Problem 107 - Algebra 1: Concepts and Skills

Question

Subtract. Write the answer as a fraction in simplest form. (Skills Review p. 768) $$ \frac{3}{4}-15 \% $$

Step-by-Step Solution

Answer

The result is \(\frac{3}{5}\).

Step 1: Convert Percentage to Fraction

To convert a percentage into a fraction, divide the percentage by 100. So, 15% becomes \(\frac{15}{100}\) or \(\frac{3}{20}\) when simplified.

Step 2: Subtract the Fractions

To subtract \(\frac{3}{4}\) - \(\frac{3}{20}\), find a common denominator, which in this case is 20. So, \(\frac{3}{4}\) becomes \(\frac{15}{20}\). Then subtract \(\frac{15}{20}\) - \(\frac{3}{20}\) to get \(\frac{12}{20}\).

Step 3: Simplify the Result

The result of the fraction subtraction is \(\frac{12}{20}\). By dividing both the numerator and denominator by their greatest common factor, which is 4, the simplest form of the fraction is \(\frac{3}{5}\).

Key Concepts

Converting Percentages to Fractions

Understanding how to convert percentages to fractions is an essential math skill that makes solving problems much simpler. To start, a percentage represents a number out of 100. Therefore, converting a percentage to a fraction simply involves taking that percentage value and placing it over 100. For example, to convert 15% to a fraction, you write it as \( \frac{15}{100} \).

The next step is to simplify this fraction by looking for the greatest common factor between the numerator and the denominator. In the case of \( \frac{15}{100} \), both numbers are divisible by 5, which reduces the fraction to \( \frac{3}{20} \). Simplification is a crucial step because it makes further arithmetic operations easier and the results clearer to understand.

Finding Common Denominators

When you're faced with the task of subtracting fractions, such as \( \frac{3}{4} \) and \( \frac{3}{20} \) in our example, it's imperative to find common denominators before you can perform the subtraction. A common denominator is a common multiple of the two denominators. To find it, you can list multiples of each denominator and find the smallest number that appears in both lists, or simply multiply the two denominators if finding a smaller common multiple proves challenging.

In this problem, the smaller number that 4 and 20 both divide into without a remainder is 20. Therefore, you convert \( \frac{3}{4} \) into twentieths by multiplying both the numerator and denominator by 5, turning it into \( \frac{15}{20} \). With the fractions now sharing a common denominator, you can subtract them directly.

Simplifying Fractions

The final and often overlooked step is simplifying our fraction, the mathematical way of tidying up. Simplification makes numbers easier to work with and understand. To simplify a fraction, we want to find the largest number that divides evenly into both the numerator and denominator—this is known as the greatest common factor (GCF). For \( \frac{12}{20} \), the GCF is 4. We then divide both the top and bottom numbers by 4 to get our simplest form:

\[\frac{12\div4}{20\div4} = \frac{3}{5}\]

Why Simplify?

• Simplifying makes it easier to see at a glance what part of a whole we have.
• It helps in comparing sizes of different fractions.
• Simple fractions are often needed for further calculations in equations, algebra, and beyond.

Always remember, the simplest form of a fraction is the one where the numerator and denominator can no longer be divided by the same number, except for one.