Problem 112 - Algebra 1: Concepts and Skills

Question

Subtract. Write the answer as a fraction in simplest form. (Skills Review p. 768) $$ 75 \%-\frac{3}{4} $$

Step-by-Step Solution

Answer

The answer to the subtraction problem is \(0\).

Step 1: Convert percentage to fraction

The percentage \(75\%\) can be written as a fraction by dividing the percentage by \(100\). So, \(75\% = \frac{75}{100}\) or simplified to \(\frac{3}{4}\)

Step 2: Subtract fractions

Subtract the given fraction \(\frac{3}{4}\) from the fraction we just calculated \(\frac{3}{4}\). This operation will look as follows: \(\frac{3}{4} - \frac{3}{4} = 0\)

Step 3: Result

The result of the subtraction is \(0\), which is already in its simplest form.

Key Concepts

Percentage to Fraction Conversion

Understanding percentage to fraction conversion is pivotal when dealing with a variety of mathematical problems. The term 'percentage' literally means 'per hundred,' which is a clue to its conversion process: divide the percentage by 100. For example, to convert 75% to a fraction, divide 75 by 100, resulting in the fraction \( \frac{75}{100} \).

However, it's essential to simplify the fraction to its lowest terms. In this case, both numerator and denominator are divisible by 25, so \( \frac{75}{100} \) simplifies to \( \frac{3}{4} \). The simplification process may require identifying the greatest common divisor (GCD) of both the top and bottom numbers and then dividing by it. This step is crucial for making fractions easier to work with in subsequent calculations.

Fraction Subtraction

Subtracting fractions involves finding a common denominator and then subtracting the numerators while keeping the denominator constant. When the fractions already have the same denominator, like in the given problem \( \frac{3}{4} \) and \( \frac{3}{4} \), one simply subtracts the second fraction's numerator from the first's. The operation is thus \( \frac{3}{4} - \frac{3}{4} \), which equals zero.

If the denominators were different, one would need to find equivalent fractions with a common denominator before subtracting. This ensures accuracy in the subtraction process. Remember, only the numerators are subtracted; the denominator remains unchanged unless the final fraction can be simplified further.

Simplifying Fractions

Simplifying fractions, also known as reducing fractions, is the process of finding the simplest form of a fraction by ensuring that the numerator and denominator have no common factors other than 1. To do this, one must find the greatest common divisor (GCD) of both the numerator and the denominator and then divide both by this number.

Take the fraction \( \frac{75}{100} \) from our earlier example; its simplification to \( \frac{3}{4} \) involves dividing both the top and bottom by their GCD, which is 25. Once simplified, the fraction becomes easier to use in further mathematical operations. It is a fundamental skill in mathematics that enables more straightforward computations and cleaner results, as seen when the simplified fraction subtracts effortlessly to zero.