Solved Exercises on Decimal Operations with Rational Numbers in Grade 7

Master decimal operations with rational numbers: addition, subtraction, multiplication, division through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Decimal Addition
Exercise 1
Calculate: \( 3.75 + 2.48 \)
Definition:

Rational number: A number that can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\)

Decimal addition method:
  1. Align decimal points vertically
  2. Add as if they were whole numbers
  3. Place decimal point in the result aligned with the others
Original Expression
\(3.75 + 2.48\)
Align Decimals
\(3.75\)
+\(2.48\)
Add
\(6.23\)
Step 1: Align decimal points

\(3.75\)

\(2.48\)

\(\underline{+\,\,\,\,\,\,\,}\)

Step 2: Add from right to left

Hundredths place: \(5 + 8 = 13\), write 3, carry 1

Tenths place: \(7 + 4 + 1 = 12\), write 2, carry 1

Units place: \(3 + 2 + 1 = 6\)

Step 3: Place the decimal point

Line up with the original decimal points

Step 4: Write the final answer

\(6.23\)

\( 3.75 + 2.48 = 6.23 \)
Final answer:

\( 3.75 + 2.48 = 6.23 \)

Applied rules:

• Decimal alignment: Always align decimal points before adding

• Place value: Add digits in the same place value columns

• Carrying: When sum exceeds 9, carry to the next column

2 Decimal Subtraction
Exercise 2
Calculate: \( 8.63 - 4.27 \)
Definition:

Decimal subtraction: Align decimal points and subtract as with whole numbers

Original Expression
\(8.63 - 4.27\)
Align Decimals
\(8.63\)
-\(4.27\)
Subtract
\(4.36\)
Step 1: Align decimal points

\(8.63\)

\(4.27\)

\(\underline{-\,\,\,\,\,\,\,}\)

Step 2: Subtract from right to left

Hundredths place: \(3 - 7\), borrow from tenths: \(13 - 7 = 6\)

Tenths place: \(5 - 2 = 3\) (after borrowing)

Units place: \(8 - 4 = 4\)

Step 3: Place the decimal point

Line up with the original decimal points

Step 4: Write the final answer

\(4.36\)

\( 8.63 - 4.27 = 4.36 \)
Final answer:

\( 8.63 - 4.27 = 4.36 \)

Applied rules:

• Decimal alignment: Always align decimal points before subtracting

• Borrowing: When top digit is smaller, borrow from the next column

• Place value: Subtract digits in the same place value columns

3 Decimal Multiplication
Exercise 3
Calculate: \( 2.5 \times 1.4 \)
Definition:

Decimal multiplication: Multiply as whole numbers, then count decimal places in factors

Original Expression
\(2.5 \times 1.4\)
Multiply as Whole Numbers
\(25 \times 14 = 350\)
Count Decimal Places
\(1 + 1 = 2\)
Place Decimal Point
\(3.50\)
Step 1: Ignore decimal points initially

Multiply \(25 \times 14\)

Step 2: Count decimal places in factors

\(2.5\) has 1 decimal place

\(1.4\) has 1 decimal place

Total: \(1 + 1 = 2\) decimal places

Step 3: Place decimal point in the product

Count 2 places from the right: \(350 \rightarrow 3.50\)

Step 4: Write the final answer

\(3.50\) or \(3.5\)

\( 2.5 \times 1.4 = 3.5 \)
Final answer:

\( 2.5 \times 1.4 = 3.5 \)

Applied rules:

• Multiply first: Ignore decimal points and multiply as whole numbers

• Count decimals: Total decimal places in factors equals decimal places in product

• Place decimal: Count from right of product

Rules and methods, laws,...
\( a.b \times c.d = \frac{ab \times cd}{100} \)
Decimal Multiplication
Decimal Addition
\( a.b + c.d = \) Align decimal points
Add as whole numbers, place decimal point
Decimal Subtraction
\( a.b - c.d = \) Align decimal points
Subtract as whole numbers, place decimal point
Decimal Multiplication
\( a.b \times c.d = \) Count decimal places
Multiply as whole numbers, place decimal point
Key definitions:

