Problem 5 - Calculate With Confidence

Question

Complete the table below. Express each of the following measures in their equivalents where indicated. Reduce fractions and ratios to lowest terms; round decimals to hundreths. $$ \begin{array}{|l|l|l|l|l|} \hline & \text { Percent } & \text { Ratio } & \text { Fraction } & \text { Decimal } \\ \hline 5 . & & & & 0.06 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

6%, \(\frac{3}{50}\), 3:50.

Step 1: Convert Decimal to Percent

To convert the decimal to a percent, multiply the decimal by 100 and add a percent sign. For 0.06, it is calculated as \( 0.06 \times 100 = 6\% \).

Step 2: Convert Decimal to Fraction

To convert a decimal to a fraction, place the decimal over 1, and then multiply the numerator and the denominator by 10 until the decimal becomes a whole number. For 0.06, it becomes \( \frac{6}{100} \). Simplify by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This results in \( \frac{3}{50} \).

Step 3: Convert Percent to Ratio

A percent can be directly converted into a ratio by interpreting the percent as part out of 100. Since the percent is 6%, this converts to the ratio \( 6:100 \). By simplifying the ratio using the greatest common divisor of 6 and 100, which is 2, the ratio becomes \( 3:50 \).

Key Concepts

Decimal to Percent Conversion

Converting decimals to percents is a straightforward process due to a simple mathematical rule. A decimal essentially represents a part of a whole, much like a percent does, but in decimal format. To change a decimal into a percent:

• Multiplies the decimal by 100
• Add a percentage sign (%)

For instance, if you have a decimal of 0.06, simply multiply it by 100. This gives you 6. Thus, 0.06 as a decimal is equivalent to 6% as a percent. This conversion is helpful in various contexts, such as when discussing discounts, probability, or statistics.

Fraction Simplification

Fractions can often be simplified to make them easier to understand and work with. Simplification refers to reducing the fraction to its lowest terms. You achieve this by finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this number.Consider the fraction resulting from the decimal 0.06, which is initially expressed as \(\frac{6}{100}\). To simplify this:

• Find the GCD of 6 and 100, which is 2.
• Divide both the numerator and the denominator by 2 to get \(\frac{3}{50}\).

Simplified fractions are particularly useful in making calculations easier and in presenting numbers in their most readable form.

Ratio and Fraction Conversion

The relationship between ratios and fractions makes conversion between them a seamless process. Both express relationships, but in slightly different formats.A percentage, like 6%, can be seen as part of a ratio. To express it as a ratio, consider it as 6 out of 100, which gives the ratio \(6:100\). However, just like fractions, ratios can often be simplified.To simplify \(6:100\):

• Find the GCD of the two numbers, which is 2.
• Divide each part of the ratio by 2, resulting in the simplified ratio \(3:50\).

Understanding this conversion is essential, especially in subjects like algebra, where relationships between quantities are frequently analyzed.

Basic Mathematical Operations

Mathematical operations allow for the manipulation of numbers across various forms, like decimals, fractions, and ratios. Basic operations include addition, subtraction, multiplication, and division, but also involve conversions, such as those between different forms of numerical representation. For decimals and fractions, operations might include:

• Converting decimals to fractions for easier computation
• Simplifying fractions before performing division

When working with ratios or percents, operations often involve conversion to uncover the best form for calculation. For example, multiplying a ratio by a number to scale it up or down, based on the needs of the problem. Mastering these operations enhances problem-solving skills and is fundamental to understanding more complex mathematical concepts.