Problem 69 - Algebra 1: Concepts and Skills

Question

You are volunteering for a charity that is operating a haunted house. You are sent to the store to buy 5 bags of cotton balls that will be used to make spider web decorations. Each bag costs $2.09. Use the distributive property to mentally calculate the total cost of the cotton balls.

Step-by-Step Solution

Answer

The total cost for the 5 bags of cotton balls is $10.45.

Step 1: Identify the Values

The first step is to identify the values in the problem. There are 5 bags of cotton balls and each costs $2.09.

Step 2: Break Down the Decimal

The next step is to break down $2.09 into two parts: $2.00 (which is a whole number) and $0.09 (which is a part of one, or decimal part).

Step 3: Apply the Distributive Property

The distributive property allows to multiply 5 separately with each part. Therefore, \(5 * 2.00\) and \(5 * 0.09\) are calculated separately.

Step 4: Calculate the Multiplication

Multiplying \(5 * 2.00\) gives $10.00, and \(5 * 0.09\) gives $0.45.

Step 5: Sum Up the Results

Finally, sum up the results of the two multiplications. So, \(10.00 + 0.45 = 10.45\).

Key Concepts

Mental Math

Mental math is a useful skill that allows you to perform calculations without using a calculator or paper. It involves breaking down numbers into simpler parts to make arithmetic operations easier. This process helps in rapid calculations, especially in everyday situations like shopping or splitting bills. For example, when you want to calculate 5 times $2.09 mentally, you can leverage mental math techniques.
One strategy in mental math involves using approximation and adjustment, which means rounding numbers to the nearest whole number and adjusting after performing the math. However, when precision is necessary, like in decimals, breaking down numbers into components is key.

• Separate the whole number portion from the decimal.
• Deal with each part individually using simple multiplication.
• Add the results together for the complete answer.

By regularly practicing mental math, you'll improve your number sense and gain confidence in tackling numerical problems in your head.

Multiplication of Decimals

The process of multiplying decimals, such as $2.09, involves a few straightforward steps to ensure accuracy. Decimals can often seem tricky, but by breaking them into parts, the task becomes manageable.
Here's a step-by-step guide:

• Step 1: Break down the decimal number into its constituent whole and fractional parts. For example, $2.09 becomes $2.00 and $0.09.
• Step 2: Use the distributive property to multiply each part separately by the multiplier—in this case, 5.
• Step 3: Multiply the whole number component separately from the decimal part. Multiply 5 by $2, resulting in $10, and 5 by $0.09, resulting in $0.45.
• Step 4: Finally, add the two products together: $10 + $0.45 = $10.45.

This breakdown helps students see how working in parts can simplify decimal multiplication. Practicing with various numbers will strengthen your understanding of this fundamental math skill.

Problem-Solving in Mathematics

Problem-solving in mathematics not only involves finding the correct answer but also understanding the best approach to tackle a problem. Let's look at how problem-solving incorporates mathematical properties and strategies.
A common technique is the use of properties like the distributive property. This mathematical property helps distribute a multiplier over terms inside parentheses. In our exercise, it efficiently solved for the cost of multiple bags of cotton by breaking down $2.09 and handling each component separately.

• Identify: Begin by clearly understanding the numbers and operations involved.
• Break Down: Simplify the numbers using mathematical properties, making calculations easier.
• Apply Strategies: Use strategies like breaking down decimals or using estimation for rough calculations.
• Verify: Finally, double-check computations to ensure accuracy.

Learning to solve problems step-by-step helps build logical thinking. It turns math from a daunting task into a puzzle that can be tackled systematically and calmly. By developing these strategies, students can apply them beyond mathematics and into real-world scenarios.