Problem 20 - Elementary Algebra Within Reach

Question

You are shopping for DVDs. You want to buy 7 DVDs for \(\$ 19.99\) each. Show how you can use the Distributive Property to find the total cost of the DVDs mentally.

Step-by-Step Solution

Answer

The total cost of buying 7 DVDs is \$139.93.

Step 1: Write out the basic calculation

The basic calculation to find the total cost of DVDs would be \(7 * 19.99\).

Step 2: Apply the Distributive Property

We can use the Distributive Property to break down this operation into simpler terms. Distributive Property states that \(a(b + c) = ab + ac\). Applying this to our calculation, we split \$19.99 into \$20.00 - \$0.01 to give: \(7 * 19.99 = 7 * (20.00 - 0.01)\).

Step 3: Multiply the individual terms

Next step is to carry out the individual multiplication operations. This gives: \(7*20.00 - 7*0.01 = 140.00 - 0.07\).

Step 4: Final computation

Subtracting 0.07 from 140.00, we get the final amount as \$139.93.

Key Concepts

Mental Math Strategies

Performing calculations in your head can be a powerful skill, especially when shopping or budgeting. One effective mental math strategy is using the Distributive Property, which simplifies complex multiplication problems. Instead of multiplying directly, break down the numbers into simpler components.
For example, when calculating the total cost for multiple items such as DVDs, break down numbers into round figures, as we did in the exercise by turning \(19.99\) into \(20.00 - 0.01\).

• Simplifies complex numbers into manageable parts
• Reduces potential errors compared to long calculations
• Increases speed of performing calculations

Practical application involves recognizing and rearranging components so that even intricate numbers transform into easy-to-handle parts.

Elementary Algebra

At its core, elementary algebra deals with the basic concepts of using variables and performing operations. The Distributive Property, illustrated in this problem, is a fundamental algebraic property. It states that multiplying a number by a sum is the same as multiplying each addend by the number and then adding the results.
When we decompose \(19.99\) into \(20.00 - 0.01\) for the multiplication \(7 * 19.99\), we are applying this concept.

• Helps in breaking down and simplifying expressions or equations
• Essential for more complex algebraic operations
• Forms the foundation for learning higher-level mathematics

By mastering elementary algebra, students can approach a wide array of mathematical problems with greater confidence and understanding.

Multiplication

Multiplication is a basic arithmetic operation that involves combining equal groups. In the exercise, we had to multiply \(7\) DVDs, each priced at \(19.99\). By expressing \(19.99\) as \(20.00 - 0.01\), the distributive property helps simplify the multiplication process.

• Breaks down complex numbers into simpler units
• Allows for easy calculation of products

Understanding multiplication is crucial, as it is used in various real-life scenarios, from shopping to dividing workloads.
Why is breaking down numbers helpful? It makes the process more intuitive, reducing cognitive load, and helps us retain accuracy in our mental calculations.