Quantitative Reasoning Study Guide for the CLT

Mathematical Reasoning

Mathematical reasoning is the process of drawing logical conclusions from a set of given facts or premises using established rules of mathematics. It involves determining what must be true, what cannot be true, and what conclusions follow logically from the information presented. You will encounter problems that require analyzing patterns, applying formulas, and deducing answers based on constraints. Being able to systematically work through these steps helps ensure accuracy in problem-solving.

Logic

Logic is a central element of mathematical reasoning and plays a key role in solving a variety of math problems. It involves understanding the relationships between statements and ensuring that conclusions follow logically from the premises provided. Logical reasoning ensures that your approach to problem-solving is structured, consistent, and accurate.

Conditions

Conditions in mathematical reasoning refer to specific rules or criteria that a number or solution must meet. These problems often involve sets of numbers and require you to evaluate which numbers satisfy one or more conditions. You might be asked to find numbers that are both divisible by a certain value and lie within a given range, or to determine which values satisfy inequalities.

To solve problems involving conditions:

Identify and break down the given conditions.
Eliminate options that don’t satisfy each condition individually.
Verify that the remaining solutions meet every requirement.

For example, consider a problem that requires finding numbers less than $20$ that are divisible by both $3$ and $5$.

To solve this, you must first identify numbers divisible by $3$ and $5$ (i.e., multiples of $15$) and then check which of those numbers are less than $20$. In this case, the answer would be $15$, as it meets both conditions.

Counterexamples

A counterexample is a specific case that disproves a general statement or rule. In many mathematical reasoning problems, you may be asked to identify counterexamples to demonstrate that a statement is not universally true. Counterexamples are essential for understanding the limitations of generalizations and are frequently used in problems requiring you to think critically about the validity of statements.

For example, if a statement says, “All numbers divisible by $4$ are also divisible by $8$,” a counterexample would be the number $4$. Although $4$ is divisible by $4$, it is not divisible by $8$, which proves that the original statement is false.

Finding counterexamples is a key skill in logic and reasoning. It requires you to test general statements by considering specific cases where the statement may not hold true. Once you find a counterexample, you can confidently conclude that the statement is not universally valid.

For example:

statement—”All prime numbers are odd.”
counterexample—The number $2$ is prime but not odd.

Another one could be:

statement—”The sum of any two even numbers is odd.”
counterexample—Since $2 + 4 = 6$, the statement is false.

Note: Sometimes there will be only one counterexample, as with the first example, and sometimes there will be many counterexamples, as with the second.

True and False Statements

When evaluating true and false statements in mathematical reasoning, you must carefully analyze each statement against known mathematical principles or given information. Determining whether a statement is true or false involves verifying that the logic and conditions within the statement hold under all circumstances. If a statement aligns with mathematical rules, it is true; if it contradicts established facts or has exceptions, it is false.

For example:

statement—”All even numbers are divisible by $2$.”
This is true, because even numbers are defined as numbers divisible by $2$.

Another one:

statement—”All multiples of $3$ are odd.”
This is false, because multiples of $3$, such as $6$, $12$, and $18$, are even.

In another type of problem, you might be given a set of numbers and asked to identify which statements about them are true or false based on their properties. For instance:

\[\{2,\,7,\,9\}\]

“All of these numbers are prime.” (This is false because $9$ is not prime).
“At least one number is even.” (This is true because $2$ is even).

The key to solving true/false statement problems is to methodically check each condition or fact and ensure the statement is correct in all cases or identify where it fails.

A similar strategy is useful when dealing with problems that ask about which statement must be true or false. For example, consider the following problem:

A new trash service is being used by a neighborhood. The trash service establishes the following conditions:

Trash pickup will be every other Thursday.
If a holiday falls on the day of a trash pickup, pickup will be moved to the next day.

This week, there is a holiday on Thursday. Which of the statements, if any, must be true?

I. Trash pickup came last week. II. Trash pickup will come on Friday this week. III. Trash pickup came on Friday last week.

Since there is a holiday on Thursday this week, we know that if trash pickup was supposed to happen this week, then it will come on Friday. However, we do not know if pickup is supposed to happen this week. Now, let us look at the provided statements considering that the only thing we know is that there is a holiday on Thursday this week:

“Trash pickup came last week.”

It is possible that trash pickup came last week. However, based only on the given information, this statement does not have to be true.

“Trash pickup will come on Friday this week.”

