Chapter Notes: Subtracting Decimals (Hundredths)

Table of Contents
1. Understanding Decimal Place Values
2. Reading and Writing Decimal Numbers to the Hundredths
3. The Connection Between Decimals and Money
4. Preparing to Subtract Decimals
5. Step-by-Step Process for Subtracting Decimals
6. Subtracting Decimals Without Regrouping
7. Subtracting Decimals With Regrouping
8. Subtracting When Decimal Places Don't Match
9. Checking Your Work
10. Common Mistakes to Avoid
11. Real-World Applications
12. Mental Math Strategies
13. Building Confidence with Decimal Subtraction
View more

Learning to subtract decimals is an important skill that helps you solve real-world problems every day. When you shop with money, measure ingredients for a recipe, or track distances on a map, you often need to subtract decimal numbers. In this chapter, you will learn how to subtract decimals that go to the hundredths place. Hundredths are the second digit to the right of the decimal point, like the pennies in a dollar amount. Once you master this skill, you'll be able to subtract any decimal numbers with confidence!

Understanding Decimal Place Values

Before we subtract decimals, we need to understand what each digit in a decimal number represents. A decimal number is a number that includes a decimal point, which separates the whole number part from the fractional part.

Think of the decimal point as a dividing line. Everything to the left of the decimal point is a whole number. Everything to the right of the decimal point is a part of a whole.

Let's look at the number 3.47:

• The 3 is in the ones place (whole number)
• The 4 is in the tenths place (the first digit after the decimal point)
• The 7 is in the hundredths place (the second digit after the decimal point)

We can also think of 3.47 as 3 ones, 4 tenths, and 7 hundredths. Another way to say this: 3.47 equals 3 + 0.4 + 0.07.

Imagine a dollar bill. The whole number part (before the decimal) represents the number of whole dollars. The tenths place is like dimes (10 cents). The hundredths place is like pennies (1 cent). So $3.47 is 3 dollars, 4 dimes, and 7 pennies!

Reading and Writing Decimal Numbers to the Hundredths

When you read a decimal number aloud, you say the whole number part first, then the word "and" for the decimal point, then the fractional part.

Here are some examples:

• 5.23 is read as "five and twenty-three hundredths"
• 12.08 is read as "twelve and eight hundredths"
• 0.56 is read as "fifty-six hundredths" (no whole number part)
• 7.30 is read as "seven and thirty hundredths" or "seven and three tenths"

Notice that when we write decimal numbers, zeros at the end after the decimal point don't change the value. For example, 7.3 and 7.30 represent the same amount. However, a zero between the decimal point and another digit is very important! The number 0.08 (eight hundredths) is very different from 0.8 (eight tenths).

The Connection Between Decimals and Money

Money is one of the best ways to understand decimals to the hundredths place. When you work with dollars and cents, you are already working with hundredths!

• 1 dollar = 100 cents
• 1 cent = 0.01 dollars (one hundredth of a dollar)
• 1 dime = 0.10 dollars (one tenth of a dollar)

So when you see $4.52, you know this means 4 dollars and 52 cents. The 5 in the tenths place represents 5 dimes (50 cents), and the 2 in the hundredths place represents 2 pennies (2 cents).

Subtracting money amounts is exactly the same process as subtracting decimals to the hundredths!

Preparing to Subtract Decimals

Subtracting decimals is very similar to subtracting whole numbers. The most important rule to remember is this: always line up the decimal points.

When the decimal points are lined up vertically, all the place values automatically line up correctly:

• Ones line up with ones
• Tenths line up with tenths
• Hundredths line up with hundredths

This is just like when you subtract whole numbers-you line up the ones with ones, the tens with tens, and so on. The decimal point helps you do this correctly.

Adding Zeros as Placeholders

Sometimes one decimal number has more digits after the decimal point than the other. When this happens, you can add zeros to the end of the shorter number to make both numbers have the same number of decimal places.

For example, if you need to subtract 2.5 - 1.23, you can rewrite 2.5 as 2.50. This doesn't change the value, but it makes the subtraction easier:

Original problem: 2.5 - 1.23
Rewritten: 2.50 - 1.23

Now both numbers go to the hundredths place, and you can subtract each place value easily.

Step-by-Step Process for Subtracting Decimals

Follow these steps every time you subtract decimals:

1. Write the problem vertically with the larger number on top (if you know which is larger)
2. Line up the decimal points exactly one above the other
3. Add zeros if needed so both numbers have the same number of decimal places
4. Subtract each column from right to left, just like with whole numbers
5. Regroup (borrow) when necessary
6. Bring the decimal point straight down into your answer

Let's see this process in action with several examples.