Rational number: A number that can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\)

Decimal number: A number that uses a decimal point to separate the whole number part from the fractional part

Terminating decimal: A decimal that ends after a finite number of digits

Repeating decimal: A decimal with a pattern that repeats indefinitely

Decimal operation methods:
  1. Addition/Subtraction: Align decimal points, perform operation, place decimal point in result
  2. Multiplication: Multiply as whole numbers, count total decimal places, place decimal point
  3. Division: Move decimal points to make divisor a whole number, divide normally
Tip 1: Always align decimal points for addition and subtraction
Tip 2: Count decimal places carefully for multiplication
Tip 3: For division, move decimal points to make the divisor whole
Tip 4: Estimate first to check if your answer is reasonable
Common errors: Misaligning decimal points, miscounting decimal places in multiplication, forgetting to place decimal point.
Exam preparation: Practice all four operations, memorize the decimal placement rules, estimate answers.
Formulas to know by heart:

• Decimal addition: Align decimal points and add

• Decimal subtraction: Align decimal points and subtract

• Decimal multiplication: Multiply as whole numbers, count total decimal places

• Decimal division: Move decimal points to make divisor whole

Solution: Exercises 4 to 5
4 Decimal Division
Exercise 4
Calculate: \( 8.4 \div 2.1 \)
Definition:

Decimal division: Move decimal points to make divisor a whole number, then divide normally

Original Expression
\(8.4 \div 2.1\)
Move Decimal Points
\(84 \div 21\)
Divide
\(4\)
Step 1: Move decimal points to make divisor whole

\(8.4 \div 2.1\)

Move decimal point 1 place right in both: \(84 \div 21\)

Step 2: Perform the division

\(84 \div 21 = 4\)

Step 3: Write the final answer

\(4\)

Step 4: Verify the answer

\(4 \times 2.1 = 8.4\) ✓

\( 8.4 \div 2.1 = 4 \)
Final answer:

\( 8.4 \div 2.1 = 4 \)

Applied rules:

• Decimal division: Move decimal points equally in dividend and divisor

• Make divisor whole: Easier to divide when divisor is a whole number

• Verification: Multiply quotient by divisor to check

5 Mixed Operations
Exercise 5
Calculate: \( (3.2 + 1.8) \times 2.5 - 4.6 \)
Definition:

Order of operations: PEMDAS - Parentheses, Exponents, Multiplication/Division, Addition/Subtraction

Original Expression
\( (3.2 + 1.8) \times 2.5 - 4.6 \)
Parentheses First
\( 5.0 \times 2.5 - 4.6 \)
Multiplication Next
\( 12.5 - 4.6 \)
Subtraction Last
\( 7.9 \)
Step 1: Solve operations inside parentheses first

\(3.2 + 1.8 = 5.0\)

Step 2: Perform multiplication

\(5.0 \times 2.5 = 12.5\)

Step 3: Perform subtraction

\(12.5 - 4.6 = 7.9\)

Step 4: Write the final answer

\(7.9\)

\( (3.2 + 1.8) \times 2.5 - 4.6 = 7.9 \)
Final answer:

\( (3.2 + 1.8) \times 2.5 - 4.6 = 7.9 \)

Applied rules:

• PEMDAS: Follow order of operations strictly

• Parentheses first: Always solve innermost parentheses first

• Multiplication before addition: Apply order of operations

Key Concepts: Laws, Methods, Rules, Definitions
\( a.b \times c.d = \frac{(ab) \times (cd)}{100} \)
Decimal Multiplication Rule
Key definitions:

Rational number: A number that can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\)

Decimal number: A number that uses a decimal point to separate the whole number part from the fractional part

Terminating decimal: A decimal that ends after a finite number of digits (like 0.75)

Repeating decimal: A decimal with a pattern that repeats indefinitely (like 0.333...)

