Subtracting Decimals with Regrouping - Solved Exercises

Master subtracting decimals with regrouping: understanding decimal place values, borrowing across decimal points, and solving complex subtraction problems through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Basic regrouping
Exercise 1
Subtract:
\(5.23 - 1.78\)
Definition:

Decimal subtraction with regrouping: When a digit in the minuend is smaller than the corresponding digit in the subtrahend, we borrow from the next higher place value.

Subtraction method:
  1. Align decimal points vertically
  2. Start subtracting from the rightmost digit (smallest place value)
  3. If a digit in the minuend is smaller than the subtrahend digit, regroup/borrow from the next higher place value
  4. Continue subtraction for each place value
Minuend
\(5.23\)
Subtrahend
\(1.78\)
Difference
\(3.45\)
Step 1: Align the numbers

\(\begin{array}{r} 5.23 \\ - 1.78 \\ \hline \end{array}\)

Step 2: Subtract hundredths (rightmost)

We need \(3 - 8\), but 3 < 8, so we regroup: Borrow 1 from tenths (2 becomes 1), and hundredths becomes 13
Now: \(13 - 8 = 5\)

Step 3: Subtract tenths

We need \(1 - 7\), but 1 < 7, so we regroup: Borrow 1 from ones (5 becomes 4), and tenths becomes 11
Now: \(11 - 7 = 4\)

Step 4: Subtract ones

Now: \(4 - 1 = 3\)

\(\begin{array}{r} \phantom{0}4\phantom{.}1\phantom{0}13\\ \cancel{5}.\cancel{2}\cancel{3} \\ - 1.78 \\ \hline 3.45 \\ \end{array}\)
\(5.23 - 1.78 = 3.45\)
Final answer:

\(5.23 - 1.78 = 3.45\)

Applied rules:

• Decimal alignment: Always align decimal points vertically

• Regrouping: When minuend digit is smaller than subtrahend digit, borrow 1 from next higher place value

• Place value: Each borrowed unit equals 10 units of the next lower place value

2 Advanced regrouping
Exercise 2
Subtract:
\(8.05 - 3.67\)
Definition:

Advanced regrouping: Sometimes multiple place values need regrouping in sequence, especially when dealing with zeros in the minuend.

Minuend
\(8.05\)
Subtrahend
\(3.67\)
Difference
\(4.38\)
Step 1: Align the numbers

\(\begin{array}{r} 8.05 \\ - 3.67 \\ \hline \end{array}\)

Step 2: Subtract hundredths

We need \(5 - 7\), but 5 < 7, so we regroup: Borrow 1 from tenths (0 becomes 9), and hundredths becomes 15
Wait! We can't borrow from 0 tenths, so we need to borrow from ones first.

Step 3: Regroup from ones to tenths

Borrow 1 from ones (8 becomes 7), and tenths becomes 10
Now we have 7.10.05

Step 4: Continue with regrouping

Now borrow 1 from tenths (10 becomes 9), and hundredths becomes 15
Calculate: \(15 - 7 = 8\) hundredths

Step 5: Subtract tenths

Calculate: \(9 - 6 = 3\) tenths

Step 6: Subtract ones

Calculate: \(7 - 3 = 4\) ones

\(\begin{array}{r} \phantom{0}7\phantom{.}9\phantom{0}15\\ \cancel{8}.\cancel{0}\cancel{5} \\ - 3.67 \\ \hline 4.38 \\ \end{array}\)
\(8.05 - 3.67 = 4.38\)
Final answer:

\(8.05 - 3.67 = 4.38\)

Applied rules:

• Zero regrouping: When borrowing from zero, continue to the next non-zero place value

• Sequential regrouping: Multiple borrowings may be needed in one problem

• Place value relationships: Each place value is 10 times the next smaller place value

3 Word problem application
Exercise 3
Sarah has $12.50. She buys a book for $7.85. How much money does she have left?
Definition:

Word problem application: Real-world scenarios require translating verbal descriptions into mathematical operations, often involving decimal subtraction with regrouping.

Starting amount
\$12.50
Amount spent
\$7.85
Remaining
\$4.65
Step 1: Set up the subtraction

We need to calculate: \$12.50 - \$7.85

Step 2: Align the numbers

\(\begin{array}{r} 12.50 \\ - 7.85 \\ \hline \end{array}\)

Step 3: Subtract hundredths

We need \(0 - 5\), but 0 < 5, so we regroup: Borrow 1 from tenths (5 becomes 4), and hundredths becomes 10
Now: \(10 - 5 = 5\)

Step 4: Subtract tenths

We need \(4 - 8\), but 4 < 8, so we regroup: Borrow 1 from ones (2 becomes 1), and tenths becomes 14
Now: \(14 - 8 = 6\)

Step 5: Subtract ones

We need \(1 - 7\), but 1 < 7, so we regroup: Borrow 1 from tens (1 becomes 0), and ones becomes 11
Now: \(11 - 7 = 4\)

Step 6: Subtract tens

Now: \(0 - 0 = 0\)

\(\begin{array}{r} \phantom{0}11\phantom{.}14\phantom{0}10\\ \cancel{1}\cancel{2}.\cancel{5}\cancel{0} \\ - 7.85 \\ \hline 4.65 \\ \end{array}\)
Sarah has \$4.65 left
Final answer:

Sarah has \$4.65 left after buying the book.

