Problem 66 - Algebra 1: Concepts and Skills

Question

Subtract. Write the answer as a fraction or as a mixed number in simplest form. (Skills Review p.764) $$ \frac{7}{8}-\frac{2}{3} $$

Step-by-Step Solution

Answer

The result of the subtraction \( \frac{7}{8} - \frac{2}{3} \) is \( \frac{5}{24} \).

Step 1: Find Common Denominator

To subtract fractions, they must have the same denominator. The least common multiple of 8 and 3 is 24, so this will be our common denominator. So, rewrite the fractions as follows: \( \frac{7}{8} = \frac{21}{24} \), obtained by multiplying both numerator and denominator of the first fraction by 3, and \( \frac{2}{3} = \frac{16}{24} \), obtained by multiplying both numerator and denominator of the second fraction by 8.

Step 2: Subtract Fractions

Now that the fractions have the same denominator, you can subtract the fractions by subtracting the numerators and using the common denominator: \( \frac{21}{24} - \frac{16}{24} = \frac{21-16}{24} = \frac{5}{24} \).

Step 3: Simplify The Result

In this case, the result, \( \frac{5}{24} \), is already in its simplest form. That is because 5 and 24 have no common factors other than 1.

Key Concepts

Common Denominator

When subtracting fractions, it's crucial to have common denominators. This means both fractions must have the same bottom number. Without this shared number, the fractions aren't easily comparable in terms of how much they represent.
The common denominator basically acts like a translator, transforming the fractions so we can work with them smoothly. Let's look at an example of \(\frac{7}{8}\) and \(\frac{2}{3}\). To find a shared or common denominator, we look for the least common multiple (LCM) of the denominators, which are 8 and 3 in this case. The smallest number both can divide into evenly is 24.
To make \(\frac{7}{8}\) have a denominator of 24, multiply top and bottom by 3, resulting in \(\frac{21}{24}\). Similarly, multiply the top and bottom of \(\frac{2}{3}\) by 8 to get \(\frac{16}{24}\). Now, both fractions are speaking the same denominator-language, 24!

Simplifying Fractions

Simplifying fractions is like tidying up to make them look as neat as possible. When you simplify, you look for common factors between the numerator (top number) and the denominator (bottom number). The goal is to divide them both by their greatest common factor (GCF).
In our example, after subtracting the fractions, you get \(\frac{5}{24}\). Check if you can divide both 5 and 24 by a number greater than 1.

• Both only share the number 1 as a common factor, which means you're already in simplest form.

Simplifying not only makes the fraction easier to understand, but verifies you're expressing the smallest and most precise version of your answer. Remember, some fractions are already simplified, like \(\frac{5}{24}\) here.

Fraction Subtraction

Now let's piece it all together—how exactly do you subtract these fractions?First, with your common denominator established, subtract the numerators. The shared denominator simply tags along since it remains unchanged.
For example, starting with the fractions \(\frac{21}{24}\) and \(\frac{16}{24}\), subtract the numerators: \(21 - 16\). This gives you a new fraction, \(\frac{5}{24}\).

• Always check if your result needs simplifying, but remember that tiring is worth it for clarity and neatness.

Understanding how to subtract fractions can seem tricky initially, but once you practice turning to a common denominator and simplifying, it'll swiftly become second nature.