Place value: The value of a digit based on its position in a number

Estimation: Finding an approximate answer to check reasonableness

Complete decimal operation methodology:
  1. Addition: Align decimal points, add as whole numbers, place decimal point
  2. Subtraction: Align decimal points, subtract as whole numbers, place decimal point
  3. Multiplication: Multiply as whole numbers, count total decimal places, place decimal point
  4. Division: Move decimal points to make divisor whole, divide normally
  5. Order of operations: Follow PEMDAS sequence
  6. Verification: Check with estimation or reverse operations
Tip 1: Always align decimal points for addition and subtraction - this is critical!
Tip 2: For multiplication, count decimal places in the factors to place the decimal in the product
Tip 3: For division, move decimal points equally in dividend and divisor to make divisor whole
Tip 4: Always estimate first to check if your final answer is reasonable
Common errors: Misaligning decimal points, counting wrong number of decimal places in multiplication, forgetting to move decimal points in division.
Exam preparation: Practice all operations, memorize decimal placement rules, use estimation to verify answers.
Formulas to know by heart:

• Decimal addition/subtraction: Align decimal points

• Decimal multiplication: Multiply as whole numbers, count total decimal places

• Decimal division: Move decimal points to make divisor whole

• Order of operations: PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)

• Estimation: Round numbers to check reasonableness

Exercise with Visualization: Decimal Operations
Exercise 6: Decimal Operations Comparison
Compare these decimal operations:
\( 2.5 + 1.75 \)
\( 2.5 \times 1.75 \)
\( 2.5 - 1.75 \)
\( 2.5 \div 1.75 \)

Analysis: The chart shows how different operations affect decimal values.

  • \( 2.5 + 1.75 = 4.25 \): Addition increases the value
  • \( 2.5 \times 1.75 = 4.375 \): Multiplication increases the value
  • \( 2.5 - 1.75 = 0.75 \): Subtraction decreases the value
  • \( 2.5 \div 1.75 \approx 1.43 \): Division reduces the value

Questions & Answers

Question: I keep making mistakes when multiplying decimals. How do I know where to put the decimal point?

Answer: The decimal point placement in multiplication follows a specific rule:

  1. Multiply as whole numbers: Ignore the decimal points initially
  2. Count decimal places: Count how many digits come after the decimal point in each factor
  3. Total decimal places: Add up the decimal places from both factors
  4. Place the decimal: Starting from the right of your product, count left that many places

Example: For \(2.3 \times 1.45\)

  • Multiply: \(23 \times 145 = 3335\)
  • Count decimal places: \(2.3\) has 1, \(1.45\) has 2, total = 3
  • Place decimal: Count 3 places from the right: \(3.335\)

Always double-check by estimating: \(2 \times 1.5 = 3\), so \(3.335\) is reasonable!

Question: Why do I need to move decimal points in division? It confuses me.

Answer: Moving decimal points in division is just a shortcut to make the calculation easier:

  • When you move the decimal point in both the dividend and divisor by the same amount, the value of the quotient stays the same
  • It's much easier to divide by a whole number than by a decimal

Example: For \(8.4 \div 2.1\)

  • Move decimal point 1 place right in both: \(84 \div 21\)
  • Now it's \(84 \div 21 = 4\)

This works because \(\frac{8.4}{2.1} = \frac{8.4 \times 10}{2.1 \times 10} = \frac{84}{21}\)

The mathematical value is preserved, but the calculation becomes simpler!

Question: How can I quickly check if my decimal operation answers are reasonable?

Answer: Estimation is your best tool for checking reasonableness:

  1. Round to simpler numbers: Round decimals to whole numbers or simple decimals
  2. Calculate mentally: Do the operation with rounded numbers
  3. Compare: See if your exact answer is close to your estimate

Examples:

  • For \(3.75 + 2.48\): Estimate \(4 + 2 = 6\), so answer near 6.23 is reasonable
  • For \(2.5 \times 1.4\): Estimate \(2.5 \times 1.5 = 3.75\), so answer near 3.5 is reasonable
  • For \(8.4 \div 2.1\): Estimate \(8 \div 2 = 4\), so answer of 4 is reasonable

This quick check helps catch major errors like misplacing decimal points!