ATI TEAS Mathematics Review (Complete)

The second section of the TEAS covers Mathematics and features 32 scored items. There are two categories of Mathematics objectives for the TEAS. The test items are divided among the Mathematics objectives as follows:

M.1 NUMBER AND ALGEBRA — 23 QUESTIONS
M.1.1 Convert among non-negative fractions, decimals, and percentages.
M.1.2 Perform arithmetic operations with rational numbers.
M.1.3 Compare and order rational numbers.
M.1.4 Solve equations in one variable.
M.1.5 Solve real-world one- or multi-step problems with rational numbers.
M.1.6 Solve real-world problems involving percentages.
M.1.7 Apply estimation strategies and rounding rules to real-world problems.
M.1.8 Solve real-world problems involving proportions.
M.1.9 Solve real-world problems involving ratios and rates of change.
M.1.10 Translate phrases and sentences into expressions, equations, and inequalities.

M.2 MEASUREMENT AND DATA —9 QUESTIONS
M.2.1 Interpret relevant information from tables, charts, and graphs.
M.2.2 Evaluate the information in tables, charts, and graphs using statistics.
M.2.3 Explain the relationship between two variables.
M.2.4 Calculate geometric quantities.
M.2.5 Convert within and between standard and metric systems.

In addition, the TEAS Mathematics section features four unscored items as a pretest. These items can address objectives from either of the above categories.

M.1 NUMBER AND ALGEBRA
M.1.1 CONVERT AMONG NON-NEGATIVE FRACTIONS, DECIMALS, AND PERCENTAGES

Numerical values are expressed in many different forms, including as fractions, decimals, and percentages. On the Mathematics section of the TEAS exam, you must show that you understand each of these forms and how to convert from one to another.
A fraction expresses a ratio, written in the form where a and b are integers. The two parts of a fraction are the numerator and the denominator.



The value of a fraction is found by dividing the top number (the numerator) by the bottom number (the denominator). A fraction also tells how many parts of a whole there are. For example, the fraction means three parts of a whole with four total parts.

A decimal is a number expressed in place values based on powers of 10. This is called decimal notation.

Decimal notation includes the decimal point. Digits to the left of the decimal point show values greater than one. Digits to the right of the decimal point show values less than one.

Look at the chart of place value below. The value of a digit in a number depends on its position or place value. The chart shows place values for the number 24,628,341.789. Notice that the decimal point separates the ones place from the tenths place.


This number can be read as twenty-four million, six hundred twenty-eight thousand, three hundred forty-one and seven hundred eighty-nine thousandths. Notice that the decimal point is read as “and.”
The place value of a decimal tells what the fractional denominator will be.

Here, the last digit of the decimal is in the thousandths place. So the fractional equivalent is or 678 thousandths.
To keep the digits in a decimal in the correct place, fill in zeroes. For example, 8 thousandths is written as 0.008, with the 8 in the third place to the right of the decimal point.

If the decimal point is moved to the right, the value of the decimal is increased by a factor of 10. For example, if the decimal point in 0.062 is moved one place to the right, the result is 0.62, which is 10 times larger than the original decimal. Moving the decimal point 2 places to the right multiplies the original decimal by 100, 3 places multiplies by 1,000, and so on.

If the decimal point is moved to the left, the value of the decimal is decreased by a factor of 10. For example, if the decimal point in 0.35 is moved one place to the left, the result is 0.035, which is 10 times smaller than the original decimal. Moving the decimal point 2 places to the left divides the original decimal by 100, 3 places by 1,000, and so on.

A percentage, or percent, is a special ratio that compares a quantity to 100. For example, 5% means “5 out of 100,” 37% means “37 out of 100,” and 62.5% means “62.5 out of 100.”



To convert among non-negative fractions, decimals, and percentages, remember how the numerator relates to the denominator in a fraction; review place value within decimals; and be able to express a percentage as a numerical part of one hundred.
Fractions, decimals, and percentages all express ratios. The chart below shows how a number of ratios are expressed in these other forms.

Percent to decimal ÷ 100
Decimal to percent × 100
For the TEAS Mathematics section, you must be able to convert among fractions, decimals, and percentages.

Converting a fraction to a decimal is done most easily on a calculator. For example, to find the decimal value of divide the numerator 5 by the denominator 8 to get 0.625.

To convert a decimal to a fraction, consider two main situations—when the value is less than 1 and when the value is greater than 1.
- Less than 1: Convert 0.84 to a fraction.
First, write 0.84 divided by 1:
Next, multiply both numerator and denominator by 100 to make the numerator a whole number. (The decimal point must be moved two places to the right.) This gives you .
Finally, simplify the fraction by dividing top and bottom by
- Greater than 1: Convert 3.42 to a fraction.
First, move the decimal point to the right to create a whole number: 342. This is the numerator.
Next, express the denominator as 1 plus the number of zeroes equal to the number of places you moved the decimal point: 100.
The fraction form of 3.42 is This can be reduced to

To convert a fraction to a percentage, first divide the numerator by the denominator. Then multiply the resulting value by 100 to get the percentage.

= 0.553 × 100 = 55.3 = 55.3%, which can be rounded to 55%.

To convert a decimal to a percentage, multiply the decimal by 100 by moving the decimal point two places to the right. For example, 0.64 converts to 64%.

M.1.1 PROBLEM 1
How do you write the fraction as a decimal?
(A)40.31
(B)40.031
(C)40.0031
(D)40.00031

STRATEGY
Use a place value frame.

THINK
- The denominator is 10,000, so write the 1 in the ten thousandths place.
- Write a 3 to the left of the 1.
- Fill in the tens and ones places with 4 and 0, respectively.
- The tenths and hundredths have no digit, so fill them in with zeroes. So The correct response is (C).



M.1.1 PROBLEM 2
How do you write the decimal number 0.016 as a fraction?



STRATEGY
Place value tells you how to write the fraction.

THINK
- The digit farthest to the right in the decimal tells you the place value.
- The 6 is in the thousandths place. So the denominator of the fraction should be 1,000.
- Write the numerator of 16 into the fraction, giving you (answer choice C).

M.1.1 PROBLEM 3
What is the decimal value of 3.5%?
(A)0.35
(B)3.5
(C)0.035
(D)0.0035

STRATEGY
Use the special “out of 100” ratio of percentages.

THINK
- For this problem, you are going from 3.5% to a decimal, so divide by 100.
- 3.5 ÷ 100 = .035 (answer choice C).

M.1.2 PERFORM ARITHMETIC OPERATIONS WITH RATIONAL NUMBERS
Performing basic calculations by hand is a necessary skill for nurses, even with a calculator readily available. On the TEAS exam, you must demonstrate the ability to complete computations with the four basic operations on integers, decimals, fractions, and mixed numbers. You also must follow the order of operations when simplifying a mathematical expression.
The four basic mathematical operations are addition, subtraction, multiplication, and division.

Basic Operations with Integers
Below are the basic number facts for addition and subtraction. To use the table, find the intersection of a row and column. For example, 7 + 6 = 13 is shown below.

An excellent learning method for number facts is to create individual flash cards and to drill with them for a few minutes each day.