This could be true. We are told that there is a holiday this Thursday. Therefore, if trash pickup was scheduled for this week, then it would come on Friday. However, since it comes every other week, it is possible that trash pickup occurred last week. Therefore, this statement does not have to be true.

“Trash pickup came on Friday last week.”

This statement could be true if trash pickup was scheduled for last week and there was a holiday last Thursday. However, we don’t know if either of these things are true. Therefore, it could be the case that this statement is not true.

After analyzing each statement, we can conclude that none of them must be true.

Unknown Integer

Sometimes, you will need to deduce the value of an unknown integer based on certain clues or mathematical properties. These types of problems provide information about the unknown number, such as its divisibility, parity (whether it is even or odd), or relationship to other numbers. Your goal is to use logical reasoning to narrow down the possibilities and arrive at the correct solution.

For example, you may be given clues such as:

“The unknown number is divisible by both $3$ and $5$.”
“The number is greater than $20$ but less than $40$.”

To solve this, you first identify the common multiples of $3$ and $5$ (e.g., $15$, $30$, $45$), then eliminate those that do not fall within the range of $20$ to $40$. The answer is $30$, as it meets both conditions.

Here is another example:

“The unknown integer is even, less than $50$, and divisible by $6$.”

To solve, list the multiples of $6$ until you reach $50$: $6, \, 12, \, 18, \, 24, \, 30, \, 36, \, 42, \, 48$. Now, confirm that these meet all the given conditions. As you can see, sometimes multiple numbers will fit the criteria.

Word Problems

On the CLT, some questions will be straightforward, presenting you with only the numbers needed to solve the problem. However, many others will begin with a scenario or story, requiring you to interpret the information before performing any calculations. In these cases, it’s important to identify key details, understand the problem’s structure, and apply the appropriate math skills. These types of problems are designed to assess not only your mathematical ability but also your problem-solving and critical thinking skills.

Drawing Conclusions

When solving word problems, you are often required to draw logical conclusions based on real-life conditions. This involves reading the given information carefully, analyzing it, and deducing the next steps or outcomes. You cannot make assumptions beyond the provided details but you can use clear, logical thinking about those details to arrive at a conclusion.

Here are two examples to illustrate this.

A local courier company guarantees same-day delivery between the hours of $9$ a.m. and $5$ p.m. as long as the package is ready for pickup by $1$ p.m. Starting at $9$ a.m., the company processes $40$ packages per hour to get them ready for pickup. A customer’s package was processed at $11\text{:}30$ a.m. Will this customer’s package definitely be delivered on the same day if the driver requires four hours to complete the deliveries after processing?

Solution

The package was processed at $11\text{:}30$ a.m., meaning that is when the delivery process will begin.

Adding the four-hour delivery time to $1\text1\text{:}30$ a.m. gives a delivery time of $3\text{:}30$ p.m., which is well before the cutoff for same-day delivery.

Thus, even if the delivery takes the full four hours, the package will be delivered on the same day.

This problem demonstrates how to draw logical conclusions based on real-life constraints. By carefully analyzing the timeframes provided and eliminating unnecessary information, you can arrive at the correct conclusion about the delivery timing.

A charity event is selling tickets online for $\$25$ each. If fewer than $100$ tickets are sold, the event organizer plans to cancel the event. As of $2$ p.m. on the day of the event, $75$ tickets had been sold. An average of five tickets are sold per hour.

Based on this trend, if ticket sales close at $6:00$ p.m., will the event be canceled?

Solution

There are four hours remaining before ticket sales close (from $2$ to $6$ p.m.).

At an average rate of five tickets per hour, an additional $4 \times 5 = 20$ tickets will be sold.

Adding the $20$ tickets to the $75$ already sold gives $75 + 20 = 95$ tickets.

Since fewer than $100$ tickets will be sold, the event will be canceled.

This example emphasizes the importance of analyzing trends and using logical thinking to predict outcomes.

Truth Value of Statements

Determining the truth value of statements means identifying whether a given statement is true or false under specific conditions. You must use the provided facts or conditions to decide if a statement can be accepted as true or rejected as false. Let’s look at a couple examples.

condition—All squares are rectangles. condition—Every angle of a rectangle is $90^{\circ}$.

statement—Every angle of a square is $90^{\circ}$.

truth value—This statement is true because every angle of a rectangle is $90^{\circ}$, and since every square is a rectangle, every angle of a square is also $90^{\circ}$. Furthermore, this truth value matches our intuition about what should be the truth value.

