Problem 18 - Prealgebra

Question

Express each percent as a fraction or mixed number in simplest form and as a decimal. $$42 \%$$

Step-by-Step Solution

Answer

42% is \(\frac{21}{50}\) or 0.42.

Step 1: Understand Percent as a Fraction

A percent represents a part out of 100. Therefore, 42% can be expressed as the fraction \(\frac{42}{100}\). This is because 42% literally means 42 out of 100.

Step 2: Simplify the Fraction

Identify the greatest common divisor (GCD) of the numerator and denominator. The GCD of 42 and 100 is 2. Divide both the numerator and the denominator by 2: \(\frac{42 \div 2}{100 \div 2} = \frac{21}{50}\). Thus, 42% in fraction form is \(\frac{21}{50}\).

Step 3: Convert to Decimal

To convert a fraction to a decimal, divide the numerator by the denominator. Divide 21 by 50: \(21 \div 50 = 0.42\). Thus, 42% as a decimal is 0.42.

Key Concepts

Understanding Fractions

In mathematics, fractions are so much more than just numbers. They represent the way we communicate parts of a whole. A fraction consists of two essential parts: the numerator and the denominator. The numerator, which is the number above the line, symbolizes how many parts we have. Meanwhile, the denominator, positioned below the line, indicates the total number of equal parts the whole is divided into.
The notion of a fraction is deeply intertwined with the concept of division. When you think of the fraction \(\frac{42}{100}\), it's like asking how many hundredths fit into 42 parts.
This foundational understanding of fractions is crucial, especially when converting percentages to fractions. Since a percent is simply a way of expressing parts per hundred, you can quickly turn any percentage into a fraction with a denominator of 100. For example, 42% effortlessly becomes \(\frac{42}{100}\). This fraction can then be simplified to its simplest form for easier mathematical handling.

Decimals Explained

Decimals serve as another way of expressing fractions, but they emphasize the position of numbers relative to the decimal point. They provide an excellent means of displaying fractions with denominators that are powers of ten.
When converting a fraction such as \(\frac{21}{50}\) into a decimal, the key is to perform a division. Think of it as distributing the 21 units evenly among 50 parts to find how much each part is worth. Since \(21 \div 50 = 0.42\), we express this value as 0.42 in decimal form.
Decimals are particularly handy because they allow for straightforward addition, subtraction, multiplication, and division without needing to align fractions. Moreover, they provide a quick and easy way to compare numbers as they clearly indicate value positions after the decimal point.

Simplifying Fractions

Simplifying fractions is akin to finding a clearer way to view the same fraction. It's the process of reducing the fraction to its simplest form while retaining the same value.
To simplify a fraction like \(\frac{42}{100}\), you start by determining the greatest common divisor (GCD) of both the numerator and the denominator. The GCD is the highest number that divides both numbers evenly. For 42 and 100, this number is 2.
Divide both the numerator and the denominator by their GCD to simplify the fraction: \(\frac{42 \div 2}{100 \div 2} = \frac{21}{50}\).

By simplifying, you achieve the most reduced form of the fraction, making it both easier to comprehend and utilize in further mathematical calculations. A simplified fraction ensures effective communication of the quantity or concept you are dealing with.