EXAMPLE FLASH CARDS

Note that knowing an addition fact means that you also know the corresponding subtraction fact by thinking “backward.” For example, if you know 3 + 6 = 9, you also know
6 + 3 = 9 → 9 – 6 = 3 → 9 – 3 = 6
Similarly, because 7 + 4 = 11:
4 + 7 = 11 → 11 – 7 = 4 → 11 – 4 = 7
For numbers with more than one digit, addition and subtraction are done column by column. For example, for 17 + 32,


If any column adds to more than 9, the value of the tens column in the answer is carried over to the next column to the left. For example, for 27 + 45,


Similarly, if a larger digit is being subtracted from a smaller digit in any column, you have to “borrow” from the column to the left. For example, for 63 – 38, the 3 borrows 10 from the 6. The 3 becomes 13, and the 6 becomes 5.



 

To perform arithmetic operations with rational numbers, you should be able to do computations using the four basic operations with integers, decimals, fractions, and mixed numbers. You should also know and follow the order of operations.
The relationship between addition and subtraction works as a check on any sum or difference that you have calculated.
For example, does 90 – 34 = 56? Check: 34 + 56 = 90
Because the sum of 34 and 56 is 90, the answer checks.
In the same way, to check 74 – 45 = 29, add 45 + 29 to see whether the sum is 74. It is, so the answer checks.
The number facts for multiplication and division are just as essential as the addition and subtraction facts. Again, to use the table, find the intersection of a row and column. For example, find 7 × 4 = 28.

Take some time to make flashcards if you don’t know these facts.
FLASHCARDS

The backward relationship works for multiplication and division just as it does for addition and subtraction. For example, if you know 8 × 7 = 56, you also know
7 × 8 = 56 → 56 ÷ 7 = 8 → 56 ÷ 8 = 7
To use the table for division to find, for example, 32 ÷ 4, find the first number, 32, in the body of the table, and then find the number to be divided into it on either the top row or left-hand column. Follow the intersection to find the answer. So 32 ÷ 4 = 8.
For numbers with more than one digit, multiplication is done by aligning the columns by place value. For example, for 143 × 2,


If any two numbers multiply to more than 9, the value of the tens column in the answer is carried over to the next column to the left, to be added to the next multiplication. Multiply first by the ones column and align that number, then multiply by the tens column and align that number with the tens column, and so on.

For example, for 36 × 17, the first multiplication is 7 × 6 (= 42), so the 2 is placed in the ones column and the 4 carries over for when 7 is multiplied by 3. This gives (21 + 4) = 25, so the first multiplication is 7 × 36 = 252.
Then for the second multiplication, 1 × 36 = 36. That answer is aligned under the 1 (not the 7) because that is the number that was used in the second multiplication. Now add the two results. (Note that a 1 has to be carried over to the left column when 5 and 6 are added.)


Division for numbers with more than one digit is done by setting up the problem as described here. Put the number being divided (dividend) under the division sign. Put the number doing the dividing (divisor) to the left of the sign. For example, 832 ÷ 16 is set up for long division as follows.


Divide the divisor (16) into the first digit of the dividend (8). 16 doesn’t go into 8, so include the next digit. 16 does go into 83; it goes 5 times (although there will be something left over). So the first number of the answer (quotient) is a 5. Be sure to align the 5 over the 3, not the 8—it is the quotient of 83, not 8. So you have


Now multiply the quotient by the divisor (5 × 16 = 80) and place this under the 83. Subtract and bring down the next digit (2). The calculation now looks like this:


Divide the 16 into 32, which goes 2 times. The 2 goes in the quotient above the 2 in the dividend. When the 2 in the quotient is multiplied by the divisor, the answer is 32. The final long division looks like this:


You can check if the answer is correct by multiplying the quotient 52 by the divisor 16. Indeed, 52 × 16 = 832.

Basic Operations with Fractions
When adding and subtracting fractions, you need the fractions to have the same denominator. If the denominators are the same for the fractions, simply add or subtract the numerators and keep the same denominator. Simplify the answer if necessary.

If the denominators of the fractions are different, use the least common multiple (LCM) to find the least common denominator and convert each fraction to an equivalent fraction with this denominator. For example, if you are adding the fractions you first must find the LCM of 8 and 6, which is 24. Then you can convert Finally, you have

You can also add and subtract mixed numbers. A mixed number includes a whole number and a fraction, like If a problem involves mixed numbers, add or subtract the whole numbers first and then the fractions (as shown above). Then combine the two results. Simplify the answer if necessary.

Multiplying and dividing fractions is actually easier than adding and subtracting them because the fractions can have different denominators.
To multiply fractions, first multiply the numerators to get the numerator of the answer. Then multiply the denominators to get the denominator of the answer. Simplify the answer if necessary. For example,


To multiply mixed numbers, convert them to improper fractions first. An improper fraction has a numerator that is larger than its denominator. For example,



Multiplying fractions can be made easier by canceling. If any numerator has a common factor with any denominator, you can divide each by that factor, which simplifies the math at the end. For example, by canceling 4 out of you get which results in the answer of
To divide fractions, you can “invert and multiply,” which means finding the reciprocal of the second fraction and then multiplying as shown above. A reciprocal of a fraction is the fraction “flipped over”—with the numerator and denominator swapped. The reciprocal of and the reciprocal of is 8. Here is an example of invert and multiply:



Basic Operations with Decimals
When adding and subtracting decimals, it is very important to align the decimal points. Fill in zeroes if necessary to keep the digits in a decimal in the correct place. Then add or subtract as you usually would.
When multiplying and dividing decimals, you do not have to align the decimal points.

Multiply decimals as though the factors were whole numbers, but keep the decimal points where they are. Then follow this procedure to put the decimal point in the correct place in the product.
- Count the total number of digits to the right of the decimal point in both factors.
- Count back to the left from the last digit in the product that same number of places. Place the decimal point there.
- Fill in zeroes as necessary.

Divide decimals as though the factors were whole numbers, but keep the decimal points where they are. Follow this procedure to put the decimal point in the correct place in the quotient.
- Count the total number of digits to the right of the decimal point in the divisor.
- Count over this many digits to the right from the decimal point in the dividend.
- Place the decimal point in the quotient at this point.
- Fill in zeroes as necessary.

Order of Operations
Order of operations tells you the order in which to simplify a complex expression. The simplest way to remember order of operations is the acronym PEMDAS:
P: Perform calculations inside parentheses
E: Simplify exponents

MD: Multiply and divide, left to right, before you
AS: Add and subtract, left to right.
This acronym can also be remembered as Please Excuse My Dear Aunt Sally. (Exponents are powers—raised numbers as shorthand to tell how many times a number or variable is a factor of itself. Exponents are not covered on the TEAS exam.)

Here is an example of using order of operations.
(12 – 5) × 4 = (12 – 5) × 4    First, perform operations in parentheses.
= 7 × 4          Next, multiply.
= 28

M.1.2 PROBLEM 1
Find the quotient:



STRATEGY
To divide fractions, invert the divisor (the number you are dividing by) and multiply.

THINK
- Invert the divisor; then multiply the fractions.
- Cancel factors, as in multiplication.
- After multiplying, check your answer to make sure that it is in lowest terms.
- The correct response is (C).



M.1.2 PROBLEM 2
What is the correct answer to this expression: 4 + (18 ÷ 3) × 2 = ?
(A)14
(B)16
(C)20
(D)48

STRATEGY
To find the correct answer, use order of operations.

THINK
- 4 + (18 ÷ 3) × 2 = 4 + (18 ÷ 3) × 2 First, perform operations in parentheses.
= 4 + 6 × 2     Then multiply.
= 4 + 12        Then add.
= 16
- Answer (B) is correct.

M.1.2 PROBLEM 3
Find the sum:



STRATEGY
Use lowest common multiple to find the sum.

