Problem 109 - Algebra 1: Concepts and Skills

Question

Subtract. Write the answer as a fraction in simplest form. (Skills Review p. 768) $$ \frac{1}{2}-39 \% $$

Step-by-Step Solution

Answer

The answer to the subtraction \(\frac{1}{2} - 39\%\) is \(\frac{11}{100}\).

Step 1: Convert the percentage to a fraction

Percentage means 'per 100'. Therefore, 39% can be written as \(\frac{39}{100}\).

Step 2: Subtract the fractions

Now that both numbers are in fractional form, we can subtract: \(\frac{1}{2} - \frac{39}{100}\). However, to do this, we need to find a common denominator. The common denominator of 2 and 100 is 100. So, we rewrite \(\frac{1}{2}\) as \(\frac{50}{100}\). Now we can subtract these fractions: \(\frac{50}{100} - \frac{39}{100} = \frac{11}{100}\).

Step 3: Simplify the fraction

In this case, the fraction \(\frac{11}{100}\) is already in the simplest form, so no further simplification is necessary. That is the final answer.

Key Concepts

Simplifying Fractions

Simplifying fractions, also known as reducing fractions, is the process of expressing a fraction using the smallest possible numerator and denominator. The primary goal is to find the greatest common divisor (GCD) of both numbers, which is the highest number that divides both without leaving a remainder. For example, consider the fraction \( \frac{8}{20} \). The GCD of 8 and 20 is 4. To simplify this fraction, both the numerator and the denominator should be divided by the GCD: \( \frac{8 \div 4}{20 \div 4} = \frac{2}{5} \).

In the context of subtracting fractions, it's important to simplify the result to ensure that it is in its most basic form, making it easier to understand and use in further calculations. As seen in the exercise, the solution \( \frac{11}{100} \) cannot be simplified further as 11 and 100 share no common factors other than 1.

Converting Percentages to Fractions

Understanding how to convert percentages to fractions is essential in mathematics, as it allows for the consistent treatment of numerical values in calculations. To convert a percentage to a fraction, simply use the definition of percent, which is 'per hundred'. This means dividing the percentage number by 100. For instance, to convert 39% to a fraction, divide 39 by 100, yielding \( \frac{39}{100} \).

After converting, you should also simplify the fraction if possible. However, in our exercise example, the fraction \( \frac{39}{100} \) is already in the simplest form because 39 is a prime number and doesn't share any common divisors with 100, apart from 1.

Finding Common Denominators

One cannot perform subtraction or addition with fractions that have different denominators without first finding a common denominator. A common denominator is a shared multiple of the original denominators that allows fractions to be combined in a straightforward manner. To find the common denominator, determine the least common multiple (LCM) of the denominators.

In our subtracting fractions exercise, the denominators are 2 and 100. The LCM of 2 and 100 is 100 since 100 is the smallest number that both 2 and 100 can divide into without leaving a remainder. Thus, the fraction \( \frac{1}{2} \) needs to be converted to have a denominator of 100, resulting in \( \frac{50}{100} \). Now, with a common denominator, the fractions \( \frac{50}{100} \) and \( \frac{39}{100} \) can be easily subtracted.