Applied rules:

• Problem translation: Convert word problems to mathematical expressions

• Money applications: Dollars and cents work the same as any decimal number

• Multiple regroupings: Complex problems may require regrouping at multiple place values

Decimal Place Value Chart & Regrouping
Ones.Tenths.Hundredths
Decimal Place Values
Ones Place
1 = 10 tenths
Value: 1
Tenths Place
0.1 = 10 hundredths
Value: 1/10
Hundredths Place
0.01 = 10 thousandths
Value: 1/100
Ones
.
Tenths
Hundredths
Regrouping Relationships:

• 1 One = 10 Tenths

• 1 Tenth = 10 Hundredths

• 1 Hundredth = 10 Thousandths

5 . 2 3
Tip 1: Always align decimal points when subtracting decimals.
Tip 2: When borrowing from 0, keep going to the next non-zero digit.
Tip 3: Each borrowed unit equals 10 of the next smaller place value.
Borrowing Process
↓
1 Ten = 10 Ones
↓
1 One = 10 Tenths
↓
1 Tenth = 10 Hundredths
Key Concepts & Methods
Minuend - Subtrahend = Difference
Basic Subtraction Formula
Key definitions:

Minuend: The number being subtracted from

Subtrahend: The number being subtracted

Difference: The result of subtraction

Regrouping: Converting a unit from a higher place value to 10 units of the next lower place value

Subtraction with regrouping steps:
  1. Align numbers: Line up decimal points vertically
  2. Compare digits: Check if top digit is smaller than bottom digit in each place value
  3. Regroup if needed: Borrow 1 from the next higher place value
  4. Adjust digits: Increase the current place value by 10 and decrease the higher place value by 1
  5. Subtract: Perform the subtraction operation
  6. Repeat: Continue for each place value moving left
Common errors: Misaligning decimal points, forgetting to adjust higher place values after borrowing, subtracting bottom from top instead of top from bottom.
Success strategies: Always verify by adding the difference to the subtrahend to get the minuend, draw lines to separate place values, use placeholders for missing decimal places.
Important rules to remember:

• Decimal alignment: Always align decimal points when setting up the problem

• Borrowing rule: When borrowing, take 1 from higher place value and add 10 to current place value

• Zero handling: When borrowing from 0, continue to the next non-zero digit

• Verification: Check your answer by adding the difference to the subtrahend

Questions & Answers

Question: Why do I need to regroup when subtracting decimals? It seems complicated compared to regular subtraction.

Answer: Great question! Regrouping is needed whenever a digit in the top number (minuend) is smaller than the digit below it (subtrahend) in the same place value. This happens in both whole numbers and decimals.

For example, in \(5.23 - 1.78\), when we try to subtract hundredths (3 - 8), we can't do 3 - 8 because 3 is smaller than 8. So we need to borrow 1 tenth (which equals 10 hundredths) to make the hundredths digit larger.

Think of it like exchanging money: if you have 5 dollars and 23 cents and want to give someone 1 dollar and 78 cents, you don't have enough cents, so you change one dollar bill into 100 cents, giving you 123 cents total. Then you can subtract 78 cents from 123 cents.

Regrouping ensures we maintain the correct place value relationships while performing subtraction.

Question: My child gets confused when there are zeros in the decimal number. How do I explain borrowing from zeros?

Answer: Borrowing from zeros is indeed tricky! Here's how to explain it:

When you need to borrow from a zero, you can't take anything from zero. So you need to go to the next non-zero digit to the left. Think of it like a chain reaction:

For example, in \(8.05 - 3.67\), when subtracting hundredths (5 - 7), we need to borrow from the tenths place, but it's 0. So we go to the ones place (8), borrow 1 to make it 7, then the 0 tenths becomes 10 tenths. Now we can borrow 1 from those 10 tenths to make the hundredths 15.

You can visualize this as: 8.05 = 7.10.05 (we borrowed 1 from the 8 to make 10 tenths, then we'll borrow from those 10 tenths).

It's like a domino effect - each place value helps the next smaller one when needed.

Question: How can I check if my answer is correct when subtracting decimals with regrouping?

Answer: There are several ways to check your decimal subtraction answer:

1. Addition Check: Add your difference back to the subtrahend. If you get the original minuend, your answer is correct.
For example: If \(5.23 - 1.78 = 3.45\), check: \(3.45 + 1.78 = 5.23\) ✓

2. Estimation Check: Round the numbers to estimate the answer. For \(5.23 - 1.78\), round to \(5 - 2 = 3\). Since our answer is 3.45, it's reasonable.

3. Digit Check: Make sure your answer has the same number of decimal places as the original numbers (or fewer, if trailing zeros are dropped).

4. Reasonableness Check: Ask yourself if the answer makes sense in the context of the problem.

The addition check is the most reliable method since it reverses the operation.