Word Problems, Percentages, and Decimals

A number system based on the number 10. Each unit is measured by how far to the right of the decimal point its place value represents.

Decimal Fraction

A fraction whose denominator is a power of 10; it expresses the position by units to the right of the decimal point.

Decimal Point

Used to distinguish decimal fractions or to distinguish the whole number from a decimal fraction.

Percentage, or Percent

A representation of a proportion based on 100. Percentage can represent the amount of change (positive or negative) based on the total amount involved.

Objective 1

Math plays a vital role in the delivery of services in a health care environment. Sometimes the math appears in an obvious, straightforward statement, but some orders are more complicated because of the available form of medication, fluids, or treatment modality that has been prescribed. One thing is certain—most mathematical problems appear in the form of a word problem.

How many of you just groaned? I am sure it was quite a few of you. For some reason, most people are not comfortable with solving word problems. They give such reasons as “I’m not sure if I performed the proper operation,” “My answers never seem to make sense,” and “I was never good with word problems in school.” Yet, when I ask a student to estimate how much it will cost to fill up his or her 20-gallon gas tank at $3.00 per gallon, the student can quickly come up with the answer of $60.00. Students argue with me that this is not a word problem or not really math, just a factor of everyday life.

We perform mathematical calculations throughout every day. Many are in the form of word problems, and the most common word problems in health care have to do with medication calculations. Why is this? Because many medications do not come in the exact dosage that the physician has prescribed, or the dose is based on the person’s weight.

Take a moment to write down a statement or two regarding how you feel about word problems. Are you comfortable solving them? Or do you feel uncertain about your skills? The first step in working through math anxiety is to identify the cause of the anxiety.

MATH IN THE REAL WORLD 4-1
Word Problems of Everyday Life

• How much tip should you add to the bill?

• How much will it cost to fill up your gas tank?

• Did you collect the 20% coinsurance?

• Do you have enough to cover the cost of groceries if the bill is $275.00?

• Did you receive the correct change?

• Do you have enough time to run an errand while your child attends a 30-minute music lesson?

Objective 2

Understanding word problems

What is it about word problems that gives people the most difficulty? When I ask this question in class, the most common answer is, “I’m so worried about the numbers that I miss the key words in the problems.” Most of the time, we do not communicate by writing math problems. We use words to describe the relationship between the numbers and the information we are trying to obtain (Boxes 4-1 and 4-2 and Table 4-1).

BOX 4-1 Key Terms for Addition and Subtraction Word Problems

Addition Problems	Subtraction Problems
Increase	Decrease
Plus	Minus
Additional*	Difference
More	Fewer
Total	Remainder
Sum	Less
	Reduced

^*See also Table 4-1.

BOX 4-2 Key Terms for Multiplication and Division Word Problems

Multiplication Problems	Division Problems
Of	Ratio
Twice	Half
Product	Quotient
Times	Divided by
At	Divided into
Total

TABLE 4-1

Key Phrases for One or More Computations

Key Phrase	Operation
What is the total?	Addition
What is the average?	Addition and division
What is the total cost?	Addition
What is the discount?	Multiplication
What are the hidden charges?	Depending on how the problem is worded, involves addition and subtraction
What is the total discount?	Depending on how the problem is worded, involves multiplication, addition, or subtraction

Objective 2

How to solve a word problem

Depending on what the question is asking, many of us can solve simple or straightforward word problems in our heads. The difficulty occurs when a fellow student or coworker asks you to explain how you came up with your answer. Many of us find it hard to explain the steps we performed to obtain the correct answer because, to us, the answer is obvious. For example, your lunch is $12.50 with tip. You give the waiter a $20.00 bill. How much is the change? Without writing anything down, many of you came up with the answer $7.50. How did you come up with the answer?

The first step in solving any word problem is to read the problem completely. Circle key terms that will help, identify the operation you must perform to obtain the answer, and check to make sure your answer satisfies the question. Does the answer make sense?

STRATEGY 4-1
Solving Word Problems

1. Read the question.

2. Some students find it helpful to underline what the question is asking.

3. Identify what you are trying to find.

4. Pull out important information in the problem.

5. Identify the mathematical operation necessary to answer the question.

6. Perform the mathematical calculation accurately.

7. Label the answer.

8. Check to make sure the answer satisfies the question that was asked in the problem.

9. Ensure that the answer makes sense.

Use the steps in the box Strategy 4-1 to solve the following word problem: Kevin spends a total of $30.00 for lunch during a 5-day workweek. What is the average amount he spends on lunch each day?

