Mathematics: Quantitative Reasoning Study Guide for the TSIA2

Rational and Irrational Numbers

All numbers on a number line, whether integers or between integers, are either rational numbers or irrational numbers.

Rational numbers are those that can be expressed as a fraction with integers for a numerator and a denominator.

Examples: $\dfrac{5}{12}, \ \dfrac{184}{3}, \ \dfrac{-20}{1}, \ \sqrt{16}, \ 7$

Note that any integer can be written as a fraction with a denominator of $1$, so all integers are rational.

Irrational numbers are those that can’t be expressed exactly as a fraction made with integers.

Take $\sqrt{2}$ for example. There is no fraction made with integers that can be squared to give exactly two. A calculator will tell you the $\sqrt{2}$ is approximately $1.414$, but if you change that to a fraction ($\frac{1414}{1000}$) and square it, you won’t get $2$. (You’ll get $1.99396$.)

Other examples of irrational numbers: $\sqrt{3}, \ \sqrt{5}, \ \sqrt{6}, \ \sqrt{7}, \ \pi$

Math Operations with Rational Numbers

These are the common arithmetic operations with whole numbers, integers, fractions, and decimals that were covered earlier in this guide.

Math Operations with Irrational Numbers

The math operations with irrational numbers are similar to those of rational numbers, but are complicated a bit by having to use additional methods to simplify the results when square roots are involved.

Ratios, Proportions, and Percents

You need to be able to write ratios and percents, and write and solve proportions, especially to solve problems.

Ratios

A ratio is a two-termed expression that is used to compare one value to another. A ratio is commonly written using a colon such as $3:5$, but can just as well be written as a fraction, $\frac{3}{5}$. In fact, ratios written as fractions are generally easier to do calculations with than the versions with a colon.

A percent can also be thought of as a ratio. Gasoline that is $10 \%$ alcohol, can be said to have an alcohol to fuel ratio of $10:100$ or $1:10$.

Example 1

Suppose we have a car that uses $11.5$ gallons of gasoline to drive $264.5$ miles. Find the ratio of miles driven to gallons of gas and simplify the ratio.

In the ratio statement “miles driven to gallons,” the quantity named first, miles, is written as the numerator and the second named quantity is written as the denominator.

$\dfrac{264.5}{11.5} = \dfrac{23}{1} = 23$ miles per gallon.

Although it’s not as commonly done, we could do the reverse ratio. That is, we could find the number of gallons used per mile.

$\dfrac{11.5}{264.5} = \dfrac{1}{23} = 0.043$ gallon per mile.

Example 2

If $6 \text{ ft}^3$ of seawater weighs $384 \text{ lb}$, what is the ratio of lb to cubic feet?

The quantity lb is named first, so it will be the numerator, and $\text{ ft}^3$ will be in the denominator.

\[\dfrac{384\text{ lb}}{6\text{ ft}^3}\] \[\dfrac{64 \text{ lb}}{\text{ ft}^3}\]

This ratio has a physical meaning: a cubic foot of seawater weighs $64$ lb.

We could just as well invert the ratio to give us the volume of $1$ lb of seawater.

\[\dfrac{6\text{ ft}^3}{384\text{ lb}}\] \[\dfrac{0.016\text{ ft}^3}{\text{ lb}}\]

Proportions

A proportion is simply an equation with a ratio on each side. This shows that the two ratios are equivalent.

Example: $\dfrac{3}{8} = \dfrac{15}{40}$

The most common use of a proportion is where a ratio is known, but we are looking for a new term to make a second ratio equal to the given one. A proportion is a very useful tool in problem solving.

Using a Proportion to Solve a Problem Example 1

To solve a proportion, cross multiply and divide both sides of the equation by the coefficient of $x$.

\[\dfrac{7}{12} = \dfrac{x}{18}\] \[12 x =126\] \[x = \dfrac{126}{12} = 10.5\]

Using a Proportion to Solve a Problem Example 2

When solving word problems, be careful to write the proportion terms in the right order.

The shadow length of an object is proportional to its height. If a $16$ ft pole casts a $24$ foot shadow, how tall is a tree that casts a $40$ ft shadow?

Write the proportion so that you keep the order the same on both sides of the equal sign. For example, you could write it this way:

\[\dfrac{\text{pole height}}{\text{pole shadow}}=\dfrac{\text{tree height}}{\text{tree shadow}}\] \[\dfrac{16}{24} = \dfrac{x}{40}\] \[24 \times x = 40 \times 16\] \[x = \dfrac{40 \times 16}{24}\] \[x = 26.7 \text{ft}\]

Ratios and Percents

Ratios and percents are easily converted back and forth. To convert percent to a ratio, just remember that percent means per hundred. That means that $5$% is the same as $\frac{5}{100}$ or, simplified, $\frac{1}{20}$.

Changing a ratio to a percentage is a tiny bit more work. First, divide the denominator into the numerator, then move the decimal point two places to the right.

\[\dfrac{6}{8} = 0.75 = 75 \%\]

Solving Problems

A common use for percents is in calculating the amount of interest earned on investments or paid out for loans. One type is called simple interest, and it is calculated using $I=prt$, where:

$I$ = interest
$p$ = principal (the amount borrowed or invested)
$r$ = interest rate (as a decimal)
$t$ = time (the number of times interest is paid)

Other financial calculations sometimes use simple linear equations to deal with money issues.

