Problem 41 - Prealgebra

Question

Express each percent as a fraction or mixed number in simplest form and as a decimal. $$330 \%$$

Step-by-Step Solution

Answer

330% is \(3\frac{3}{10}\) as a fraction and 3.3 as a decimal.

Step 1: Convert Percent to Fraction

To convert 330% to a fraction, first recognize that percent means per hundred. So 330% is equal to \(\frac{330}{100}\).

Step 2: Simplify the Fraction

Next, simplify the fraction \(\frac{330}{100}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 10. This gives us \(\frac{33}{10}\).

Step 3: Convert to Mixed Number

Since \(\frac{33}{10}\) is an improper fraction, convert it to a mixed number. Divide 33 by 10 to get 3 as the whole number and 3 as the remainder, which means \(3\frac{3}{10}\).

Step 4: Convert Percent to Decimal

To convert 330% to a decimal, divide 330 by 100. So, \(\frac{330}{100} = 3.3\).

Key Concepts

Fractions and Decimals

Understanding the relationship between fractions and decimals is essential in mathematics. Both fractions and decimals are ways to express numbers that are not whole numbers. A fraction has two parts: a numerator (top number) and a denominator (bottom number). For example, in the fraction \(\frac{330}{100}\), 330 is the numerator, and 100 is the denominator.

To convert a fraction to a decimal, you divide the numerator by the denominator. So for \(\frac{330}{100}\), dividing 330 by 100 gives the decimal 3.3. Similarly, converting a decimal to a fraction involves reversing this process—instead of dividing, you turn the decimal into a fraction that is expressed as a number over powers of ten, such as \(\frac{33}{10}\) for 3.3.

When working with percentages, remember that a percent is just a fraction with a denominator of 100. So 330% means 330 per 100, which can be directly expressed as \(\frac{330}{100}\).

Mixed Numbers

A mixed number combines a whole number with a fraction. Mixed numbers are especially useful when dealing with improper fractions, where the numerator is larger than the denominator.

Take \(\frac{33}{10}\) as an example. This improper fraction indicates that 33 is divided by 10, which results in a whole number of 3 and a remainder of 3.

So, to convert \(\frac{33}{10}\) to a mixed number, you perform the division: 33 divided by 10 equals 3, with a remainder of 3. Therefore, \(\frac{33}{10}\) becomes the mixed number \(3\frac{3}{10}\).

Mixed numbers present a straightforward way of expressing quantities in everyday life scenarios, such as in measurements or more intuitive contexts where a fraction part alone may not convey the exact full amount.

Simplifying Fractions

Simplifying fractions is the process of reducing the fraction to its simplest form, where the numerator and denominator have no common factors other than 1.

To simplify a fraction, you need to find the greatest common divisor (GCD) of both the numerator and the denominator. In our example, \(\frac{330}{100}\), the GCD of 330 and 100 is 10. By dividing both the numerator and the denominator by 10, you simplify the fraction to \(\frac{33}{10}\).

Simplifying is important because it makes fractions easier to read and compare. It also helps in performing arithmetic operations, such as addition or multiplication. Remember: a fraction in its simplest form is equivalent to its original form, meaning it has the same value. Thus, \(\frac{330}{100}\) and \(\frac{33}{10}\) represent the same numerical quantity, just in different forms.