If you’re struggling with ATI TEAS Math: Numbers and Algebra, don’t worry – you’re not alone!

This section of the test can be tricky, but with a little bit of practice, you can boost your scores.

In this blog post, we will discuss some tips that will help you improve your performance on the ATI TEAS Math: Numbers and Algebra section. We’ll also provide a few practice problems to help get you started.

## Objectives for Numbers and Algebra

**Total scored items on ATI TEAS:** 18 questions out of 34

#### Table of Contents

## Converting among non-negative fractions, decimals, and percentages

### Relationship between the numerator and denominator in a fraction

Fractions can be written in the form a/b, where a and b are integers, and b is not equal to zero. Integers are the set of whole numbers and their opposites.

The bottom integer is called the **denominator**, and the top integer is called the **numerator**. The line between them represents division: a/b means “a divided by b.”

For example, in the fraction ¾, the numerator is three (it is on top), and the denominator is four (it is on the bottom).

### How to Calculate a Percentage

A percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign: %.

For example, 35% is equivalent to the decimal 0.35, or the fraction 35/100. To calculate a percentage, multiply the decimal by 100, or divide the fraction by 100 and reduce it to the lowest terms.

### Identifying Place Values within Decimals

Every digit in a number has a place value associated with it. The place value of a digit tells you what that digit is worth in relation to other digits in the number.

For example, the number 1,234 has four digits with place values of ones, tens, hundreds, and thousands.

The digit in the one’s place is worth one (4), the digit in the tens place is worth ten (3), the digit in the hundreds place is worth one hundred (2), and the digit in the thousands place is worth one thousand (1).

To find the place value of a digit, look at its position in the number. The place value of the digit will be the base (ten, one hundred, one thousand, etc.) to which that position corresponds.

The decimal part of any number can be read using place values as well. The place values of the digits to the right of the decimal are tenths, hundredths, thousandths, and so on.

For example, in the decimal 0.35, the digit in the tenths place is worth one-tenth (0.3), the digit in the hundredths place is worth one-hundredth (0.05), and the digit in the thousandths place is worth one-thousandth.

### Converting Fractions to Decimals and Percentages

**To convert a fraction to a decimal, divide the numerator by the denominator.**

For example, to convert the fraction ¾ to a decimal, divide three by four:

¾ = 0.75 (the answer is a decimal)

**To convert a fraction to a percentage, multiply the decimal by 100.**

For example, to convert the decimal 0.75 to a percent, multiply by 100:

0.75 = 75% (the answer is a percent)

### Converting Decimals to Fractions

**To convert a decimal to a fraction, divide the decimal by the place value of the decimal part and remove the decimal from the numerator.**

For example, to convert the decimal 0.75 to a fraction, divide 0.75 by 100 (the last whole number is in the hundredths place value):

0.75 = 75/100 (the answer is a fraction).

Simplify the fraction into the lowest form: 75/100 = 3/4.

Another example is to covert the decimal 0.584 to a fraction, and divide 0.584 by 1000 (the last whole number is in the thousandths place).

0.584 = 584/1000 (the answer is a fraction).

### Converting Percentages to Decimals and Fractions

**To convert a decimal to a percentage, multiply by 100.**

For example, to convert 75% to a decimal, divide 75 by 100:

75% = 0.75 (the answer is a decimal)

**To convert a percentage to a fraction, divide by 100 and remove the percent sign from the numerator.**

For example, to convert 75% to a fraction, divide 75 by 100:

75% = 75/100 (the answer is a fraction).

Simplify the fraction into the lowest terms: 75/100 = ¾.

Remember, a percentage is a number or ratio expressed as a fraction of 100. So, when you see the symbol %, think “divide by 100.”

## Performing arithmetic operations with rational numbers (both positive and negative numbers)

### Order of Operations

The Order of Operations is a set of rules that determine the order in which operations (addition, subtraction, multiplication, division, etc.) are performed in an expression.

**The Order of Operations is often abbreviated as PEMDAS:**

P = Parentheses first

E = Exponents (ie Powers and Square Roots, etc.)

MD = Multiplication and Division (left-to-right)

AS = Addition and Subtraction (left-to-right)

For example, consider the expression: 32 + 2^{2}

Which operation should be performed first, the addition or the exponentiation?