Subtracting Decimals Without Regrouping

The simplest decimal subtraction problems are those where you don't need to borrow from another place value.

Example:  Sarah had $5.87 in her piggy bank.
She spent $2.43 on a book.
How much money does she have left?

How much money does Sarah have left?

Solution:

We need to subtract: 5.87 - 2.43

Step 1: Write the problem vertically and line up the decimal points:

  5.87
- 2.43
______

Step 2: Subtract the hundredths place: 7 - 3 = 4

  5.87
- 2.43
______
     4

Step 3: Subtract the tenths place: 8 - 4 = 4

  5.87
- 2.43
______
   .44

Step 4: Subtract the ones place: 5 - 2 = 3

  5.87
- 2.43
______
  3.44

Step 5: Bring down the decimal point (already shown above)

Sarah has $3.44 left in her piggy bank.

Example:  A recipe calls for 6.95 cups of flour.
You have already added 4.52 cups.
How much more flour do you need to add?

How much more flour is needed?

Solution:

We need to subtract: 6.95 - 4.52

Step 1: Line up the decimal points:

  6.95
- 4.52
______

Step 2: Subtract hundredths: 5 - 2 = 3

Step 3: Subtract tenths: 9 - 5 = 4

Step 4: Subtract ones: 6 - 4 = 2

  6.95
- 4.52
______
  2.43

You need to add 2.43 more cups of flour.

Subtracting Decimals With Regrouping

Often when subtracting decimals, you'll need to regroup (or borrow) from the next place value to the left, just like when subtracting whole numbers. This happens when a digit in the top number is smaller than the digit below it.

Regrouping from Tenths to Hundredths

Example:  Marcus ran 8.34 miles this week.
Last week he ran 5.68 miles.
How many more miles did he run this week?

How many more miles did Marcus run this week?

Solution:

We need to subtract: 8.34 - 5.68

Step 1: Line up the decimal points:

  8.34
- 5.68
______

Step 2: Look at the hundredths place. We need to subtract 8 from 4, but 4 is smaller than 8. We must regroup!

Step 3: Borrow 1 tenth from the tenths place. The 3 tenths becomes 2 tenths. The 4 hundredths becomes 14 hundredths (we added 10 hundredths):

  8.2¹⁴
- 5.68
______

Step 4: Now subtract hundredths: 14 - 8 = 6

Step 5: Subtract tenths: 2 - 6 won't work! We need to regroup again from the ones place.

Step 6: Borrow 1 one from the ones place. The 8 ones becomes 7 ones. The 2 tenths becomes 12 tenths:

  7.¹²2¹⁴
- 5.68
______

Step 7: Subtract tenths: 12 - 6 = 6

Step 8: Subtract ones: 7 - 5 = 2

  8.34
- 5.68
______
  2.66

Marcus ran 2.66 more miles this week than last week.

Regrouping with Zeros

Sometimes you need to regroup from a zero, which requires borrowing from the place value even further to the left.

Example:  A board is 7.05 meters long.
A carpenter cuts off 3.28 meters.
How long is the remaining piece of board?

What is the length of the remaining board?

Solution:

We need to subtract: 7.05 - 3.28

Step 1: Line up the decimal points:

  7.05
- 3.28
______

Step 2: Hundredths place: 5 - 8 won't work. We need to borrow from the tenths place, but there's a 0 there!

Step 3: We must first borrow from the ones place. Borrow 1 one from 7, making it 6. The 0 tenths becomes 10 tenths:

  6.¹⁰05
- 3.28
______

Step 4: Now borrow from the tenths place. Take 1 tenth from 10 tenths, making it 9 tenths. The 5 hundredths becomes 15 hundredths:

  6.⁹0¹⁵
- 3.28
______

Step 5: Subtract hundredths: 15 - 8 = 7

Step 6: Subtract tenths: 9 - 2 = 7

Step 7: Subtract ones: 6 - 3 = 3

  7.05
- 3.28
______
  3.77

The remaining piece of board is 3.77 meters long.

Subtracting When Decimal Places Don't Match

When the two numbers have different numbers of decimal places, add zeros to make them match before subtracting.

Example:  Jenna bought a snack for $5.00.
The snack cost $3.47.
How much change did she receive?

How much change did Jenna receive?

Solution:

We need to subtract: 5.00 - 3.47

Step 1: Both numbers already have two decimal places, so we can line them up:

  5.00
- 3.47
______

Step 2: Hundredths: 0 - 7 won't work. Borrow from tenths. But tenths is 0!