Solution for the example question:

1. Read the question.

Did you read the entire question? If not, go back and reread the question.

2. Underline what the question is asking.

What is the average amount Kevin spends on lunch each day?

3. Identify what you are trying to find.

What is the average amount Kevin spends on lunch each day?

4. Pull out important information in the problem.

a. Kevin spends a total of $30.00 for lunch in a 5-day workweek.

b. What is the average amount Kevin spends on lunch each day?

5. Identify the mathematical operation necessary to answer the question.

To find the average, divide the total amount spent by the number of days in the workweek (30 ÷ 5).

6. Perform the mathematical calculation accurately.

30 ÷ 5 = 6

7. Label the answer.

$6.00

8. Check to make sure the answer satisfies the question that was asked in the problem.

Kevin spends on average $6.00 a day for lunch.

9. Ensure that the answer makes sense.

Yes, the answer makes sense.

Let’s try one more example: Amy is ordering supplies for the physician’s office. Figure out the total cost of supplies based on the information given:

Gloves	$3.50/box	Need 3 boxes
Gauze	$4.00/box	Need 2 boxes
Needles	$14.00/box	Need 3 boxes

Solution for the example question:

1. Read the question.

Did you read the entire question? If not, go back and reread the question.

2. Underline what the question is asking.

What is the total cost of the supplies Amy needs to order?

3. Identify what you are trying to find.

What is the total cost of the supplies Amy needs to order?

4. Pull out important information in the problem.

Identify the total cost of each supply.

5. Identify the mathematical operation necessary to answer the question.

a. Determine the total cost of each supply

b. Add the total cost of each supply to obtain the grand total of the order.

6. Perform the mathematical calculation accurately.

Gloves	$3.50/box	Need 3 boxes
$3.50 × 3 = $10.50
Gauze	$4.00/box	Need 2 boxes
$4.00 × 2 = $8.00
Needles	$14.00/box	Need 3 boxes
$14.00 × 3 = $42.00
Add the sums to obtain the total cost: $10.50 + $8.00 + $42.00 = $60.50

7. Label the answer.

$60.50

8. Check to make sure the answer satisfies the question that was asked in the problem.

The total cost of Amy’s supplies is $60.50.

9. Ensure that the answer makes sense.

Yes, the answer makes sense.

As with any skill, start out slow. Take your time in identifying the key words. If necessary, you could write the symbol for the operation that the word represents. Furthermore, you could rewrite the word problem into a mathematical expression and then solve it. As your confidence builds with word problems, so will your ability to perform “mental math” calculations. What is mental math? Mental math involves being able to solve math problems (in any format) in your head and produce the correct answer.

PRACTICE THE SKILL 4-1

Perform the proper function to solve each word problem

1. Ryan is running errands around town. He drives 12 miles to the grocery store, 8 miles to the dry cleaners, 30 miles to work, and 14 miles to meet friends at a local restaurant. How many miles does Ryan drive today? _____________________

2. Mary Ann receives the following IV fluids: 150 cc of Ancef, 250 cc of blood, and 3000 cc of 0.9NS. What is the total amount of fluids Mary Ann receives? _____________________

3. Shawn has been instructed to take 500mg of medication 3 times a day for 10 days. What is the total amount of medication Shawn will have taken when he completes the prescription? _____________________

4. You are taking your family (yourself and three others) to the baseball game today. Based on the following information, how much will you spend during the baseball game? _____________________

Tickets	$15.00 each
Hot dogs	$2.50 each and everyone has two hot dogs
Soda	$4.00 each and everyone has two sodas
Ice cream	$3.50 each

5. Your physician orders 1000 cc of intravenous fluids for you because you are dehydrated. The IV is running at 150 cc/hour. How long will it take to infuse 1000 cc? _____________________

6. Kevin spent $32.50 at the pharmacy. He gave the cashier a $50.00 bill. How much change did Kevin receive? _____________________

7. The medication bottle contains 100 cc. The first prescription requires 66 cc, and the second prescription takes 17 cc. How much remains in the medication bottle? _____________________