THINK
- To find the sum, first find the LCM of 8 and 4, which is 16.
- Convert both fractions to equivalent fractions with the denominator 16. Then add the numerators and simplify.


- Answer choice (D) is correct.

M.1.3 COMPARE AND ORDER RATIONAL NUMBERS
A rational number is any number that can be expressed in fraction form, including decimals, percents, and mixed numbers. You can compare rational numbers using inequality symbols and put them in the correct numeric order. On the TEAS Mathematics exam, you must compare and order rational numbers using the correct terms and symbols.

To compare the numeric values of rational numbers, you must use signs or symbols. These ranking signs include:
<     less than
>     greater than
=     equal to
≠     not equal to
≤     less than or equal to
≥     greater than or equal to

These signs (minus the equal sign) are also called inequality symbols.
Comparing numeric values can be shown on a number line. Values to the left on a number line get smaller. Values to the right get larger.

Negative integers are values to the left of 0 on a number line. All negative integers are less than any positive integer. Notice that these negative numbers appear to get larger as you move to the left, from –4 to –5, for example. However, because “negative” can be interpreted as “less than,” the values actually do get smaller to the left on the number line. So even though 5 is greater than 3, –5 is less than –3.

To compare the values of fractions, first look at the denominators. If two fractions have the same denominator, compare them by comparing the numerators. For example,



Fractions that have different denominators are more difficult to compare. Use the LCD (least common denominator) to convert them to fractions with the same denominator. For example, to find whether is greater or less than find the LCD by finding the lowest common multiple of 8 and 10:
8: 8, 16, 24, 32, 40, . . .
10: 10, 20, 30, 40, . . .
So multiply the numerators and denominators of each fraction by a number that will make the denominator 40, which is the LCD.


Then, compare the equivalent fractions.




To compare and order rational numbers, you must know the symbols for comparing rational numbers or putting them in numeric order from least to greatest and from greatest to least

Any positive fraction is greater than any negative fraction. For example,


To compare and order numbers in decimal form, use place value. You can stack the numbers 7.143, 0.756, and 7.046 to compare them by place value.
7.143
0.756
7.046
Comparing them this way, you can order them from least to greatest:
0.756 < 7.046 < 7.143
Where does the fraction fit in this order? Divide 7 by 8 to get a decimal: 0.875. Then place the decimal where it belongs in the inequality.
0.756 < 0.875 < 7.046 < 7.143
Note that the decimals can also be ordered from greatest to least:
7.143 > 7.046 > 0.875 > 0.756

M.1.3 PROBLEM 1
Which of the numbers in the following series has the greatest value: –7, 0, –2.1, –0.8?
(A)–7
(B)–2.1
(C)0
(D)–0.8

STRATEGY
When ranking or ordering integers and negative numbers, think of a number line.

THINK
- Here are the four numbers from the series above on the number line.


- You can see that 0 is farthest to the right on the number line. So 0 has the greatest value. Answer (C) is correct

M.1.4 SOLVE EQUATIONS IN ONE VARIABLE
An algebraic equation is a mathematical expression that contains one or more variables. Solving equations is not difficult once you know the proper steps to follow. On the TEAS Mathematics exam, you must be able to solve different kinds of mathematical equations with one variable.

A variable is a term that stands for a number or quantity. Any letter can be used for a variable in algebra, but x, y, z, a, b, and c are used most frequently.
A constant is a number that is not linked to a variable.
An expression is a mathematical phrase that contains constants, variables, and symbols such as +, –, ×, and ÷. An expression can contain parentheses or other symbols as well. Numbers and variables that are separated by + or – signs are called terms. Those that are part of multiplication are called factors.

Multiplication can be indicated by ×, - ,or *, or by parentheses, as in (2)(3x). It can also be indicated like this: xy, which means x multiplied by y.
An example of a simple algebraic expression is just the variable x. An expression may contain two terms, such as 2a + 3b, or it can be quite complicated, with many letters, numbers, and symbols.

An equation is essentially two expressions that are equal. Usually, a variable is included in one or both expressions. Solving the equation means finding a value for the variable that makes the equation true.
Since the two sides of an equation are equal, whatever you do to one side of the equation you must do to the other side. For example, if you add a number to one side, you must add the same number to the other side.

The goal in solving an equation is to put the variable on one side of the equal sign and its value on the other. You do this by using inverse operations. Addition and subtraction are inverses of each other, and multiplication and division are inverses of each other.

Here are some examples of how to solve equations by isolating the variable on one side.
- Solve for x: x + 4 = 10.
You must get rid of the 4 on the variable side of the equation. Since 4 is added to the variable, you must subtract 4 on both sides of the equation. This results in:
x + 4 – 4 = 10 – 4
x = 6
To check, substitute x = 6 into the original equation.
6 + 4 = 10
10 = 10
The answer checks out, so it is correct.
- Solve for y: y – 3 = 7.

Likewise, if a value is subtracted from the variable, you should add that number to both sides. For example,
y – 3 + 3 = 7 + 3
y = 10
- Solve for a: 7a = 14.
You must get rid of the 7 on the variable side of the equation. Since the variable is multiplied by 7, you must divide (inverse of multiply) both sides of the equation by 7.


a = 2
To check, substitute a = 2 into the original equation.
7(2) = 14
14 = 14
- Solve for
You must get rid of the 3 on the variable side of the equation. Since the variable is divided by 3, you must multiply both sides of the equation by 3.


b = 18
To check, substitute b = 18 into the original equation.


6 = 6

An equation may combine all of the above examples. The thing to remember is that whatever you do to one side of the equation must be done to the other side. Here is a more difficult equation with the same variable on both sides
- Solve for
First, put all of the variable terms on one side by subtracting from both sides.


Next, to put all of the constant (number) terms on the right-hand side, subtract 5 from both sides.


Multiply all terms on both sides by 3 to get rid of the fraction.


To check, substitute x = 3 into the original equation.




To solve an equation in one variable, you must understand what a variable is and use inverse operations to solve for the variable in an equation.

M.1.4 PROBLEM 1
Solve the equation 5x = 65 for x.
(A)13
(B)–13



STRATEGY
To solve the equation, apply operations to both sides of the equation in order to isolate the variable on one side of the equation.

THINK
- You want to remove 5, which is the coefficient of the variable.
- To do this, divide both sides of the equation by 5.


Answer (A) is correct.

M.1.4 PROBLEM 2
Solve the equation y – 45 = 0 for y.


(C)45
(B)–45
(D)90

STRATEGY
Apply the same operation on both sides of the equation to isolate the variable.

THINK
- To solve, add 45 to both sides of the equation.
y – 45 + 45 = 0 + 45
y = 45
- Answer (C) is correct.

M.1.4 PROBLEM 3
Solve the equation for n.
(A)n = 28


(C)n = 9
(D)n = 36

STRATEGY
Isolate the variable by applying the same operations on both sides of the equation.

THINK
- To solve, isolate the variable using subtraction, multiplication, and division.
- First, subtract 7 from both sides of the equation to isolate the variable.


- Next, put the result in fractional form.


- Multiply both sides by 4.


- Then divide both sides by 3.


- Answer (D) is correct.

M.1.5 SOLVE REAL-WORLD ONE- OR MULTI-STEP PROBLEMS WITH RATIONAL NUMBERS
As a nurse, you often must translate a real-world situation into the correct mathematical equation. Word problems about real-world professional situations include words like reduction, double, total, and distribute that indicate what mathematical operations are required. On the TEAS exam, you must be able to decide on the necessary operations to solve word problems dealing with rational numbers.

