Problem 64 - Algebra 1: Concepts and Skills

Question

SUBTRACTING FRACTIONS Subtract. Write the answer as a fraction or as a mixed number in simplest form. $$ 17 \frac{5}{8}-10 \frac{3}{8} $$

Step-by-Step Solution

Answer

The answer is \(7 \frac{1}{4}\)

Step 1: Subtract the Fraction Parts

Start by subtracting the fraction parts of the mixed numbers: \(17 \frac{5}{8}-10 \frac{3}{8} = 17 \frac{5}{8}-10 \frac{3}{8} = 17 \frac{2}{8}-10\). The fraction \(\frac{2}{8}\) can be further simplified to \(\frac{1}{4}\), leaving us with \(17 \frac{1}{4}-10\).

Step 2: Subtract the Integer Parts

Next, subtract the integer parts of the mixed numbers: \(17 \frac{1}{4}-10 = 7 \frac{1}{4}\). The operation results in the mixed number \(7 \frac{1}{4}\).

Step 3: Simplify for Final Answer

Our answer, following simplification, is \(7 \frac{1}{4}\)

Key Concepts

Subtracting Fractions

Subtracting fractions is a fundamental skill in math. To subtract fractions with like denominators (the bottom part of the fraction), simply subtract the numerators (the top part) and keep the denominator the same. For example, to solve
\(17 \frac{5}{8} - 10 \frac{3}{8}\), you focus on the fractions:
\(\frac{5}{8} - \frac{3}{8} = \frac{2}{8}\).
However, when subtracting mixed numbers, which include both a whole number and a fraction, subtract the whole numbers and fractions separately as shown in the solution above. It's important to remember to simplify the fractions if possible, which leads us to our next concept.

Simplifying Fractions

Simplifying fractions, also known as reducing fractions, is the process of making a fraction as simple as possible.A fraction is simplified when the numerator and denominator are both as small as possible, while still maintaining the same value. This means finding the greatest common divisor (GCD) for both numbers and dividing them by it. In our example,
\(\frac{2}{8}\) can be simplified because 2 and 8 have a GCD of 2.

By dividing the numerator and denominator by 2, we get:\(\frac{2 \div 2}{8 \div 2} = \frac{1}{4}\).
Simplifying makes the fraction easier to work with and understand. Always aim to simplify fractions as part of finding the final answer in any math problem.

Mixed Numbers

Mixed numbers combine a whole number with a fraction, representing a value greater than a whole but not quite reaching the next whole number. The steps in the given solution showcase how to handle mixed numbers: first, separate the whole numbers and the fractions when performing arithmetic operations.

For subtraction, this means dealing with the whole number parts independently from the fractions. Once you have the results, combine them for the final mixed number. If the fraction can be simplified, as with the \(\frac{2}{8}\) in our solution, do so before finalizing the mixed number. This helps in concluding with a neat, easily interpreted answer. In the case of our exercise, our mixed number was \(7 \frac{1}{4}\) after simplifying the fraction.