8. The pharmacy stocked the medication cabinet with 35 vials of hepatitis B vaccine. Ann removes three vials at 9:00 a.m., two vials at 10:30 a.m., and four vials at 12:00 noon. Betty removes two vials at 12:30 p.m. and one vial at 2:00 p.m. Lori removes three vials at 2:30 p.m. How many vials remain? _____________________

9. Ben has been instructed to wear his headgear a total of 12 hours per day. Ben wears his headgear while he sleeps and after school. Ben gets 7.5 hours of sleep each night. How many additional hours does Ben need to wear his headgear after school? _____________________

10. Gas prices are $3.35 a gallon. You have a 14-gallon tank in your car. How much will it cost to fill your completely empty gas tank? _____________________

Objectives 3, 4

Decimals

“Decimals—something I feel comfortable with when it comes to math.” I have heard this phrase many times during my teaching career. Most students say they feel at ease with decimals because they can associate them with money. In the medical field, decimals are used to describe:

• Laboratory values

• Fluid concentrations

• Measurement of incisions and wounds

• Currency amounts for payment of services

As a reminder, when you are reading and writing decimals, whole numbers are written to the left of the decimal point and place values are written to the right.

The same strategy used for rounding whole numbers can be used for rounding decimals. Refer to the box Strategy 4-2 when completing the Practice the Skill problems that follow it.

STRATEGY 4-2
Rounding Decimals

1. Identify the place value to which you are rounding.

2. Look to the immediate right (one place value) to determine how to round.

3. If this number is 5 or greater, round the place value up by one digit and drop all the remaining numbers to the right.

4. If the number is 4 or less, leave the number alone and drop all the numbers to the right.

*Example:*
Round 3.4567 to the nearest hundredth.
1. Identify the place value to which you are rounding. Identify the hundredths place: 3.4567 2. Look to the immediate right (one place value) of the place value you are rounding. Look at the number 6, which is the number immediately to the right of the place value you are rounding. 3. If this number is 5 or greater, round the place value up by one digit and drop all the remaining numbers to the right. Since the number 6 is greater than 5, it is necessary to raise the 5 (found in the hundredths place) to a 6 and drop the remaining numbers to the right.
Thus, your answer is 3.46.

HUMAN ERROR 4-1

To avoid confusion, place a 0 before the decimal point to show that there is not a whole number in front of it. Example: 0.5 or 0.05.

PRACTICE THE SKILL 4-2

Round the following decimals to the nearest thousandth

1. 2.3579 _____________________________

2. 15.153246 _____________________________

3. 12.9723 _____________________________

4. 0.972354 _____________________________

5. 7.13244 _____________________________

Round the following decimals to the nearest hundredth

6. 97.5326 _____________________________

7. 12.3673 _____________________________

8. 0.981423 _____________________________

9. 14.3275 _____________________________

10. 0.252643 _____________________________

Round the following decimals to the nearest tenth

11. 13.462 _____________________________

12. 75.631 _____________________________

13. 0.98263 _____________________________

14. 2.2534 _____________________________

15. 109.8736 _____________________________

Round the following decimals to the nearest whole number

16. 2.254 _____________________________

17. 89.45 _____________________________

18. 97.78 _____________________________

19. 1.025 _____________________________

20. 0.654 _____________________________

If you are having difficulties with this section, review the concept of rounding numbers in Chapter 2 (p. 34).

Addition and Subtraction of Decimals

Do you remember the trick with adding and subtracting decimals? Always line your decimals up in a vertical manner. If you have uneven numbers, you can add zeros to the end. Once your math problem is formatted, all you need to do to solve the problem is perform the requested operation. The way your answer will be used determines whether you should round the number to a whole number, tenths, hundredths, or thousandths place value.

STRATEGY 4-3
Addition and Subtraction of Decimals

1. Write the problem in a vertical format.

2. Line the decimals up vertically.

3. Add zeros to fill in any uneven place values.

4. Perform the required mathematical operation.

*Example:*
45.679 + 2.35 + 0.2431 =
1. Write the problem in a vertical format. 45.6792.35+0.2431 2. Line the decimals up vertically. 45.6792.35+0.2431 3. Add zeros to fill in any uneven place values. 45.67902.3500+0.2431 4. Perform the required mathematical operation. 45.67902.3500+0.243148.2721