So, it will take \(24\) minutes for the tank to be filled to its maximum capacity.

Time and Distance

Time and distance problems are a type of rate problem and are common in mathematical assessments, including the CCAT. These questions gauge your ability to handle the relationship between time spent traveling and the corresponding distance covered. Essentially, you’ll be dealing with scenarios involving speed, time, and distance. Let’s break down this concept with a straightforward example.

Ruby and Theodore are each driving their cars across the country. If Ruby covers a distance of \(288\) miles in four hours and Theodore covers a distance of \(375\) miles in five hours, who is driving faster and by how much?

Solution

This is the basic formula that relates speed, time, and distance:

\[\text{distance} = \text{rate} \times \text{time}\]

First, let’s find Ruby’s rate (speed):

\[\text{rate} = \frac{\text{distance}}{\text{time}}\] \[\text{rate} = 288 \div 4 = 72\,\text{mph}\]

Now, let’s calculate Theodore’s rate:

\[\text{rate} = 375 \div 5 = 75\,\text{mph}\]

So, Theodore’s rate is faster than Ruby’s by \(75 - 72 = 3\) miles per hour.

Combinations

Combinations are a fundamental concept of probability and counting. Combinations represent the number of ways to select items in a set without considering the order of selection. This concept is particularly useful in scenarios where the arrangement of elements is irrelevant and the focus is solely on choosing a group of items from a larger collection. Combinations find practical use in various real-world scenarios, such as team formation, choosing representatives, and creating committees.

To find all possible combinations that can be made by taking one item each from multiple sets, simply multiply the total number of items in each set by each other.

Let’s try an example.

An exercise instructor is going to choose one item from each of four sets of similar equipment for each of her students. How many different combinations can she make with this array of equipment choices?

\(4\) balls \(3\) striking objects \(5\) ropes \(6\) elastic bands

Solution

Because order does not matter, we will multiply the number of ways to choose a single item from each of the four sets:

There are four ways to choose a ball, three ways to choose a striking object, five ways to choose a rope, and six ways to choose an elastic band. Multiplying these together, we get:

\[4 \times 3 \times 5 \times 6 = 360\]

Therefore, there are \(360\) possible combinations.

Logical Thinking

The assessment of your logical thinking skills is another part of the CCAT. Questions of this type assess your ability to analyze information, recognize patterns, and draw accurate conclusions based on provided statements or assumptions. These test questions are of two distinct types, each challenging the test-taker’s ability to infer logical connections and outcomes.

The first type of question involves assumptions, and you must identify whether two given assumptions logically lead to a specific conclusion. The second type of question, statements, requires evaluating the logical consequences of two provided statements. Let’s explore each of these question types to understand their nuances.

Assumptions

In this question type, you’ll find scenarios with two given assumptions and a conclusion. You are tasked with determining, given both assumptions are true, if the conclusion is thereby true. You will be asked to determine if the conclusion is correct, incorrect, or cannot be determined based on the available information.

Let’s work through an example.

Based on the given assumptions, determine if the conclusion is correct, incorrect, or cannot be determined?

Assumptions:

• Everyone who completes the math course receives a certificate.
• Susan completed the math course.

Conclusion:

• Susan received a certificate.

Explanation

In this scenario, we have two assumptions: the first states that everyone who completes the math course receives a certificate, and the second states that Susan completed the math course. The conclusion states Susan received a certificate.

The conclusion is correct.

We can confirm this by turning the conclusion and assumptions into “if/then” statements (“if a, then b”) and see if they contradict. Note that we are using the conclusion as the “then” part of the statement:

• If Susan completed the math course, then Susan received a certificate.
• If everyone who completes the math course receives a certificate, then Susan received a certificate.

Not only do the if/then statements not contradict each other, but when taken together they are true.

Let’s try another example.

Based on the given assumptions, determine if the conclusion is correct, incorrect, or cannot be determined?

Assumptions:

• All Americans listen to country music and watch football.
• Tom watches football but only listens to rock music.

Conclusion:

• Tom is an American.

In this scenario, we have two assumptions: the first states that all Americans listen to country music and watch football, and the second states that Tom watches football but only listens to rock music. Now, we must evaluate the conclusion, which is that Tom is an American.

The conclusion is incorrect.

Again, by creating if/then statements (using the conclusion as the “then” part), we can confirm this answer:

• If all Americans listen to country music and watch football, then Tom is an American.
• If Tom watches football but only listens to rock music, then Tom is an American.

These two statements contradict each other. Tom watches football, but we are told he only listens to rock music, not country, so he can’t be an American based on the given assumptions.

Statements

In statement questions, test-takers must ascertain whether a specific conclusion follows or doesn’t follow logically from the statements, or if it cannot be determined based on the information given. This type of question assesses the individual’s ability to deduce logical relationships and draw valid conclusions from the information presented. Let’s look at a simple example.

Based on the given statements, determine if the conclusion follows, does not follow, or cannot be determined from the given statements?

• Everyone who ate cake at Rhoda’s party enjoyed the party.
• Mark did not eat cake at Rhoda’s party.

Conclusion:

• Mark did not enjoy Rhoda’s party.

Explanation

We can answer this question by thinking of the different statements as creating sets and subsets. From the first statement, we know there is a subset of people who went to Rhoda’s party and enjoyed the party. That is the group who ate cake. However, just because that subset of people enjoyed the party, that does not mean they were the only people who enjoyed the party. Therefore, even though Mark is not part of the subset of people who ate cake, we don’t know if he did or did not enjoy the party.

The conclusions cannot be determined from the given information.