Prealgebra Chapter 5: Decimal Multiplication and Applications

Decimal Multiplication Principles and Procedures

• Algorithm for Multiplying Two Decimals:

  • Step 1: Multiply the numbers as if they were integers, ignoring the decimal points during initial calculation.

  • Step 2: Place the decimal point in the product so that the number of decimal places equals the combined number of decimal places in both factors.

  • Special Case Procedure: Zeros may need to be inserted to the left of the whole-number product if additional decimal places are required to achieve the correct place value.

Step-by-Step Examples of Decimal Multiplication

• Basic Decimal Multiplication:

  • Problem: Multiply $11.914 \times 0.8$.

  • Step 1: Determine decimal place counts:

  • $11.914$ has $3$ decimal places.

  • $0.8$ has $1$ decimal place.

  • Combined total decimal places needed: $3 + 1 = 4$ decimal places.

  • Step 2: Multiply the integer values:

  • $11914 \times 8 = 95312$

  • Carrying details during multiplication:

    • $8 \times 4 = 32$ (write $2$, carry $3$)

    • $8 \times 1 + 3 = 11$ (write $1$, carry $1$)

    • $8 \times 9 + 1 = 73$ (write $3$, carry $7$)

    • $8 \times 1 + 7 = 15$ (write $5$, carry $1$)

    • $8 \times 1 + 1 = 9$ (write $9$)

  • Step 3: Insert decimal point $4$ digits from the right:

  • Final product: $9.5312$

• Skill Practice Problem 1:

  • Problem: Multiply $(19.7)(4.1)$.

  • Analysis:

  • $19.7$ has $1$ decimal place.

  • $4.1$ has $1$ decimal place.

  • Total required decimal places: $1 + 1 = 2$

  • Calculation:

  • Integer multiplication: $197 \times 41 = 8077$

  • Placing decimal point: $80.77$

Estimating Products with Front-End Rounding

• Front-End Rounding Concept:

  • Front-end rounding is used to estimate products and verify decimal point placement.

  • Each factor is rounded to its left-most nonzero digit.

  • This process determines whether the actual product's highest place value matches the estimated product's magnitude.

• Example of Decimal Multiplication with Estimation Check:

  • Problem: Multiply $(29.3)(2.8)$ and check using estimation.

  • Direct Multiplication Procedure:

  • $29.3$ has $1$ decimal place.

  • $2.8$ has $1$ decimal place.

  • Total required decimal places: $1 + 1 = 2$

  • Partial products:

    • $29.3 \times 8 = 2344$

    • $29.3 \times 20 = 5860$

    • Sum of partial products: $2344 + 5860 = 8204$

  • Applying $2$ decimal places: $82.04$

• Estimation Verification:

  • Round $29.3$ to front-end digit: $30$

  • Round $2.8$ to front-end digit: $3$

  • Multiply estimates: $30 \times 3 = 90$

• Evaluation: The first digit for both the actual product ($82.04$) and the estimate ($90$) is in the tens place, confirming that the decimal point is placed correctly.

  • Skill Practice Problem 2:

• Problem: Multiply $(1.9)(29.1)$ and check answer using estimation.

• Direct Calculation:

  • $1.9$ ($1$ decimal place)

  • $29.1$ ($1$ decimal place)

  • Integer product: $19 \times 291 = 5529$

  • Product with $2$ decimal places: $55.29$

• Estimation Check:

  • $1.9 \rightarrow 2$

  • $29.1 \rightarrow 30$

  • Estimated product: $2 \times 30 = 60$

  • Tens place consistency confirms correct decimal placement.

Practical Applications and Word Problems

• Multi-Step Consumer Calculation:

  • Scenario: Jane Marie bought $8$ cans of tennis balls for $\$3.79$ each and paid $\$1.97$ in sales tax. Calculate the total bill.

  • Step 1: Calculate the item cost for tennis balls:

  • $\text{Cost} = 8 \times \$3.79$

  • $3.79 \times 8 = 30.32$

  • Cost of tennis balls = $\$30.32$

• Step 2: Compute total bill by adding tax:

  • Formula: $\text{Total Cost} = (\text{Cost of tennis balls}) + (\text{Tax})$

  • Substitution: $\text{Total Cost} = \$30.32 + \$1.97 = \$32.29$

• Skill Practice Problem 7:

  • Scenario: A book club ordered $12$ books for $\$8.99$ each. Shipping cost was $\$4.95$. Find the total bill.

  • Calculation Steps:

  • Book subtotal: $12 \times \$8.99 = \$107.88$

  • Total bill: $\$107.88 + \$4.95 = \$112.83$

Geometric Applications: Area Calculation

• Formula for Rectangular Area:

  • Area $A$ of a rectangle equals length $l$ multiplied by width $w$:     $A = l \times w$

• Area of the Mona Lisa Painting:

  • Context: The Mona Lisa was painted by Leonardo da Vinci between $1503$ and $1506$ and hangs in the Louvre in Paris, France.

  • Dimensions: Length $l = 30\,\text{in.}$ and width $w = 20.875\,\text{in.}$

  • Step 1: Set up the area formula:     $A = (30\,\text{in.}) \times (20.875\,\text{in.})$

• Step 2: Perform vertical multiplication:

  • $20.875$ ($3$ decimal places)

  • $30$ ($0$ decimal places)

  • Multiplication steps: $20.875 \times 30 = 626250$

  • Applying $3$ decimal places: $626.250\,\text{in.}^2 = 626.25\,\text{in.}^2$

• Final Area: $626.25\,\text{in.}^2$

  • Skill Practice Problem 8:

• Scenario: The IMAX movie screen at the Museum of Science and Discovery measures $18\,\text{m}$ by $24.4\,\text{m}$. Find its area.

• Calculation:     $A = (18\,\text{m}) \times (24.4\,\text{m})$     $A = 439.2\,\text{m}^2$

Additional Problem Solutions

• Problem 51: Athletic Department Purchase:

  • Items Purchased:

  • $20$ pizzas at $\$12.95$ each

  • $10$ Greek salads at $\$5.95$ each

  • $60$ soft drinks at $\$1.29$ each

  • Sales tax: $\$27.71$

  • Step 1: Calculate subtotal for each item type:

  • Pizzas: $20 \times \$12.95 = \$259.00$

  • Salads: $10 \times \$5.95 = \$59.50$

  • Soft drinks: $60 \times \$1.29 = \$77.40$

  • Step 2: Sum item subtotals:     $\text{Subtotal} = \$259.00 + \$59.50 + \$77.40 = \$395.90$

  • Step 3: Add sales tax to get total bill:     $\text{Total Bill} = \$395.90 + \$27.71 = \$423.61$

• Problem 53: Tire Store Savings Comparison:

  • Pricing Options:

  • Single tire cost: $\$220.40$

  • Set of four tires cost: $\$769.99$

  • Step 1: Compute total cost of four single tires:     $\text{Cost of 4 single tires} = 4 \times \$220.40 = \$881.60$

  • Step 2: Compute savings by subtracting set price:     $\text{Savings} = \$881.60 - \$769.99 = \$111.61$

  • Result: Buying the set saves $\$111.61$.