Here’s another one:

condition—All lions are cats. condition—Some cats are black.

statement—All lions are black.

truth value—This statement is false because while it could be true that some lions are black, we cannot conclude based on the condition that all lions are black. Again, this matches our intuition, since most lions we can think of are not black.

In each of these cases, the truth of the statement is evaluated based on logical deductions drawn from the given conditions.

Spatial Reasoning

Spatial reasoning involves understanding and manipulating shapes and figures in space, often requiring you to visualize the relationship between objects or their movements. This is important in solving geometric problems that involve positioning, orientation, and measurements of objects. For example, determining how shapes fit together, finding missing dimensions, or calculating the area or volume are common tasks that rely on spatial reasoning.

Let’s look at two examples.

Determine the number of cubes that can fit inside a rectangular box with dimensions of $10$ centimeters by $15$ centimeters by $20$ centimeters and a side length of $5$ centimeters.

Solution

First, find how many cubes fit along each dimension of the box:

In the $10\,\text{cm}$ dimension:

\[10 \div 5 = 2\]

In the $15\,\text{cm}$ dimension:

\[15 \div 5 = 3\]

In the $20\,\text{cm}$ dimension:

\[20 \div 5 = 4\]

Now, multiply the number of cubes along each dimension to find the total number of cubes:

\[2 \times 3 \times 4 = 24\]

Thus, $24$ cubes fit inside the box.

A right triangular garden has sides measuring $5$ meters, $12$ meters, and $13$ meters. Find the area of the garden.

Solution

Since the triangle has sides $5$, $12$, and $13$, this is a right triangle ($5-12-13$ is a Pythagorean triplet).

20 Pythagorean Triplet.png

To find the area of a right triangle, use the formula for the area of a triangle:

\[A = \frac{1}{2}bh\]

Here, the base ($b$) is $12$ meters and the height ($h$) is $5$ meters:

\[A = \frac{1}{2} \times 12 \times 5 = 30 \,\text{m}^2\]

Thus, the area of the triangular garden is $30$ square meters.

Percent of Increase and Decrease

Understanding percent increase and decrease is essential for solving real-world problems involving financial transactions, population changes, and other scenarios where quantities change over time. By calculating these percentages, you can determine how much a value has grown or shrunk compared to the original amount. Let’s look at two examples.

A laptop was originally priced at $\$800$. After a sale, its price increases to $\$960$. What is the percent increase in price?

Solution

To find the percent increase ($PI$) from the old value ($OV$) to the new value ($NV$), use this formula:

\[PI = \left( \frac{NV - OV}{OV} \right) \times 100\]

Substituting in the values, we find the percent increase:

\[\left( \frac{960 - 800}{800} \right) \times 100 = \left( \frac{160}{800} \right) = \left( \frac{1}{5} \right) \times 100 = 20\%\]

A concert ticket originally sold for $\$120$ but is now on sale for $\$90$. What is the percent decrease in the ticket price?

Solution

To find the percent decrease ($PD$), use the formula:

\[PD = \left( \frac{OV - NV}{OV} \right) \times 100\]

Substituting in the values to find the answer:

\[\left( \frac{120 - 90}{120} \right) \times 100 = \left( \frac{30}{120} \right) = \left( \frac{1}{4} \right) \times 100 = 25\%\]

Proportional Concepts

Proportional concepts are important for understanding the relationships between quantities in various mathematical scenarios. In this section, we will look at problems involving proportions, ratios, and rates, all of which are common in real-life situations.

Ratio

A ratio is a comparison of two quantities, showing the relative sizes of two or more values. Ratios can be expressed as fractions, decimals, or with a colon. Understanding ratios is essential in many fields, where comparisons of different quantities are common. Let’s do a problem related to ratio.

A car travels a distance of $120$ miles in two hours. What is the ratio of distance to time?

Solution

The ratio of distance to time can be calculated as follows:

\[D:T = \frac{120 \,\text{m}}{2 \,\text{h}} = 60 \,\text{mph}\]

Proportion

A proportion is an equation that states that two ratios are equal. Proportions are used to find unknown values when two quantities have a constant ratio. To clarify, let’s explore a couple examples.

A recipe requires three cups of flour for every four cups of sugar. If you use six cups of flour, how many cups of sugar should you use?

