# Mathematics: Quantitative Reasoning Study Guide for the TSIA2

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## General Information

There is only one math test as part of each of the TSIA2 CRC and Diagnostic Tests, but the questions on both tests assess skills in four areas of math:

- Quantitative Reasoning
- Algebraic Reasoning
- Geometric and Spatial Reasoning
- Probabilistic and Statistical Reasoning

Depending on your score on the CRC test, you may or may not have to take the Diagnostic Test in math immediately following it. The skills tested are the same on both tests, so it’s best to prepare as if you’ll need to take both.

This study guide provides an outline of and basic information about the **quantitative** area of math. There are **six questions** from this area of math on the **Math CRC Test** and **12** of them on the **Diagnostic Test**. Be sure to access additional help if you need it and use our other three math study guides for complete preparation. We also have free practice questions and flashcards in each math area to aid in your studies.

All items on the math tests are **multiple-choice questions**.

## Math Operations

The math operations below include common **adding**, **subtracting**, **multiplying**, and **dividing** of integers, fractions, percents, and decimals.

### Math Symbols

Here is a table of symbols that may be used in this section of the test. You should know what they all mean.

Symbol | Meaning | Symbol | Meaning | Symbol | Meaning |
---|---|---|---|---|---|

\(+\) | Plus | \(<\) | Less than | \(=\) | Is equal to |

\(-\) | Minus | \(>\) | Greater than | \(\neq\) | Is not equal to |

\(\times\) or \(\cdot\) | Times | \(\leq\) | Less than or equal to |
\(\sqrt{}\) | Square root of |

\(\div\) | Divided by | \(\geq\) | Greater than or equal to |
\(|\ |\) | Absolute value of |

### Whole Numbers

Whole numbers **start at 0 and continue upward forever**. They are the numbers you would naturally use to **count** things with.

{0, 1, 2, 3, 4, 5, 6 … }

They *don’t* include negative numbers, fractions, or decimals.

Being able to perform the four basic operations (addition, subtraction, multiplication, and division) with whole numbers forms the basis for all other math operations. You should be extremely comfortable with this, but if you’d like to check your skill or know you need some review, see pages 2 and 3 of our Math Basics Numbers and Operations Study Guide for complete procedures and details.

### Integers

Integers include all of the **whole numbers plus their negative counterparts**. In other words, they start at 0 and increase in the positive direction and decrease in the negative direction forever.

{ … -4, -3, -2, -1, 0, 1, 2, 3, 4 …}

Like whole numbers, they *don’t* include fractions or decimals.

#### Addition and Subtraction of Integers

Know how to add and subtract positive and negative integers. This is sometimes very difficult for students to consistently get right at first. Below is a condensed version of how to do this.

**Addition**—If the signs of the two numbers are the **same**, simply add the two numbers and give the result the **sign that both numbers have**.

Example: \(-11 + (-9) = -20\)

If the two signs are **different**, temporarily ignore any negative sign, subtract the smaller numeral from the larger numeral and give the result the original **sign of the larger numeral**.

Example: \(-11 + 9\)

Ignore the negative sign and subtract: \(11-9=2\)

Give the \(2\) the original sign of the \(-11\) and get \(-2\).

**Subtraction**—**Change the sign of the second number**, then **add**, following the rules of addition. It may help if you remember these phrases:

**Subtracting a negative**is the same as**adding a positive**.**Subtracting a positive**is the same as**adding a negative**.

Example 1: \(15 -(-6) = 15 +(+6) = 21\)

Example 2: \(-21 -(+7) = -21 +(-7) = -28\)

#### Multiplication and Division of Integers

Know how to multiply and divide positive and negative integers. The rules can be summarized in two simple sentences:

- If the
**signs**on the two integers are the**same**, the result is**positive**. - If the
**signs**are**different**, the result is**negative**.

Example 1: \(-12 \times -7 = 84\) The two signs are the same, so the result is positive.

Example 2: \(\dfrac{28}{-7} = -4\) The two signs are different, so the result is negative.

### Fractions

A fraction is a **statement of division between two integers**, written as \(\frac{a}{b}\), where \(a\) is the **numerator** and \(b\) is the **denominator**.

#### Fraction Representation

Fractions can be represented in several forms.

In the fraction \(\frac{a}{b}\), if \(a\) is the larger of the two integers, the fraction is called an **improper fraction**.

Examples:

\[\frac{4}{3}\; \text{and} \;\frac{24}{9}\]A **mixed number** is a fraction paired with a whole number, as in these representations:

A quantity greater than 1 can be represented by either an improper fraction or a mixed number, as shown here:

\[\frac{5}{4} = 1\frac{1}{4}\]#### Conversion Between Mixed Numbers and Improper fractions

Improper fractions can always be converted to a **mixed number** (a whole number with a fraction).

To convert an improper fraction to a mixed number, divide the denominator into the numerator and you will get a quotient consisting of a whole number and a remainder. Write down the whole number and to its right write the remainder over the original denominator.