When you approach a word problem, read the entire problem very carefully before you try to solve it. Note any irrelevant or unnecessary details as you read. Focus on the relevant information to create an equation for the problem.

Assign a variable to the quantity you are looking for. The variable can be x, y, n, or any letter you choose. Write down what that variable represents in the problem.

Look for key words that indicate what operation is needed to solve the problem.
- Addition: sum, increase, total, combined, added to, more than
- Subtraction: difference, decrease, reduced by, fewer, less than, subtracted from
- Multiplication: double, triple, times, of, per, product of, rate, multiplied
- Division: half, out of, percent, quarter, distribute, quotient of, divided

Think about how to translate the word problem into an equation. Then solve the equation for the variable and check your answer. Make sure the answer is reasonable. For example, a problem that calls for the amount of time it takes to complete a task should not have a negative number for the answer.

STRATEGY
For a One-Step Problem:
Mrs. Rodriguez’s school spends $910 to buy art supplies for the 28 students in her art class. How much does the school spend on art supplies per student?

THINK
The problem is asking what part of $910 each student in art class receives for art supplies. The variable n = each student’s share of $910. The equation should read:
28n = 910

Solve:


Each student received $32.50 in funding for art supplies.

STRATEGY
For a Two-Step Problem:
So far this season Sandra has scored three times the number of goals she scored all last season. If she scores two more goals this season, she will have 20. How many goals did Sandra score last season?

THINK
The problem is asking how many goals Sandra scored last season. So the variable x = number of goals last season. This season Sandra has scored 3x goals, or three times the number she scored last year. If she scores two more goals, she will reach 20 total goals. The equation should read:
3x + 2 = 20

Solve:
3x + 2 – 2 = 20 – 2
3x = 18


x = 6
Sandra scored six goals last season.
Check:
3(6) + 2 = 20
18 + 2 = 20
20 = 20


To solve a real-world problem with rational numbers, you must read the problem carefully, note relevant information, and choose the correct operations and sequence of steps to solve the problem.

M.1.5 PROBLEM 1
Greg had $105 in his savings account. After depositing two identical weekly paychecks, he had $563 in the account. How much money does Greg earn each week?
(A)$105
(B)$229
(C)$359
(D)$458

STRATEGY
To solve this problem, find x, the amount of Greg’s weekly paycheck.

THINK
- When you subtract the total amount of Greg’s two paychecks (2x) from $563, you get $105.
- Write an equation for this situation: 563 – 2x = 105
- Solve for x:
–563 + 563 – 2x = 105 – 563
–2x = –458
(–1)–2x = –458(–1)
2x = 458
x = 229
- Greg’s weekly paycheck is $229. (B) is correct.

M.1.5 PROBLEM 2
Julio’s rock garden measures 8 feet by 4 feet. Julio wants to create a continuous border around the rock garden using square concrete paving blocks that measure 6 inches on a side. How many paving blocks should Julio purchase?
(A)24
(B)48
(C)52
(D)96

STRATEGY
To solve this problem, first find n, the number of paving blocks needed for the rock garden’s perimeter.

THINK
- The total perimeter of the rock garden is 2(8 + 4).
- To convert to inches, multiply the perimeter, which is in feet, by 12: 2(8 + 4) × 12.
- Since the square paving blocks are 6 inches on a side, the total needed for the perimeter can be expressed as 6n.
- The equation for this problem is the following:

6n = 2(8 + 4) × 12
Simplify:
6n = 2(12) × 12
6n = 24 × 12
6n = 288


n = 48
- Finally, add 4 paving blocks for the four corners of the rock garden’s rectangular border. (You can sketch a diagram to see why these are needed.) That will make the border continuous. 48 + 4 = 52. Julio should purchase 52 paving blocks. (C) is correct.

M.1.6 SOLVE REAL-WORLD PROBLEMS INVOLVING PERCENTAGES
Percentage problems crop up frequently in real-world situations. They can involve anything from salary increases to price reductions. On the TEAS Mathematics exam, you will use the concept of percent of a fixed quantity and percentage increase and decrease to solve word problems.

There are three basic types of percentage problems that all fit the same basic framework. That framework has three parts: PERCENT, PART, and TOTAL.

If the variable, or the answer you are seeking, is the PART that is a certain PERCENT of the TOTAL, the equation is:
PART = PERCENT × TOTAL or x = PERCENT × TOTAL

If the variable is the PERCENT, the equation is:


If the variable is the TOTAL, the equation is:



Here are some reminders about working with percentages.

Quick Tips: Percentages
- Find 10% of a number: Move the decimal point 1 place to the left.
- Find 50% of a number: Divide in half.
- Find 1% of a number: Move the decimal point 2 places to the left.
- Find 25% of a number: Divide in half. Then divide in half again.
- Find 20% of a number: Move the decimal point 1 place to the left to find 10%. Then double that figure to make 20%.
- Find 5% of a number: Move the decimal point 1 place to the left to find 10%. Then find half of that figure to make 5%.
- 100% of any number is the number itself.

Now look at a sample word problem.
Shelley’s basketball team has made 120 three-point baskets this season. If Shelley has made 45% of these baskets, how many three-point baskets has she made?

THINK
- This problem is asking for the PART of the TOTAL number. The equation to use is: PART = PERCENT × TOTAL.
y = .45 × 120
y = 54
- Shelley has made 54 three-point baskets.
- When checking your answer, remember that 45% is less than half, so the correct answer is slightly less than half of 120. Thus 54 is a reasonable answer.



To solve a real-world problem involving percentages, you must use percent of a quantity or percent increase and decrease to calculate the answer.

M.1.6 PROBLEM 1
Last year, 128 babies were born on the fifth floor of the hospital. This year, 160 babies were born on the fifth floor. What was the percentage increase in babies born on the fifth floor?
(A)20%
(B)28%
(C)25%
(D)16%

STRATEGY
Create a basic type of percentage framework comparing the increase with the original number.

THINK
- You want to find the
- The PART is the change, 160 – 128 = 32. Always calculate change as the new amount minus the original amount. If it is positive, it is an increase; if it is negative, it is a decrease.
- The TOTAL is the original amount, 128.
- So (answer choice C).

M.1.6 PROBLEM 2
At an electronics store sale, Jasmine purchased a computer tablet that was marked down 23% to $346.50. What was the original price of the tablet?
(A)$450
(B)$506.50
(C)$875
(D)$1,506.52

STRATEGY
This problem is asking you to find the TOTAL, or original price before the discount.

THINK
- Use the equation
- Now remember that $346.50 is the price after a 23% discount. So $346.50 is 77% of the original price, or total (100 – 23 = 77).
- Plug in the values to write an equation:


- Solve: x = 450
- The original price of the tablet was $450. (A) is correct.

M.1.7 APPLY ESTIMATION STRATEGIES AND ROUNDING RULES TO REAL-WORLD PROBLEMS
As a nurse, you will primarily use metric system measurements in your work. Thus it is important to be able to estimate these measurements in length, area, volume, and weight. Estimating strategies can also help you solve real-world problems more quickly. On the TEAS exam, you will demonstrate the use of estimation strategies and rounding rules to solve real-world problems.
Remember these approximations for units of measurement in the metric system.

