Problem 22 - Prealgebra

Question

Express each percent as a fraction or mixed number in simplest form and as a decimal. $$150 \%$$

Step-by-Step Solution

Answer

150% is \( \frac{3}{2} \) or \( 1 \frac{1}{2} \) and 1.5 as a decimal.

Step 1: Convert Percentage to Fraction

To express a percentage as a fraction, you divide by 100. So, \( 150\% \) becomes \( \frac{150}{100} \).

Step 2: Simplify the Fraction

Simplify \( \frac{150}{100} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 50. \( \frac{150 \div 50}{100 \div 50} = \frac{3}{2} \).

Step 3: Convert the Improper Fraction to a Mixed Number

The fraction \( \frac{3}{2} \) can be converted into a mixed number. Divide 3 by 2, which results in 1 whole and a remainder of 1, so it becomes \( 1 \frac{1}{2} \).

Step 4: Convert Percentage to Decimal

To convert a percentage to a decimal, divide by 100. Thus, \( 150\% \) as a decimal is \( 1.5 \).

Key Concepts

Simplifying Fractions

When dealing with fractions, simplifying them can make calculations more straightforward and results easier to interpret. A fraction is simplified when you can no longer divide the numerator and the denominator by the same number other than 1.
In our example, the fraction is \( \frac{150}{100} \). The process of simplifying it involves finding the greatest common divisor (GCD) of 150 and 100, which is 50. This simplifies the fraction as follows:

• Divide the numerator 150 by 50: \(150 \div 50 = 3\)
• Divide the denominator 100 by 50: \(100 \div 50 = 2\)
• This gives the simplified fraction: \( \frac{3}{2} \)

Simplifying fractions helps in making them more approachable and less bulky, which is particularly effective for further operations.

Improper Fractions

An improper fraction is a fraction where the numerator is greater than or equal to the denominator. For example, \( \frac{3}{2} \) is an improper fraction. This type of fraction often indicates that the fraction is greater than or equal to one.
Dealing with improper fractions can initially seem challenging, but they represent a quantity quite well, especially in mathematical operations.
One way to think of an improper fraction is to see it as a division problem that's not fully executed. In our example of \( \frac{3}{2} \), 3 divided by 2 equals 1, with a remainder of 1, which leads us to considering mixed numbers.

Mixed Numbers

A mixed number is a combination of a whole number and a proper fraction. Converting an improper fraction to a mixed number offers a more intuitive way to understand the quantity.
Let's take the improper fraction \( \frac{3}{2} \) and convert it to a mixed number:

• Divide the numerator by the denominator: \(3 \div 2\) equals 1 with a remainder of 1.
• The quotient (1) becomes the whole number part of the mixed number, and the remainder (1) stays over the original denominator (2), forming the fraction \( \frac{1}{2} \).
• So, \( \frac{3}{2} \) becomes \(1 \frac{1}{2} \).

Mixed numbers are especially useful for visually understanding measurements and for ease in addition and subtraction operations.

Decimal Conversion

Converting a percentage to a decimal is another crucial skill that streamlines mathematical computations. When you convert percentages to decimals, you simply divide the percentage by 100, thereby removing the percent symbol. This aligns your values for operations like multiplication and addition.
For example, converting \(150\%\) to a decimal involves:

• Divide 150 by 100: \(150 \div 100 = 1.5\)
• This gives the decimal 1.5

Decimal conversion is efficient because it allows computations without having to deal with fractions directly. It simplifies equations and calculations, making it easier to assess and manipulate numbers in everyday and complex mathematical contexts.