Step 3: Borrow from ones. 5 ones becomes 4 ones, and 0 tenths becomes 10 tenths:

  4.¹⁰00
- 3.47
______

Step 4: Now borrow from tenths. 10 tenths becomes 9 tenths, and 0 hundredths becomes 10 hundredths:

  4.⁹0¹⁰
- 3.47
______

Step 5: Subtract hundredths: 10 - 7 = 3

Step 6: Subtract tenths: 9 - 4 = 5

Step 7: Subtract ones: 4 - 3 = 1

  5.00
- 3.47
______
  1.53

Jenna received $1.53 in change.

Example:  A water bottle contains 1.5 liters of water.
After drinking some, 0.83 liters remain.
How much water was consumed?

How many liters of water were consumed?

Solution:

We need to subtract: 1.5 - 0.83

Step 1: Add a zero to 1.5 so both numbers have two decimal places: 1.50 - 0.83

  1.50
- 0.83
______

Step 2: Hundredths: 0 - 3 won't work. Borrow from tenths. 5 tenths becomes 4 tenths, 0 hundredths becomes 10 hundredths:

  1.⁴5¹⁰
- 0.83
______

Step 3: Subtract hundredths: 10 - 3 = 7

Step 4: Tenths: 4 - 8 won't work. Borrow from ones. 1 one becomes 0 ones, 4 tenths becomes 14 tenths:

  0.¹⁴4¹⁰
- 0.83
______

Step 5: Subtract tenths: 14 - 8 = 6

Step 6: Subtract ones: 0 - 0 = 0

  1.50
- 0.83
______
  0.67

0.67 liters of water were consumed.

Checking Your Work

It's always a good idea to check your subtraction answer. The best way to check is by using addition. If your subtraction is correct, then when you add your answer to the number you subtracted, you should get the original number.

For example, if you calculated 8.34 - 5.68 = 2.66, you can check by adding:

  2.66
+ 5.68
______
  8.34 ✓

Since we got back the original number (8.34), we know our subtraction was correct!

Common Mistakes to Avoid

Here are some errors students often make when subtracting decimals:

• Not lining up decimal points: Always write the numbers so the decimal points are directly above and below each other. If they're not lined up, your place values won't match correctly.
• Forgetting to add zeros: When one number has fewer decimal places, add zeros to the end so both numbers have the same number of digits after the decimal point.
• Forgetting the decimal point in the answer: Remember to bring the decimal point straight down into your answer, keeping it in line with the decimal points above.
• Subtracting in the wrong direction: Always subtract the bottom number from the top number in each column. When you need to regroup, remember to increase the place value to the right and decrease the place value you borrowed from.

Real-World Applications

Subtracting decimals to the hundredths is something you'll use throughout your life. Here are just a few situations where this skill is essential:

• Shopping: Calculating change, comparing prices, finding discounts
• Cooking: Adjusting recipe measurements, figuring out how much more of an ingredient you need
• Sports: Finding the difference between race times, comparing scores or distances
• Science: Measuring temperature changes, calculating differences in weights or lengths
• Travel: Determining remaining distance, calculating fuel usage
• Personal finance: Tracking spending, balancing a checkbook, managing savings

Think about checking the temperature outside. If it was 68.75°F in the morning and dropped to 54.38°F in the evening, you would subtract decimals to find that the temperature dropped by 14.37°F!

Mental Math Strategies

For some decimal subtraction problems, you can use mental math shortcuts:

Friendly Numbers

When subtracting amounts close to a whole number, you can adjust both numbers to make the math easier.

For example, to solve 7.43 - 2.98:

• Notice that 2.98 is very close to 3.00
• Add 0.02 to both numbers: 7.45 - 3.00
• This is easier: 7.45 - 3.00 = 4.45

Counting Up

Another strategy is to count up from the smaller number to the larger number, especially when making change.

For example, to find 10.00 - 6.35:

• Start at 6.35
• Count up to the next whole number: 6.35 + 0.65 = 7.00
• Count up to 10.00: 7.00 + 3.00 = 10.00
• Add the two amounts: 0.65 + 3.00 = 3.65

These mental strategies are helpful when you need a quick answer and don't have paper and pencil handy!

Building Confidence with Decimal Subtraction

Subtracting decimals to the hundredths place is a skill that improves with practice. Remember these key points:

• Decimal subtraction follows the same rules as whole number subtraction
• Always line up the decimal points before you begin
• Add zeros when needed to make both numbers have the same decimal places
• Regroup (borrow) whenever a digit on top is smaller than the digit below it
• Bring the decimal point straight down into your answer
• Check your work by adding your answer to the number you subtracted

The more you practice, the more automatic these steps will become. Soon you'll be able to subtract decimals quickly and accurately in any situation. Whether you're making change at a store, measuring ingredients for your favorite recipe, or solving math problems in school, you now have the tools you need to subtract decimals with confidence!