Metric Unit - Real-World Approximation
Gram (g) - weight of one paperclip
Kilogram (kg) - weight of a kitten
Millimeter (mm) - grain of sand
Centimeter (cm) - width of a little finger
Meter (m) - height of a table
Kilometer (km) - 12-minute walk for an adult
Liter (l) - contents of a bottle of soda

Estimation is a way of simplifying numbers to make a problem easier to solve. It uses methods of rounding and mental math.
Rounding also is a way of simplifying numbers. For example, a number like 499,732 can be rounded to 500,000 for simpler calculations and to estimate answers.

The method for rounding is to look at the next digit after (to the right of) the place value to be rounded. If it is less than 5, just drop that digit and all the ones to the right (inserting zeroes if necessary).

If it is 5 or more, add 1 to the digit to be rounded. This method works whether the number is a whole number or a decimal. You only have to know which digit is being rounded.

For example, you can round 1,362 to the nearest hundred. First, look at the 6—the digit to the right of 3, which is in the hundreds place. Since 6 > 5, you change the 3 to 4 and fill in zeroes for the rest of the placeholders. So 1,362 rounded to the nearest hundred is 1400.
You can also round fractions and mixed numbers. To round a mixed number to the nearest whole number, check the numerator of its fraction. If the numerator is equal to or greater than half the fraction’s denominator, you should round up. If it is less than half the denominator, you should round down.

For example, for the mixed number the numerator of the fraction, 3, is greater than half of the denominator 4. You should round this mixed number up to 4. For the numerator 3 is less than half of the denominator 8. You should round this mixed number down to 3.


To apply estimation strategies and rounding rules to solving problems, you must know when estimations or rounding is appropriate and practice using these strategies correctly.

Mental math is a term used for any calculation you can do in your head. For example, if you have to distribute 200 pills equally among 10 containers, you know you should not put 30 pills in the first container. You can calculate right away that each container should have 20 pills.
Often mental math is a form of estimation. Say you have to distribute 96 pills equally among 6 containers. Your choices are (A) 5 pills per container, (B) 16 pills per container, (C) 21 pills per container, and (D) 43 pills per container. You can eliminate answer choices (A) and (D) as being too small and too large. But what about the answers 16 and 21? Mental math helps you determine that 21 pills is too large a number because 20 × 6 is more than 96.
M.1.7 PROBLEM 1
What is 2,346 rounded to the nearest 100?
(A)2,400
(B)2,000
(C)3,000
(D)2,300

STRATEGY
Circle the place value you are rounding to; then look to the right.

THINK
- Because you are rounding to the hundreds place, circle the 3.
- Look to next place value on the right in the tens place. If the tens place digit is 5 or greater, round up. If it is less than 5, round down.
- The 4 in the tens place is less than 5, so round down.
- Write zeroes in the places to the right of the number you are rounding.
- 2,346 rounded to the nearest hundred is 2,300. The correct response is (D).



M.1.7 PROBLEM 2
What is 19.796 rounded to the nearest hundredth?
(A)19.7
(B)19.80
(C)19.79
(D)19.8

STRATEGY
Circle the place value you are rounding to; then look to the right.

THINK
- Circle the place value you want to round to. Here, you are rounding to the hundredths.
- Look to the next place value on the right.
- The 9 rounds up, so it turns to “10.” Write a 0 and “carry” the 1 to the tenths place, turning the 7 into an 8.
- 19.796 rounded to the hundredths place is 19.80. The correct response is (B).
- Key fact: Always place a digit in the place value you’re rounding to, even if it is a zero. The value 19.80 is different from 19.8 because it tells you that the number is accurate to the hundredths place, not just the tenths place.


 

You can eliminate answer choices (A) and (D) because they are expressed to the nearest tenth, not the nearest hundredth.

M.1.7 PROBLEM 3
A 4324-lb truck needs to carry a load across a bridge with a legal limit of 6,400 lbs. Which of the following would be the largest load that the truck could legally carry on the bridge?
(A)2,100 lbs
(B)2,000 lbs
(C)2,050 lbs
(D)2,150 lbs

STRATEGY
You don’t need an exact answer for this problem. You can use estimation to find a sum that is safely less than 6,400 lbs.

THINK
- The weight of the truck plus the load must be less than the bridge’s limit. Substituting the known numbers, the weight of the load has to be less than the bridge limit minus the truck weight, or 6,400 – 4,324.
- Do the subtraction to find the maximum weight of the load: 6,400 – 4,324 = 2,076. The answer choice that is closest to this weight without going over it is 2,050, answer choice (C).
- Alternatively, using estimation, by rounding the truck weight to 4,300 pounds, the difference can be figured to be less than 2,100 pounds, so answer choices (A) and (D) are eliminated.
- Of the remaining choices, since the question asks for the largest load, use the larger number, 2,050, and check whether that would be too large: 4,324 + 2,050 = 6,374. Since this is less than the bridge limit of 6,400, the answer is (C) 2,050 lbs.

M.1.8 SOLVE REAL-WORLD PROBLEMS INVOLVING PROPORTIONS
Setting up and solving problems with proportions is an important practical skill for nurses. Proportions enable you to scale dosages up or down as needed, or prepare solutions of a desired strength. On the Mathematics section of the TEAS exam, you must set up and solve real-world problems that involve ratios and proportions.

A ratio is a fractional relationship between two quantities.

A proportion is a mathematical sentence that states two ratios are equal to each other. When you set up a proportion, be sure that the units for the values in the numerator correspond to each other, and that the same is true for the denominators. In other words, the two ratios should both compare the same two types of things.

Proportions can be written in two ways: 3 : 4 = 12 : 16 or These proportions are read as “3 is to 4 as 12 is to 16.”
Proportions with variables are called algebraic proportions. They are solved by cross-multiplication. This means that you multiply in the form of a cross (×)—each numerator times the opposite denominator. The two products are equal. Thus, for the proportion cross-multiplication yields the equation ad = bc.

Here is a proportion with a variable x. You can solve it by cross-multiplying.




To solve a real-world problem involving proportions, you must set two ratios in fraction form equal to each other and solve for the variable.

M.1.8 PROBLEM 1
A nurse adds 4 g of salt to 20 ml of water to make a saline solution. How much salt should be added to 75 ml of water to make a solution of the same strength?
(A)0.25 g
(B)10 g
(C)15 g
(D)150 g

STRATEGY
Use a ratio and a proportion to solve the problem.

THINK
- Here the ratio is 4 g salt to 20 ml water, or 1 to 5.
- Write n for the unknown quantity of salt.


- Cross-multiplying as shown yields a simple equation, 5n = 75.
- Solve the equation in the normal way. The nurse would need 15 g of salt (answer choice C).



M.1.8 PROBLEM 2
Tamara reads 40 pages of a novel in 52 minutes. How long will it take her to read a novel of 200 pages?
(A)2 hours, 40 minutes
(B)3 hours, 50 minutes
(C)4 hours, 10 minutes
(D)4 hours, 20 minutes

STRATEGY
Set up a proportion with ratios of pages/minutes to solve the problem.

THINK
- The proportion should be 40:52 = 200 : n. The variable n represents the time it takes Tamara to read 200 pages.
- Cross-multiply to get 40n = 200 × 52.
- Then solve: 40n = 10,400.


n = 260
- The answer, 260, represents number of minutes. Convert to hours and minutes: 260 minutes = 4 hours, 20 minutes. Answer (D) is correct.


An easier way to solve the problem is to look for mathematical relationships. You know that 40 × 5 = 200, so the number of minutes Tamara needs to read the novel is 5 × 52, or 260.