The answer is that the exponentiation should be done first because it has precedence over addition. That is, the operations within parentheses are always performed first, followed by exponentiation.

Therefore, the correct answer is 36.

**Multiple-Step Problems**

Here is another example:

24 ÷ 12 + 17 – 11

Which operation should be performed first, the division or addition?

The answer is that the division should be performed first because it has precedence over addition and subtraction. That is, the operations within parentheses are always performed first, followed by division, multiplication, addition, and subtraction (from left to right).

Therefore, the correct answer is 8.

**Operations with Parentheses **

Remember, the operations within parentheses are always performed first.

For example, consider the expression: (15 – 12) + 18 ÷ (21 – 11)

Which operation should be performed first?

The answer is that the operations within parentheses are always performed first. In this case, that means subtracting 12 from 15 and 11 from 21.

Therefore, the correct answer is 3 + 18 ÷ 10.

The next step is to divide 18 by 10 leaving 3 + 1.8.

The final answer is 4.8.

## Compare and order rational numbers (both positive and negative)

### Defining Rational Numbers and Irrational Numbers

**Rational numbers** are numbers that can be expressed as a fraction (a/b). A and b are both integers and the denominator is not zero. That is, they can be written as a ratio of two integers.

For example, some rational numbers are:

- – ½ (this can be written as -25/50)
- ¾ (this can be written as 75/100)
- 0.125 (this can be written as 125/1000)
- − 0.75 (this can be written as -75/100)
- − 15 (this can be written as -15/100

**Irrational numbers** are numbers that cannot be expressed as a fraction. That is, they cannot be written as a ratio of two integers.

For example, some irrational numbers are:

– π (this is approximately equal to 22/71)

– √16 (this is approximately equal to 25/41)

– e (this is approximately equal to 27/28)

### Ordering Rational Numbers

Rational numbers can be ordered from least to greatest or greatest to least.

To order rational numbers from least to greatest, line them up in order from left to right. The numbers get larger as you go from left to right.

For example, the following numbers are listed in order from least to greatest:

-15, -0.75, 0, 0.125, ½, ¾

To order rational numbers from greatest to least, line them up in order from right to left. The numbers get larger as you go from right to left.

For example, the following numbers are listed in order from greatest to least:

0.125, ½, ¾, 0, -0.75, -15

When ordering negative numbers, the smaller the number, the larger it is. That is, the absolute value of a negative number is always less than the absolute value of a positive number.

### Comparing Rational Numbers

Rational numbers can be compared by looking at their relative size. That is, which number is larger or smaller than the other?

For example, to compare the numbers -15 and -1. Because -15 is less than -1 the answer will be as follows.

-15 < -1

You can find the answer by using a number line and plotting the values. Therefore, the value that is farthest to the right on the number line is the greatest.

**Symbols and Meanings of Comparing Rational Numbers**

The following symbols are used when comparing rational numbers:

>, ≥, =, ≤, and <

The meanings of these symbols are as follows:

- > (greater than)
- ≥ (greater than or equal to)
- = (equal to)
- ≤ (less than or equal to)
- ≤ (less than)

## Solving Equations with One Variable

### Identify the Terms of an Algebraic Equation

An algebraic equation consists of terms such as a number, variables, or product of numbers. Terms can be separated by additional and subtraction signs.

**Constant**is defined as the number itself not attached to a variable- A
**Variable**is a letter that represents an unknown quantity - A
**Coefficient**is a number being multiplied by the variable

For example, the expression:

-15x – 18 = 30

has the following terms:

- The constants are – 18 and 30
- The variable is -15x
- The coefficient in this equation is -15

### Inverse Arithmetic Operations

The inverse of an arithmetic operation is the operation that “undoes” the original operation.

For example, the inverse of addition is subtraction and the inverse of multiplication is division.

The inverse operations of addition and subtraction are opposite of each other. The inverse operation of multiplication and division are also opposite of each other.

This is true because the operations “undo” each other.

### Implement A Sequence of Steps to Solve Equations

To solve an equation, you need to find the value of the variable that makes the equation true.

This can be done by using inverse operations to isolate the variable on one side of the equation.

Once the variable is isolated, you can then use a number line or other methods to find the value of the variable.

