Problem 10 - 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. $$ \begin{array}{|l|l|l|l|l|} \hline & \text { Percent } & \text { Ratio } & \text { Fraction } & \text { Decimal } \\ \hline 10 . & 2.5 \% & & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Decimal: 0.025; Fraction: \(\frac{1}{40}\); Ratio: 1:40.

Step 1: Convert Percent to Decimal

To convert from percent to decimal, divide the percentage by 100. For 2.5%, this calculation is \( \frac{2.5}{100} = 0.025 \).

Step 2: Convert Percent to Fraction

To convert 2.5% to a fraction, first express it as \( \frac{2.5}{100} \). Then multiply both the numerator and the denominator by 10 to eliminate the decimal, which leads to \( \frac{25}{1000} \). Simplify \( \frac{25}{1000} \) by dividing both the numerator and the denominator by their greatest common divisor which is 25. This gives \( \frac{1}{40} \).

Step 3: Convert Percent to Ratio

A percentage can also be expressed as a ratio. For 2.5%, consider it as a ratio to 100, which is \(2.5:100\). By eliminating the decimal, it becomes 25:1000. Simplify by dividing both parts of the ratio by 25, ending up with 1:40.

Key Concepts

Percent to Decimal Conversion

Converting a percent to a decimal is a straightforward process that involves dividing by 100. This is because "percent" means "per hundred," so you need to scale down to find the equivalent decimal.
To convert 2.5% to a decimal:

• Start with the given percent: 2.5%.
• Divide 2.5 by 100: \( \frac{2.5}{100} = 0.025 \).

As a result, 2.5% is equivalent to 0.025 as a decimal. Remember this rule: **To convert any percent to a decimal, divide by 100.**
This method works for all percentages, whether they are whole numbers or include decimals. Round the result to the nearest hundredth if necessary, but here 0.025 is already precise.

Percent to Fraction Conversion

When converting a percent to a fraction, you often need to handle decimals. The key is to first write the percent as a fraction with a denominator of 100, and then simplify if possible.
Here's how you convert 2.5% to a fraction:

• Begin by writing it as a fraction over 100: \( \frac{2.5}{100} \).
• Eliminate the decimal by multiplying both the numerator and the denominator by 10, resulting in \( \frac{25}{1000} \).
• To simplify \( \frac{25}{1000} \), find the greatest common divisor (GCD) of 25 and 1000, which is 25.
• Divide both the numerator and the denominator by their GCD: \( \frac{25}{25} = 1 \) and \( \frac{1000}{25} = 40 \).

This gives you the simplified fraction \( \frac{1}{40} \). Always remember to simplify the fraction to its lowest terms to have a precise and usable result.

Percent to Ratio Conversion

Converting a percent to a ratio is very similar to the process of converting to a fraction. The key idea is to conceptualize the percent as parts of a hundred, which can be expressed in a 1 to 100 format and then simplified.
To convert 2.5% to a ratio:

• Start with 2.5 as a percentage of 100: 2.5:100.
• Get rid of the decimal by multiplying both numbers by 10, resulting in 25:1000.
• Simplify by dividing both sides by their greatest common divisor, which is 25. This gives you 1:40.

Hence, 2.5% can be expressed as the ratio 1:40. Ratios are simplified by dividing both terms by their greatest common divisor, similar to simplifying fractions. This ensures that ratios are easy to understand and compare.