M.1.9 SOLVE REAL-WORLD PROBLEMS INVOLVING RATIOS AND RATES OF CHANGE
Proportions are often used to solve problems involving ratios and rates. How many calories a serving of food has, how many miles a vehicle goes per gallon of fuel, how much money a job pays per hour—these are all rate problems that are fairly easy to solve. On the TEAS, you must solve real-world problems involving ratios and rates of change.

A rate is a ratio expressed with numbers and units:
A unit rate is a rate expressed as a quantity of one:
A word problem on rate is solved by using a proportion. For example: A typist can type 80 words in 60 seconds. How many seconds would it take this typist to type 200 words?
Set up the proportion:
Solve for x:
Rate of change can also be used to compare two points on a graph, or two points that each have an x-coordinate and a y-coordinate. The x-coordinate is a number along the horizontal axis of the graph. The y-coordinate is a number along the vertical axis. Rate of change is measured as the slope of the line between two points on the graph. Slope is equal to the change in the y-coordinates divided by the change in the x-coordinates. Slope can thus be expressed as m = It can also be expressed as or the change in vertical position over the change in horizontal position.


To solve a real-world problem that involves rate of change, you must compare two ratios to describe how one thing changes in relation to something else.

M.1.9 PROBLEM 1
Renay rode her bike 3.2 miles in 12 minutes. At this rate, how long will it take her to ride the entire 42-mile trip from her house to Santa Fe?
(A)157.5 min
(B)1.57 hr
(C)217 min
(D)2.57 hr

STRATEGY
Use a proportion to solve the problem.

THINK
- Renay’s speed can be expressed as a ratio: 3.2 miles to 12 minutes. Express the ratio as a fraction.
- Set up a proportion using the ratio above and the 42-mile distance to Santa Fe.
- Solve as you would normally. It would take 157.5 minutes to ride to Santa Fe, making answer choice (A) the correct response.
- Key fact: In a proportion, make sure that units correspond. Here, for example, both numerators have miles, and both denominators have minutes.


M.1.9 PROBLEM 2
Look at the graph below.

What is the slope of the line AB ?

(C)2
(D)3

STRATEGY
Use the formula for slope, or rate of change, to solve this problem.

THINK
- The formula for slope is
- The coordinates for point A are (4, 4). The coordinates for point B are (1, 2). Plug the x and y coordinates into the formula.


- Simplify: The slope of the line AB is Answer (B) is correct.

M.1.10 TRANSLATE PHRASES AND SENTENCES INTO EXPRESSIONS, EQUATIONS, AND INEQUALITIES
A word problem based on a real-world situation must be translated into mathematical language to be solved. On the TEAS Mathematics exam, you must translate written language into mathematical expressions, equations, and inequalities.
A mathematical expression is a symbol or combination of symbols to show numbers, variables, operations, and grouping. Examples of expressions include x, (5 – x), and 2(5 – x). Words in problems can be translated into an expression, as follows.

nine more than twice a number      2n + 9
seven less than four times a number      4x – 7
the product of a number and 25      25y
thirty-two divided among a number t

An equation is a statement that two expressions are equal. Word problems can be translated into equations to be solved.
Two fewer than seven times a number equals forty-seven.     7n – 2 = 47
Dividing 975 by a number results in 162.5.
An inequality is a statement that two expressions are unequal. Word problems can also be translated into inequalities to be solved.
A number divided by six is greater than 17.

Three times a number is less than 124 divided by four.

To translate a word problem into expressions, equations, and inequalities, read the problem carefully and look for key words that tell the quantities, operations, and variables.

M.1.10 PROBLEM 1
Sergio had twelve fewer credits than five times what Joanna had. Which of the following expressions describes the number of Sergio’s credits?
(A)12x – 5
(B)5x + 12
(C)5(12x)
(D)5x – 12

STRATEGY
Joanna’s number of credits is unknown. It should be written as the variable x.

THINK
- “Twelve fewer” translates to –12.
- “Five times what Joanna had” translates to 5x.
- The expression for the number of Sergio’s credits is 5x – 12. Answer choice (D) is correct.

M.1.10 PROBLEM 2
Ninety-two dollars added to five times Herbert’s weekly salary is less than Mrs. Morton’s weekly salary of $1,119. Which of the following describes this situation?
(A)5s – 92 > 1119
(B)5s + 92 < 1119
(C)(5s)92 < 1119



STRATEGY
Set up this problem as an inequality between the two weekly salaries.

THINK
- “Five times Herbert’s weekly salary” translates to 5s. Ninety-two dollars added to this is 5s + 92.
- The expression above is less than Mrs. Morton’s weekly salary. The symbol for less than is <. The inequality should be 5s + 92 < 1119. Answer choice (B) is correct.

M.2 MEASUREMENT AND DATA
M.2.1 INTERPRET RELEVANT INFORMATION FROM TABLES, CHARTS, AND GRAPHS
In the nursing profession, you will see a wide variety of graphics: tables, charts, and graphs that convey all kinds of information. To use this data, you must understand how to read different kinds of graphics. On the TEAS, you will demonstrate the ability to read and interpret relevant data from tables, charts, and graphs.

A line graph displays a number of data points plotted on an axis grid and connected with lines. Line graphs are often used to show changes over time.
A bar graph uses vertical or horizontal bars on a grid to show changes over time or to compare quantities. For example, a bar graph could show the annual number of births in the U.S. from 2005 to 2015.
A table presents data arranged in rows and columns. For example, a table might list the number of goals each player on a soccer team scored during the season.
A circle graph (or pie chart) presents data as portions of a circle, or percentages of a whole. For example, a circle graph could show how much of a school’s athletic budget goes to each sports team.
Always read the title of a table, chart, or graph to see what data it presents. Also read the labels on the horizontal and vertical axes. Look for a legend, which explains the data used in the graphic.

To interpret information from tables, charts, and graphs, you must understand how each kind of graphic is set up and what it is designed to do.

M.2.1 PROBLEM 1
The circle graph shows time allocation in hours for nurses on Floor 3 at the Sound Shore Hospital. Use this graph for the following two problems.
Sound Shore Hospital: Nurse Time Per Shift (hrs)

What percentage of the time do nurses spend on prep work and paperwork?
(A)12%
(B)36%
(C)17.5%
(D)48.5%

STRATEGY
Use the data in the graph and the proportion method of finding percentages to solve the problem.

THINK
- Add to find the total number of hours in a shift: 8 hr.
- Find the total amount of time spent on paperwork and prep work: 1.4 hr.
- Use a proportion to find the percentage of time spent doing prep work and paperwork: 1.4 out of 8 = 17.5% (Choice (C)).
Total = 5.2 + 0.7 + 0.7 + 1.4
= 8 hr
Paperwork



M.2.1 PROBLEM 2
One nursing textbook states that nurses in a good-quality facility will spend at least 60% of their time in patient care, but in the best facilities, nurses devote over 70% of their time to patient care. What can you conclude about Sound Shore Hospital from the graph on the previous page?
(A)The hospital rates are substandard.
(B)The hospital is rated good but not the best.
(C)Sound Shore rates among the best hospitals.
(D)Sound Shore rates above even the best hospitals.

STRATEGY
Find the amount of time in percentage devoted to patient care using the equation method of finding percentages.

THINK


- Find the percentage.
- Evaluate the percentage. Sound Shore qualifies as good but not the best because it is more than 60% but less than 70%, making (B) the correct answer choice.