Here is an example of solving an equation:

Original equation: x + 18 = 30

Inverse operation of addition: x + 18 – 18 = 30 – 18

Isolate the variable on one side: x = 30 – 18

x = 12

The value of x is 12.

This can be verified by substituting 12 for x in the original equation.

### Solve Proportional Relationships (Equations and Inequalities) with One Variable

A proportional relationship is a relationship between two quantities in which the ratio of one quantity to another is constant, or when one fraction is equivalent to the other.

This constant is called the proportionality constant or the multiplier.

Proportional relationships can be represented in equation or inequality form.

To solve a proportion, you need to find the value of the variable that makes the proportion true.

This can be done by using inverse operations to isolate the variable on one side of the proportion.

Once the variable is isolated, you can then use a number line or other methods to find the value of the variable.

Here is an example of solving a proportion:

Original equation: 6x = 19

Inverse operation of addition: 6x ÷ 6 = 19 ÷ 6

Isolate the variable on one side: x = 19 ÷ 6

x = 19/6

The value of x is 19/6 or 3.17.

## Solve Real-World Problems using One or Multiple Step Operations with Real Numbers

### Problem Solving Plan Word Problems

The first step in solving a word problem is to read the problem carefully and identify the information that is given and the information that is needed.

The next step is to identify the problem type and the equation that needs to be solved.

Once the equation is identified, you can then solve the equation.

The last step is to check your work by substituting the answer back into the original problem.

**Here is an example of a word problem:**

A plumber charges $25 for a service call plus $50 per hour of service. Write and solve an equation to find the cost of a plumber’s service if he works for h hours.

- The first step is to read the problem and identify the given information and the needed information.
- The given information is that a plumber charges $25 for a service call plus $50 per hour of service. The needed information is the cost of the plumber’s service if he works for h hours.
- The next step is to identify the problem type and the equation that needs to be solved.
- The last step is to check your work by substituting the answer back into the original problem.

The problem type is a linear equation and the equation that needs to be solved is C = 25 + 50h.

Once the equation is identified, you can solve for h.

C = 25 + 50h

If the plumber works for h hours, the cost of his service will be $25 for the service call plus $50 per hour of service, or $25 + $50h.

For example, if the plumber works for two hours, the cost of his service will be $25 + $50(two hours) = $125.

You can check your work by substituting the value of h back into the original equation.

### Solve Word Problems Using Percentages

Percentages are a way of expressing a number as a fraction of 100.

Percentage numbers increase or decrease in word problems.

To solve a word problem that involves a percentage increase or decrease, you need to identify the following information:

- The original amount
- The percentage increase or decrease
- The new amount

Once you have this information, you can set up and solve an equation.

**Here is an example of a word problem that involves a percentage decrease: **

For example, if a store is offering a 20% discount on an item that originally costs $100, the new price of the item would be 80% of the original price, or $80.

**Here is an example of a word problem that involves a percentage increase**:

The population of a town increased by 12% from 2010 to 2011.

The population of the town in 2010 was 20,000.

What was the population of the town in 2011?

The original amount is the population of the town in 2010, which is 20,000.

The percentage increase is 12%.

The new amount is the population of the town in 2011.

To find the new amount, you need to set up and solve an equation.

The equation would be as follows:

20,000 + 12% = new amount

Once you have set up the equation, you can solve for the new amount.

20,000 + 12%(original population) = new amount

20,000 + 0.12(20,000) = new amount

20,000 + 2400 = new amount

22,400 = new amount

The population of the town in 2011 was 22,400.

## Apply Estimation Strategies and Rounding Rules to Word Problems

### Metric Measurements

The metric system is a system of measurement that is used in many countries around the world.

The most common units of measurement in the metric system are:

Length – meters (m)

Weight or Mass – grams (g)

Capacity or Volume – liters (L)

Degrees Celsius (C)

It’s important to note: area is measured by square units and volume is measured by cubic units.

### Estimation and Rounding of Numbers

One way to increase your speed and accuracy on the ATI TEAS Math Number and Algebra section is to practice estimation and rounding of numbers.

This will help you to quickly see what the answer should be in your head, without having to do a lot of calculations.

For example, if you are asked to round the number 45.678 to the nearest whole number, you would first look at the number in the ones place, which is 5.