Problem Solving Example 1

A used car that you have your eye on has a price of $\$12,000$ and a relative will lend you the money at a simple interest rate of $7 \%$ for a time of 3 years. At the end of the three years, how much will you have paid for your car?

First, calculate the amount of interest, then add it to the purchase price.

\[I=prt\] \[I = \$12,000 \times 0.07 \times 3\] \[I = \$ 2520\] \[\$12,000 + \$2520 = \$14,520\]

Problem Solving Example 2

To calculate $s$, a salesperson’s total yearly salary, a company uses the linear equation $s = 32,000 +0.25 t$, where $t$ is the total amount of sales for that person. If a salesperson wants a salary of $\$74,000$, what amount will she have to sell?

\[s = 32,000 +0.25 t\] \[74,000 = 32,000 + 0.25 t\] \[74,000 - 32,000 = 0.25 t\] \[42,000 = 0.25 t\] \[\dfrac{42,000}{0.25} = t\] \[168,000=t\] \[\$168,000\]

See the next section for a review of how to solve equations.

Linear Expressions, Equations, and Inequalities

The word linear here means that these are all relating to a graph of a straight line. It also means that these expressions, equations, and inequalities have no variable exponents greater than 1.

What Is the Difference?

Expressions have one or more variables with operations to be performed, such as $3x + 4$. An equation is a statement that two expressions are equal, such as $y + 1 = x - 2$. An inequality states a relationship where one expression is greater or less than the other. Inequalities can also include the relationships “less than or equal to” and “greater than or equal to.” Here are two inequalities:

\[2 + y \lt 42\] \[15 \le 6 + x\]

Interpretation

The algebraic properties and procedures that follow essentially lay out the rules and methods of algebra. You might say that they form the logical skeleton on which the structure of algebra is built. There is more to the skeleton, but these are the basics.

Algebraic Properties

There are a number of algebraic properties of real numbers. The three major ones are the following, in which $a, b, \text{ and } c$ are real numbers.

Commutative property for addition and multiplication: $a+b=b+a$ and $ab=ba$
In plain language, order doesn’t matter when you are adding or multiplying.
Associative property for addition and multiplication: $(a+b)+c = a+(b+c)$ and $(a \cdot b)c=a(b \cdot c)$
In plain language, how you group doesn’t matter when you are adding or multiplying.
Distributive property for multiplication over addition: $a(b+c)=ab+ac$
In plain language, the outside number is multiplied times each number in the parentheses and the results are added. Reminder it is not true that $a(b \cdot c) = a \cdot b \cdot a \cdot c$.

Algebraic Procedures

A big part of algebra is solving equations, and a big part of solving equations is manipulating them using these properties of equality:

Addition, Subtraction, Multiplication, and Division Properties of Equality

Symbolically,

If $a=b$, then $a+c=b+c$
If $a=b$, then $a-c=a-c$
If $a=b$, then $a \cdot c = b \cdot c$
If $a=b$, then $a \div c = b \div c$ where $c \neq 0$

In plain language, you can add, subtract, multiply, or divide both sides of an equation by any number (though you can’t divide by zero) and the equation is still true. These properties are also true for inequalities with one big exception. If you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol.

You can draw conclusions using the transitive property. If $a=b$ and $b=c$, then $a=c$.

Example: If $x-3=y \text{ and } y = 2x+2 \text{, then } x-3=2x+2$.

This is closely related to the substitution property: if $a=b$ and $x=b$, then $a=x$.

Example: If $y=x+8$ and $x = -11$, then $y=-11+8$.

Order of Operations

When an expression has a variety of operations, you must follow the order of operations to get the correct result. The order of operations is this:

Do everything in parentheses (P), left to right.
Evaluate any exponents (E), left to right.
Do all multiplication and division (MD), in order, left to right.
Then do all addition and subtraction (AS), in order, from left to right.

Good way to remember: Please Excuse My Dear Aunt Sally

Manipulation

As mentioned above, equations and inequalities can be manipulated using algebraic procedures, often in multiple steps using several different properties.

Example: Solve $3(x-8)=x+10$

Use the distributive property on $3(x-8)$.

\[3x-24=x+10\]

Subtract $x$ from both sides.

\[3x-x-24=x-x+10\] \[2x-24=10\]

Add $24$ to both sides.

\[2x-24+24=10+24\] \[2x=34\]

Divide both sides by $2$.

\[2x \div 2 = 34 \div 2\] \[x=17\]

Algorithms

A known series of steps to follow to accomplish a task is known as an algorithm. The method of solving the simple linear equations above uses an algorithm that could be summarized by these steps after the distributive property is used.

Use the addition and subtraction properties of equality to remove subtracted or added variable terms from the right side of the equation.
Use the addition and subtraction properties of equality to remove subtracted or added numbers from the left side of the equation.
Use the multiplication and division properties of equality to remove any variable’s divisors or factors from the left side of the equation.

These steps should leave you with a single variable on the left and a number on the right. Each step transforms the equation into a new, but equivalent equation. Equivalent means that each subsequent equation has the same solution as the original one.

More complex equations, of course, will have more complex algorithms.

Transformation

As mentioned above, solving an equation uses a series of steps that change the equation into progressively simpler forms. In mathematics this is known as transforming an equation.

Evaluation

Algebraic expressions have been described earlier, but it’s important to note that unlike equations, expressions cannot be solved, but they can be evaluated by substitution.

Example: If $x=-7$, evaluate $3(x+12)$.

\[3(x+12)\] \[3(-7+12)\] \[3(5)\] \[15\]