M.2.2 EVALUATE THE INFORMATION IN TABLES, CHARTS, AND GRAPHS USING STATISTICS
A data distribution shows how data points are clustered or spread out on a graph. This enables you to identify important trends in the data. On the TEAS Mathematics exam, you must analyze trends in data and be able to calculate measures of central tendency.

The measures of central tendency (mean, median, and mode) tell you something about the trend of the data.

The mean is the average value of the data, and it gives an idea of a typical value. To calculate the mean, add all of the data points and divide by the number of data points.

The median is the “middle” value in a set of data. It is the value at which half the data points are above and half are below. To find the median, put the data in order from smallest to largest and pick the data point that is in the middle. If there are two points in the middle, add them together and divide by 2.

The mode is the most frequent value in the data set. Be sure to use the value, not the number of times it appears.

All three measures tell something important about the data set, but they emphasize different things. Which one is used by statisticians (or news reporters) depends on the situation or the point the person is trying to make.

Data can also be displayed on a graph as a data distribution of points. The spread in a data distribution is the range of values. Typically data is distributed symmetrically in a bell curve, with data points clustered in a single peak in the middle, with fewer points to either side of the peak. Isolated data points that do not fit with the other data are called outliers. These are unexpected values that depart from the expected trend of the data.

A unimodal data distribution has one clear peak of values. A bimodal distribution has two clear peaks.

To use statistics to evaluate data in graphics, you must employ measures of central tendency, which include mean, median, and mode.

M.2.2 PROBLEM 1
The table below shows Hector’s scores from seven different judges in a gymnastics competition. Use this table to answer the next two questions.


M.2.2 PROBLEM 2
Which was greater, Hector’s median score or mode score? By how much?
(A)The mode was 0.15 greater than the median.
(B)The mode and median were both the same, 8.6.
(C)The mode was 0.15 less than the median.
(D)The mode was 0.5 greater than the median.

STRATEGY
Identify the mode and median in the data set. Then compare the two values.

THINK
- The median is the value that appears as the central value in the data set when it is arranged from least to
- greatest. So 8.6 is the median.
- The mode is the value that appears most frequently in the data set.
- So 8.6 is the mode because it appears three times.
- The median and mode are the same for this data set, 8.6, meaning that answer choice (B) is the correct response.



M.2.2 PROBLEM 3
Which score change would push the mean value of Hector’s scores to 9.0 or above?
(A)Judge 1 increasing her score to 9.5.
(B)Judge 2 increasing his score by 2.1.
(C)Judge 5 decreasing her score by 0.5 to 9.0.
(D)Judge 2 increasing his score to 9.5.

STRATEGY
The mean is the average score. Find the mean by adding to get the total of all scores and dividing by the number of scores. Find a score change that will result in a new mean of 9.0.

THINK
- Add to find the current total. Divide by 7 to get the mean: 8.7
Mean: 8.6 + 7.8 + 8.6 + 8.2 + 9.5 + 8.6 + 9.6 = 60.9 60.9 ÷ 7 = 8.7
- Now find the total that will yield a mean of 9.0
Mean of 9.0: total ÷ 7 = 9.0
n ÷ 7 = 9.0 → n = 63
- Finally, find the answer choice that will increase the total by (63.0 – 60.9 = 2.1) points.
- Answer choice (B) is the correct response.
- Check: Increasing Judge 2’s score by 2.1 gives a mean of 9.0.



M.2.3 EXPLAIN THE RELATIONSHIP BETWEEN TWO VARIABLES
Life is full of examples in which changes in one variable cause another variable to change. As you purchase more groceries, your supply of cash goes down. As you drive more miles, the gas in your tank decreases. On the TEAS exam, you must identify independent and dependent variables and how they are related.

An independent variable stands alone and is not affected by other variables you are measuring.
A dependent variable is affected by changes to the independent variable. For example, in a science experiment, several pea plants are exposed to different levels of sunlight. The lighting levels are the independent variable. The growth rate of the plants is the dependent variable. The dependent variable is what is being tested and measured.

The correlation in changes between two variables is called covariance. If both variables increase, there is a positive covariance, or positive correlation. The variables are directly related. If one variable decreases as the other increases, there is a negative covariance, or negative correlation. The variables are inversely related.

In graphing two variables, the independent variable is plotted on the x-axis and the dependent variable is plotted on the y-axis.

To explain the relationship between two variables, you must describe how changes in one variable affect changes in a second variable.

M.2.3 PROBLEM 1
Which of the following pairs of variables has a negative covariance?
(A)x = number of snowfalls in one winter, y = number of snow shovels sold
(B)x = number of hours Reza works in a month, y = amount of Reza’s paycheck
(C)x = time spent running on a treadmill, y = number of calories burned
(D)x = number of customers renting a room in a motel, y = number of rooms available

STRATEGY
To solve this problem, remember that negative covariance means that as one variable increases, the other decreases.

THINK
- Answer choices (A), (B), and (C) all show positive covariance. As one variable increases, the other variable also increases.
- For (D), as more customers rent rooms in a motel, the number of rooms available decreases. This is a negative covariance, or negative correlation. Answer choice (D) is correct.

M.2.4 CALCULATE GEOMETRIC QUANTITIES

The ability to calculate perimeter and area is a useful everyday skill. You might have to figure how much paint is needed for the dining room ceiling or how many pavers are required to line the front garden. On the Mathematics section of the TEAS exam, you must calculate perimeter and area for various figures, both regular and irregular.

The perimeter of a figure is the sum of all the sides. Perimeters are measured in units such as inches or centimeters.

The area of a figure is what is enclosed within the perimeter. Area is measured in square units, such as in2 or cm2. It is a measurement of space within a flat, two-dimensional boundary. Total surface area is the sum of the areas of all the surfaces in a three-dimensional object, such as a cube. Total surface area can also measure the surface space of a curved object, such as a sphere or cone.

A rectangle is a four-sided figure with four right angles (90°). A rectangle has a length (l) and a width (w). These opposite sides are equal. Generally the longer sides are designated l, but it actually doesn’t matter which is called which.



The perimeter of a rectangle is l + w + l + w, or
p = 2l + 2w

The area of a rectangle is length times width, or
A = l × w

A square is a special type of rectangle with all equal sides, usually called s, so its perimeter is p = 2s + 2s, or
p = 4s

The area of a square is s × s, or
A = s2.

A triangle has three sides, a, b, and c, so its perimeter is simply
p = a + b + c

The area of a triangle is essentially half of the area of a four-sided figure. However, it is important that these two measures are perpendicular to each other, not just any two sides. Therefore, the area of a triangle is given as


where the height of the triangle (h) is any length that extends from a corner to a side (called the base, b) and forms a right angle. Two types of triangles and their parts are labeled below. The small square indicates “perpendicular.”




A circle is a figure with all its points the same distance from its center.



The perimeter of a circle is called the circumference. It is calculated as
C = πd or 2πr
where π is pi (approximately 3.14); d is diameter, or the distance across the circle through its center; and r is radius, or the distance from the center to any point on the circle. Notice that the circle’s diameter is twice as long as the radius. The area of a circle is given as
A = πr 2

Some figures may be made up of several shapes, such as rectangles and triangles. For example, you can calculate the area of a figure such as


by dividing it into a rectangle and two triangles, if you are given the measurements.


In this case, the two triangle bases are equal and they total b – l, so the area is calculated by adding the area of the rectangular part (lh) and the areas of both triangles, The total area is



To calculate geometric quantities, you must be able to find the perimeter and area of regular and irregular shaped figures.

