Problem 99 - Algebra A Combined Function

Question

Which of the following are correct? a. \(6.5 \%=0.65\) b. \(7.8 \%=0.078\) c. \(120 \%=0.12\) d. \(0.35 \%=0.0035\)

Step-by-Step Solution

Answer

Option (b) and (d) are correct.

Step 1: Understand the Percentage to Decimal Conversion

To convert a percentage to a decimal, divide the percentage by 100. This means moving the decimal point two places to the left.

Step 2: Convert Option a

For option (a), convert \(6.5\%\) to a decimal. \(6.5 \div 100 = 0.065\). This means \(6.5\% = 0.065\), not \(0.65\).

Step 3: Convert Option b

For option (b), convert \(7.8\%\) to a decimal. \(7.8 \div 100 = 0.078\). This matches the given decimal, so option (b) is correct.

Step 4: Convert Option c

For option (c), convert \(120\%\) to a decimal. \(120 \div 100 = 1.2\). This means \(120\% = 1.2\), not \(0.12\).

Step 5: Convert Option d

For option (d), convert \(0.35\%\) to a decimal. \(0.35 \div 100 = 0.0035\). This matches the given decimal, so option (d) is correct.

Key Concepts

Decimal Conversion

Converting a percentage into a decimal is a straightforward task, yet it holds significant importance in mathematics. When we talk about percentages, we are referring to a number or ratio expressed as a fraction of 100. The term itself originates from Latin words “per centum,” meaning “by a hundred.” To convert a percentage into a decimal, the key operation is division. Here’s how you do it:

• Divide the percentage value by 100.
• Move the decimal point two places to the left.

These steps achieve the same result, essentially easing the human error seen in arithmetic. For example, taking the percentage 6.5% and converting it:

• 6.5 divided by 100 gives you 0.065.
• As a result, 6.5% converts to 0.065.

Understanding these mechanics can help you easily breeze through several mathematical problems involving percentages and decimals. Remember, each shift in the decimal point signifies reducing the number by a factor of 10.

Mathematics Problem Solving

Problem-solving in mathematics involves interpreting and processing numeric and symbolic information to find solutions. Applying a systematic approach helps to tackle even the most complex problems efficiently. When presented with exercises like converting percentages to decimals, it’s vital to break down the problem into manageable steps. Let’s see how this approach works:

• Identify: Recognize the form of data you are working with, which in this exercise is percentages.
• Understand: Gather information about conversion rules, such as how to transition from percentage to decimal.
• Evaluate: Convert the numbers by applying the conversion process.
• Verify: Compare your result with options given.
  Evaluate each option to determine its correctness, as demonstrated in the initial steps from the exercise.

These systematic strategies can be applied across various mathematical problems, ensuring clarity and easing the complexity involved in solving them.

Understanding Percentages

To truly master the concepts of percentages, we need to grasp what they represent and how they can easily interconnect with other numeric forms. A percentage reflects parts of a whole, much like fractions and decimals. The convenience of percentages lies in their ability to straightforwardly represent data comparisons and proportions. By understanding percentages, we can:

• Facilitate easy conversion between different numerical expressions like fractions and decimals.
• Quickly interpret statistical data, which often relies on percentages for clarity.
• Simplify real-world applications such as financial discounts, interest rates, and statistics.

For example, converting an incorrect percentage like 120% to a more meaningful decimal emerges as 1.2, not 0.12, hence presenting a whole new contextual meaning. Comprehending these relationships puts challenges like textbook problems into perspective and eases daily applications of mathematics in various fields.