Problem 10 - Algebra for College Students

Question

For Problems 1-12, perform the indicated operations involving rational numbers. Be sure to express your answers in reduced form. \(\frac{2}{3}-\frac{7}{8}+\frac{1}{4}\)

Step-by-Step Solution

Answer

The result is \( \frac{1}{24} \).

Step 1: Find a common denominator

To perform the operations with fractions, we first need to find a common denominator. The denominators here are 3, 8, and 4. The least common multiple (LCM) of these numbers is 24.

Step 2: Convert each fraction to have the common denominator

Convert \( \frac{2}{3} \) by multiplying both the numerator and the denominator by 8 to get \( \frac{16}{24} \). Convert \( \frac{7}{8} \) by multiplying both by 3 to get \( \frac{21}{24} \). Convert \( \frac{1}{4} \) by multiplying both by 6 to get \( \frac{6}{24} \).

Step 3: Perform the subtraction and addition

Now, perform the operations using the common denominator: \( \frac{16}{24} - \frac{21}{24} + \frac{6}{24} \). Start by performing the subtraction: \( 16 - 21 = -5 \), so the result is \( \frac{-5}{24} \).

Step 4: Add the next fraction

Next, add \( \frac{-5}{24} + \frac{6}{24} \). Add the numerators: \( -5 + 6 = 1 \), giving the result \( \frac{1}{24} \).

Step 5: Simplify the fraction

The fraction \( \frac{1}{24} \) is already in its simplest form, as 1 and 24 have no common factors besides 1.

Key Concepts

Least Common Multiple

When dealing with fractions that have different denominators, finding a common denominator is crucial for performing operations like addition and subtraction. The least common multiple (LCM) is the smallest number that all denominators divide into without leaving a remainder.

To find the LCM of 3, 8, and 4, we list the multiples of each number:

• Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24...
• Multiples of 8: 8, 16, 24, 32...
• Multiples of 4: 4, 8, 12, 16, 20, 24...

The smallest number that appears in all lists is 24, so 24 is the LCM. This means 24 is the smallest common denominator for these fractions, allowing us to perform operations more easily.

Fractions with Common Denominators

Once we determine the LCM, the next step is converting each fraction to have this common denominator. This involves adjusting each fraction so its denominator matches the LCM while keeping the fraction equivalent.

For example, to convert \( \frac{2}{3} \) to a fraction with a denominator of 24:

• Multiply both the numerator and denominator by 8 (since 3 × 8 = 24), resulting in \( \frac{16}{24} \).

Similarly, for \( \frac{7}{8} \):

• Multiply both by 3 (as 8 × 3 = 24) to get \( \frac{21}{24} \).

And for \( \frac{1}{4} \):

• Multiply both by 6 (because 4 × 6 = 24), resulting in \( \frac{6}{24} \).

Now all fractions have a denominator of 24, simplifying the process of subtraction and addition.

Simplifying Fractions

Simplifying a fraction means reducing it to its lowest terms. This involves dividing the numerator and the denominator by their greatest common divisor (GCD). A fraction is in its simplest form when the numerator and denominator can no longer be divided by any number other than 1.

For instance, consider \( \frac{16}{24} \). The GCD of 16 and 24 is 8. So, dividing both the numerator and denominator by 8 gives us \( \frac{2}{3} \). However, in our problem, the resulting fraction \( \frac{1}{24} \) is already in its simplest form since 1 and 24 have no common factors other than 1.

Always check your final answer to make sure it is simplified, as this is often a requirement in mathematical problems.

Subtraction and Addition of Fractions

With all fractions now having a common denominator, you can perform subtraction and addition without any hassle.

Taking our fractions \( \frac{16}{24} - \frac{21}{24} + \frac{6}{24} \):

• First, subtract \( \frac{21}{24} \) from \( \frac{16}{24} \): This becomes \( 16 - 21 = -5 \), so \( \frac{-5}{24} \).
• Next, add \( \frac{6}{24} \) to \( \frac{-5}{24} \): Add the numerators, \( -5 + 6 = 1 \), resulting in \( \frac{1}{24} \).

Since all operations were performed with fractions that have the same denominator, the denominator remains unchanged. This makes the subtraction and addition of fractions straightforward. Remember always to perform these operations from left to right according to their order in the expression.