M.2.4 PROBLEM 1
A basketball court has two areas on either end called the keys or the free throw lanes. A diagram of a free throw lane is shown below.



What is the total perimeter measurement of the free throw lane?
(A)80.84 ft
(B)68.84 ft
(C)56 ft
(D)87.68 ft

STRATEGY
Add together the perimeter of the rectangular section (minus one side) and half the circumference of the circle.

THINK
- The perimeter of the three sides of the rectangular section is 19 + 19 + 12 = 50. (The other side that measures 12 is not included.)
- Now you need a perimeter measurement for half the circle. The circle has a radius of 6. So half the circumference is


- Add 50 + 18.84 = 68.84. The total perimeter of the free throw lane is 68.84 ft. Answer choice (B) is correct.

M.2.4 PROBLEM 2
What is the total area of the free throw lane in the diagram shown a little above?
(A)228 ft2
(B)248.52 ft2
(C)284.52 ft2
(D)341.04 ft2

STRATEGY
Add together the areas of the rectangular section and the half circle.

THINK
- Use the formula for area to find the area of the rectangular section of the free throw lane.
A = l × w
= 19 × 12 = 228
- Then find the area of the half circle, which is


- Add the two areas together: 228 + 56.52 = 284.52. The total area of the free throw lane is 284.52 ft2. Answer choice (C) is correct. Notice that answer choice (D) is incorrect because it included the area of the whole circle plus the area of the rectangle.

M.2.4 PROBLEM 3
What is the area of the figure below?

(A)526 cm2
(B)574 cm2
(C)696 cm2
(D)588 cm2

STRATEGY
Use the formulas for areas of a rectangle and triangle to find the total area.

THINK
- First use the formula A = l × w to find the area of the rectangle: A = 12 × 36 = 432 cm2.
- Then use the triangle formula, to find the area of the triangle on the left. Use 16 cm as its base and 12 cm as its height:
- Use the triangle formula a second time to find the area of the triangle on the right. Use 12 cm as its base and 10 cm as its height:
- Add the three areas to get the total area: A = 432 cm2 + 96 cm2 + 60 cm2 = 588 cm2. This makes (D) the correct answer choice.

M.2.5 CONVERT WITHIN AND BETWEEN STANDARD AND METRIC SYSTEMS
A nurse must deal accurately with units of measurement every day. A mistake in converting between standard and metric units could endanger a patient’s health and safety. On the exam, you must demonstrate the ability to convert units within and between the standard and metric systems of measurement.

The system of measurement commonly used in the United States is the standard system. (It is also called United States customary units.) The standard system is summarized in the chart below.

Standard System of Measurement

Length
Weight
Volume

12 inches = 1 foot
16 ounces = 1 pound
1 cup = 8 fluid oz

3 ft = 1 yard
2000 lb = 1 ton
1 pint = 2 cups

5280 ft = 1 mile
1 quart = 2 pints
1 gallon = 4 quarts

The standard system has various conversions (seemingly unrelated), and therefore, you must refer to a table such as the one above for conversion.
The measurement system used by most countries in the world is the metric system. This system is based on 10s, the same as decimals, so you often need only move the decimal point to convert between measurement units. The metric system is summarized in the chart below.

Metric System of Measurement

Length
Weight
Volume

1000 millimeters = 1 m
1000 milligrams = 1 g
1000 milliliters = 1 ℓ

100 centimeters = 1 m
100 centigrams = 1 g
100 centiliters = 1 ℓ

10 decimeters = 1 m
10 decigrams = 1 g
10 deciliters = 1 ℓ

1 meter = 1 m
1 gram = 1 g
1 liter = 1 ℓ

1 dekameter = 10 m
1 dekagram = 10 g
1 dekaliter = 10 ℓ

1 hectometer = 100 m
1 hectogram = 100 g
1 hectoliter = 100 ℓ

1 kilometer = 1000 m
1 kilogram = 1000 g
1 kiloliter = 1000 ℓ


Sometimes it is necessary to convert between the metric and standard systems of measurement. For example, conversion is necessary when a dosage is given per body weight in kilograms, but the patient’s weight is given in pounds.
The chart below includes some conversions from the metric system to the standard system.

Metric/Standard Conversion

Length
Weight
Volume

1 km = 0.62 mile
28.35 g = 1 oz
1 liter = 33.81 fl oz

1 m = 39.37 in
1 kg = 2.205 lb
3.79 liters = 1 gallon

On the TEAS Mathematics exam, you will see conversion problems like the following:
Betsy has a 2.5-gallon jug of saline solution. How many ml does the jug contain?

STRATEGY
To solve the problem, you should use a series of ratios and then cancel units.

THINK
- Write the quantity as a fraction:
- Multiply your ratio by another identity ratio (from the table) that cancels with your units.
- Multiply by other identity ratios until you obtain the units you are looking for—milliliters.


- 2.5 gallons of saline is equal to 9475 ml.

To convert between standard and metric systems, you must know conversion factors and how to calculate equivalent values between measurement systems.

M.2.5 PROBLEM 1
A nurse poured 3.2 quarts of liquid into a container. How many fluid ounces were in the container?
(A)96 fl oz
(B)44.4 fl oz
(C)102.4 fl oz
(D)202.6 fl oz

STRATEGY
Use a series of ratios to solve the problem. Cancel units.

Think
- Write the quantity you want to convert as a fraction:
- Multiply your ratio by another identity ratio (from the table) that cancels with your units: Quarts cancel with quarts.
- Keep multiplying by different identity ratios until you reach the units you are looking for—in this case, fluid ounces.


- 3.2 quarts is equal to 102.4 fluid oz. (Answer choice (C) is correct.)
- Key facts: Units are the key to conversion problems. Start with the units you have. Keep multiplying by identities to get to the units you want. Note: If the units of your answer are not correct, your answer cannot be correct.

Use common sense and common knowledge to solve problems. For this problem, you may be aware that You can use this conversion identity directly to solve the problem rather than go through a series of conversions.

M.2.5 PROBLEM 2
45.6 cm equals how many hectometers?
(A)45,600 hm
(B)0.456 hm
(C)0.00456 hm
(D)4.56 hm

STRATEGY
Move the decimal point using powers of 10 to convert.

THINK
- Start with the units you want to convert: centimeters.
- Count the number of places you need to go to reach hectometers: 4 places.
- Move the decimal point 4 places LEFT to go from centimeters to hectometers.
Decimal point in 45.6 moved 4 places left = 0.00456
- This means that answer choice (C) is correct.

- Check: A common-sense check for a conversion problem is of critical importance. 45.6 centimeters is about the length from your fingertips to your elbow. So it must be a small fraction of a hectometer, which is 100 m in length.


A centimeter, the size of your fingernail, is much smaller than a hectometer, which is longer than a football field. So you can definitely rule out answer choices (A) and (D) as far too large to be correct.

M.2.5 PROBLEM 3
Each day, a 125-lb patient is supposed to receive 0.8 mg of medication per kilogram of body weight. Which dosage should the patient receive?
(A)4.56 mg
(B)456.4 mg
(C)45.36 mg
(D)22.5 mg

STRATEGY
Write equations to solve the problem, based on the conversion table.

THINK
- Use a proportion to convert the patient’s weight to kilograms.
- Write an equation to find the dose.
- Calculate the dose.
- The patient should receive 45.36 mg of medication, answer choice (C).