Problem 1 - Prealgebra

Question

Express each percent as a fraction or mixed number in simplest form and as a decimal. $$30 \%$$

Step-by-Step Solution

Answer

30% is \( \frac{3}{10} \) or 0.3.

Step 1: Identify the Percent as a Fraction

To express 30% as a fraction, start by recognizing that percent means 'per hundred.' Thus, 30% is \( \frac{30}{100} \).

Step 2: Simplify the Fraction

Simplify \( \frac{30}{100} \) by finding the greatest common divisor of 30 and 100, which is 10. Divide the numerator and the denominator by 10: \( \frac{30 \div 10}{100 \div 10} = \frac{3}{10} \). Thus, 30% as a fraction is \( \frac{3}{10} \).

Step 3: Convert the Fraction to a Decimal

To convert \( \frac{3}{10} \) into a decimal, divide 3 by 10, which gives 0.3. Thus, 30% as a decimal is 0.3.

Key Concepts

Simplifying Fractions

Fractions often start off looking a bit more complex than they need to be. Simplifying them makes calculations easier and keeps expressions tidy. Simplification involves reducing a fraction to its smallest form, where the numerator and the denominator share no common factors other than 1.

• Start by identifying the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both without leaving a remainder.
• Divide the numerator and the denominator by this GCD.
• The resulting fraction is equal to the original but in its simplest form.

Consider the example of 30%, expressed as the fraction \( \frac{30}{100} \). To simplify this, find the GCD of 30 and 100, which is 10. Dividing both by 10 gives \( \frac{3}{10} \). Therefore, the simplified form of \( \frac{30}{100} \) is \( \frac{3}{10} \). Simplifying fractions is an essential skill in both everyday math and advanced applications.

Decimal Conversion

Decimals are a way of expressing fractions without using a fraction bar. Converting fractions to decimals can make certain calculations and comparisons more straightforward.
Converting a fraction to a decimal involves dividing the numerator by the denominator using long division. When you divide 3 by 10, the result is 0.3. Here's how you can understand the process:

• Set up the division of the numerator (3) by the denominator (10).
• Since 3 is smaller than 10, you will get a number smaller than 1, which explains the decimal.
• Perform the division carefully, and you'll find out that \( \frac{3}{10} \) equals 0.3.

Decimals are especially helpful for representing fractions in a format compatible with digital devices and statistical data. For example, instead of \( \frac{3}{10} \), 0.3 is often easier to use in spreadsheets and calculators.

Percentages

Percentages are a way to express proportions out of a total of 100. It is one of the most common ways to describe data in both academic and real-world settings. Understanding percentages is crucial for interpreting statistics, financial data, and everyday contexts like shopping discounts or interest rates.

• "Percent" means "per hundred", so 30% is actually saying 30 out of every 100.
• To convert a percentage to a fraction, place the percentage number over 100, then simplify if possible.
• For example, 30% becomes \( \frac{30}{100} \), and, as previously explained, simplifies to \( \frac{3}{10} \).

Percentages are intuitive for grasping relative sizes, allowing quick comparisons between different data sets or values. They are often used in contexts like sales to indicate discounts, in polls to show popularity, and in finance to discuss interest rates.