You will round up if the number in the tenths place is greater than or equal to five, and you will round down if the number in the tenths place is less than five.

Next, the number in the tenths place is 6, as this number is greater than 4 we can round up our whole number to 46.

You can use estimation strategies and rounding rules to quickly solve word problems.

## Solving Word Problems involving Proportions

### Introduction to Proportions

A proportion is a ratio in fraction form equal to another ratio in fraction form.

### Writing and Solving a Proportion

The following steps will help you to set up and solve a proportion:

- Read the word problem carefully and identify all of the information that is given.
- Draw a picture or diagram to help you visualize the problem.
- Determine what quantity you are looking for in the problem (this is the unknown quantity).
- Identify two equivalent ratios in the problem.
- Write a proportion using the equivalent ratios.
- Cross multiply to solve the proportion.
- Check your answer to make sure it makes sense in the context of the problem.

**Here is an example of a word problem that can be solved using a proportion: **

The ratio of dogs to cats in a shelter is 12 to 25.

If there are 100 animals in the shelter, how many of them are dogs?

In this problem, the unknown quantity is the number of dogs in the shelter.

The two equivalent ratios are 12 to 25 and x to 100.

The proportion would be set up as follows:

12/25 = x/100

To solve the proportion, you will need to cross multiply.

12(100) = 25x

1200 = 25x

48 = x

There are 48 dogs in the shelter.

### Direct Proportions and Constant of Proportionality

A proportion is a direct proportion if the two equivalent ratios are in the form y = kx.

The constant of proportionality (k) is the number that represents the relationship between two variables.

Examples of direct proportional equations are y = 3x, y = 10x, y = x/6

Examples of not directly prpoortional equations are y = 5x + 10, y = x-15, y = 5

## Solving Word Problems using Ratios and Rates of Change

### What is a ratio?

A ratio is a comparison of two numbers by division. For example, 4:5 or 4/5.

### What is Rate, Unit Rate, and Rate of Change?

A **rate** is a ratio that is used to compare two different units.

For example, 60 miles per hour (mph) or 60 miles: An hour.

A **unit rate** is a rate in which the second number in the ratio is one unit. You can find the unit rate by dividing the numerator by the denominator.

For example, 60 miles per 2 hours or 30 miles per hour.

A **rate of change** is the speed at which something is happening. It can also be known as the unit rate.

For example, the rate of change in population, the rate of change in temperature, or the rate of change in distance.

### Using Ratio and Rate of Change to Solve Problems

When solving problems, it is often helpful to think about the relationships between different quantities in terms of ratios or rates of change.

For example, if you know that the ratio of dogs to cats in a shelter is 12 to 25, and you also know that there are 100 animals in the shelter, you can use these relationships to solve for the number of dogs in the shelter.

You can also use rates of change to solve problems.

For example, if you know that the average rate of change in population over a period of time is 0.02%, you can use this information to predict the population of a city in the future.

These problems can be easily solved with the representation of a graph to plot out your points. These points may form a slope of a line.

## Solving Word Problems using Expressions, Equations and Inequalities

### What are expressions, equations, and inequalities?

An **expression** is a mathematical phrase that can contain numbers, variables, and operators. Examples are 10-2y, x, 4(x – 16)

An **equation** is an expression that contains an equal sign. Examples are 10 – y = 20, x = 10, 3x – 10 = 4x +8

An** inequality** is an expression that contains an inequality sign (<, >, ≤, ≥).

As we have previously discussed how to solve problems with expression and equations, we will take a closer look at solving problems with inequalities.

### Solving Problems with Inequalities

Inequalities are mathematical phrases that contain an inequality sign (<, >, ≤, ≥).

They can be used to represent situations where one value is greater than or less than another value.

For example, the inequality x > y can be read as “x is greater than y.”

Inequalities can be used to solve problems by representing the relationships between different quantities in a problem.

For example, x + 2 > 12.

Subtract 2 from both sides: x + 2 – 2 = 12 – 2

Solution: x > 10

There are also times when you may reverse the direction of the inequality if we are multiplying or dividing negative numbers.

For example: -2y < -8

Divide both sides of -2 to isolate for y: -2y/-2 > -8/-2

Solution: y > 4

It is important that we reverse the inequality when we multiple or divide by a negative number.