Solution

Set up the proportion based on the given ratio:

\[\frac{3}{4} = \frac{6}{x}\]

Cross-multiply to solve for $x$:

\[3x = 4 \times 6\] \[3x = 24\] \[x = \frac{24}{3} = 8\]

Thus, you should use eight cups of sugar.

A map indicates that two inches represent $50$ miles. If two towns are five inches apart on the map, how many miles apart are they in reality?

Solution

First, we set up the proportion based on the given information:

\[\frac{2}{50} = \frac{5}{y}\]

Cross-multiply to solve for $y$:

\[2y = 50 \times 5\] \[2y = 250\] \[y = \frac{250}{2} = 125\]

Therefore, the two towns are $125$ miles apart.

Rate

A rate is a specific type of ratio that compares two different units of measurement. Rates are commonly used to express how one quantity changes in relation to another over a specific period. Let’s look at an example of speed rate.

A runner completes a $10$-kilometer race in $40$ minutes. What is the runner’s average speed in kilometers per hour?

Solution

First, convert the time from minutes to hours:

\[40 \,\text{min} = \frac{40}{60} \,\text{hr} = \frac{2}{3} \,\text{hr}\]

Now calculate the runner’s speed by dividing the distance by their speed:

\[\frac{10 \,\text{km}}{\frac{2}{3} \,\text{hr}} = 10 \times \frac{3}{2} = 15 \,\text{km/h}\]

Thus, the runner’s average speed was $15$ kilometers per hour.

Quadratic Functions

A quadratic function is a type of polynomial function that can be used to model real-life scenarios, such as projectile motion, area measurements, and certain economic situations. These functions can be expressed in the standard form $f(x) = ax^2 + bx + c$, where $a, b,$ and $c$ are constants.

The graph of a quadratic function is called a parabola, like this one:

20a Parabola.png

When $a > 0$, the graph of $f(x)$ will have a U-shape and the minimum value of $f(x)$ occurs at the vertex of the parabola. When $a < 0$, the graph of $f(x)$ will have an upside-down U-shape and the maximum value of $f(x)$ occurs at the vertex of the parabola. The vertex of $f(x)$ always has an $x$-value of $x = -\frac{b}{2a}$. Understanding how to analyze quadratic functions allows you to predict outcomes based on given variables.

With quadratic functions it also important to remember the quadratic formula:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Let’s do an example.

A rectangular garden has an area of $240$ square feet. If the length is five feet longer than the width, what are the dimensions of the garden?

Solution

Let $w$ be the width of the garden. Then the length ($l$) can be expressed as:

\[l = w + 5\]

The area of the rectangle is given by the equation:

\[A = l \times w = 240\]

Substituting for $l$ gives us:

\[(w + 5)w = 240\]

This simplifies to the quadratic equation:

\[w^2 + 5w - 240 = 0\]

Now we can use the quadratic formula to find the width:

\[w = \frac{-5 \pm \sqrt{5^2 - 4(1)(-240)}}{2(1)} = \frac{-5 \pm \sqrt{25 + 960}}{2} = \frac{-5 \pm \sqrt{985}}{2}\]

Calculating this gives two possible solutions for $w$:

\[w \approx 13\]

Note: Only the positive solution is valid because width can’t be negative.

Using the width to find the length gives us:

\[l = 13 + 5 = 18\]

Thus, the dimensions of the garden are approximately $13$ feet wide and $18$ feet long.

Rate of Work

In mathematics, the rate of work measures how quickly a task is completed. It is often expressed as a portion of the job done per unit of time. Work-rate problems commonly involve calculating the time it takes for one or more individuals, machines, or systems to complete a task when working together or separately. These types of problems require an understanding of how to combine individual rates. We’ll use an example to explain this concept further.

A machine can complete a task in three hours. If another machine can complete the same task in five hours, how long will it take for both machines to complete the task together?

Solution

First, determine the work rates of each machine:

machine A—$\frac{1}{3}$ of the task per hour
machine B—$\frac{1}{5}$ of the task per hour

Now, to determine the number of tasks per hour that machines A and B can do combined, we add their rates together:

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

To add these fractions, find a common denominator ($15$):

\[\frac{5}{15} + \frac{3}{15} = \frac{8}{15}\]

Thus, together they complete $\frac{8}{15}$ tasks per hour. Since $\frac{8}{15}$ is in tasks/hour, the reciprocal, $\frac{15}{8}$ tells us hours/task. So, it will take $\frac{15}{8} \approx 1.88$ hours for the machines to complete one task together.