Example: Convert \(\dfrac{32}{9}\) to a mixed number.

\(32\) divided by \(9\) gives \(3\) with a remainder of \(5\).

The resulting mixed number is \(3\dfrac{5}{9}\).

To convert a mixed number into a fraction, multiply the integer portion times the denominator. Then add the numerator to the result. This gives the numerator of the improper fraction. The denominator is the same as the original denominator.

Example: Convert \(2\dfrac{7}{8}\) to an improper fraction.

\(2\) times \(8\) is \(16\). Add \(7\) to get \(23\).

The resulting improper fraction is \(\dfrac{23}{8}\).

#### Addition and Subtraction of Fractions

To add or subtract fractions, they must have the **same denominator**. If they don’t, they must be rewritten so that they do. This is done by multiplying the numerator and the denominator by an integer chosen to make the denominators match. The matching denominator is the **least common multiple** of the two denominators.

Example: Add \(\dfrac{3}{4}\) and \(\dfrac{2}{3}\).

Choose 3 to multiply the top and the bottom of \(\dfrac{3}{4}\).

\[\dfrac{3}{3} \times \dfrac{3}{4} = \dfrac{9}{12}\]Now choose 4 to multiply the top and the bottom of \(\dfrac{2}{3}\).

\[\dfrac{4}{4} \times \dfrac{2}{3} = \dfrac{8}{12}\]Now add \(\dfrac{9}{12}\) and \(\dfrac{8}{12}\).

\[\dfrac{9}{12} + \dfrac{8}{12} = \dfrac{17}{12} = 1\dfrac{5}{12}\]To add mixed numbers, first add the integer portions, then add the fractional portions and then add the two results. If the fractions add to an improper fraction, convert back to a mixed number and add the integer to the sum of the other integers. The remaining fraction is the fraction portion of the mixed number

Example: \(2\dfrac{1}{3} + 5\dfrac{3}{4}\).

\[2 + 5 = 7\] \[\dfrac{1}{3} + \dfrac{3}{4} = \dfrac{4}{12} + \dfrac{9}{12} = \dfrac{13}{12} = 1\dfrac{1}{12}\] \[7 + 1\dfrac{1}{12} = 8\dfrac{1}{12}\]To subtract mixed numbers, it is best to convert them to improper fractions first and then subtract. This will avoid any complicated “borrowing” process.

Example: \(3\dfrac{2}{3} - 1\dfrac{1}{2}\)

\[\dfrac{11}{3} - \dfrac{3}{2} = \dfrac{22}{6} - \dfrac{9}{6} = \dfrac{13}{6} = 2\dfrac{1}{6}\]#### Multiplication of Fractions

Multiplying fractions is easier than adding or subtracting them. **No common denominator is needed.** All you need to do is **multiply the two numerators** to get a new numerator and **multiply the two denominators** to get the new denominator. Then, simplify your answer, if possible.

Any improper fractions you get should be changed to mixed numbers.

Any mixed numbers you start with should be changed to improper fractions before multiplying.

\[1\dfrac{2}{3} \times \dfrac{5}{6}\] \[\dfrac{5}{3} \times \dfrac{5}{6}\] \[\dfrac{25}{18}\] \[1\dfrac{7}{18}\]#### Division of Fractions

Once you have multiplication down, division is straightforward. **Invert the second fraction and multiply** the first fraction by the inverted one.

#### Reducing or Simplifying Fractions

Simplifying is also known as **reducing to lowest terms**. A fraction in lowest terms has no common factor between numerator and denominator. Simplify a fraction by finding **common factors** in both the numerator and denominator and dividing that factor into both of them.

Simplifying also means to convert an improper fraction to a mixed number. Fractions should always be simplified in the final answer to a problem.

Example:

Simplify \(\dfrac{9}{24}\).

The number \(3\) is a factor of both \(9\) and \(24\).

\(3 \times 3 = 9\) and \(3 \times 8=24\)

Dividing both \(9\) and \(24\) by \(3\) leaves you with \(\dfrac{3}{8}\).

#### Cross-Simplification

If you are multiplying it might be more convenient to leave the problem in fraction form and look to simplify with other fractions before multiplying. In a fraction multiplication problem, any numerator can be simplified with any denominator. For example:

\[\dfrac{3}{1} \times \dfrac{2}{15}\]Divide 3 in the first numerator into itself and 15 in the second denominator.

The resulting problem is:

\(\dfrac{1}{1} \times \dfrac{2}{5}\) = \(\dfrac{2}{5}\)

This method is especially helpful if the numbers in the problem are larger:

\[\dfrac{7}{4} \times \dfrac{16}{21}\]Divide 7 in the first numerator into itself and 21 in the second denominator. Divide 4 in the first denominator into itself and 16 in the second numerator.

The resulting problem is:

\(\dfrac{1}{1} \times \dfrac{4}{3}\) = \(\dfrac{4}{3}\) = 1\(\dfrac{1}{3}\)

This is much easier than trying to simplify \(\dfrac{112}{84}\)!

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