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Chapter 7: Problem 71

Add or subtract as indicated. Write each answer in lowest terms. $$ \frac{1}{2}+\frac{7}{8} $$

Short Answer

Expert verified
\frac{11}{8}

Step by step solution

01

Find the common denominator

To add fractions, the denominators must be the same. The denominators here are 2 and 8. The least common multiple (LCM) of 2 and 8 is 8.
02

Convert fractions to have the same denominator

Change the fraction \(\frac{1}{2}\) to an equivalent fraction with a denominator of 8. \(\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}\). Now we have \(\frac{4}{8} + \frac{7}{8}\).
03

Add the fractions

Now that the fractions have the same denominator, add the numerators: \(\frac{4+7}{8} = \frac{11}{8}\).
04

Simplify the fraction

The fraction \(\frac{11}{8}\) is already in its lowest terms. So, the answer is \(\frac{11}{8}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

common denominator
When adding or subtracting fractions, having a common denominator is essential. The denominator is the bottom number of the fraction, showing how many equal parts the whole is divided into.
To operate with fractions, they need the same number of equal parts, meaning their denominators must match.
For instance, if we have \(\frac{1}{2}\) and \(\frac{7}{8}\), we can't directly add them since 2 and 8 are different.
To find a common denominator, we look for a common multiple of both denominators, so both fractions can describe parts of the same whole.
least common multiple
The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both.
In the exercise, the denominators are 2 and 8. We need to find the LCM to make the denominators the same.
The multiples of 2 are 2, 4, 6, 8, 10, etc. The multiples of 8 are 8, 16, 24, etc.
The smallest common multiple is 8. This is why we change both fractions to have a denominator of 8.
Converting the fractions \(\frac{1}{2}\) and \(\frac{7}{8}\) into a common form allows us to easily add them.
simplifying fractions
Simplifying fractions means reducing them to their simplest form.
This is done by dividing both the numerator (the top number) and the denominator by their greatest common divisor (GCD).
Fractions should be simplified to make them easier to read and understand. In the exercise, we add \(\frac{4}{8} \) and \(\frac{7}{8}\) getting \(\frac{11}{8}\).
Since 11 and 8 have no common divisors besides 1, \(\frac{11}{8}\) is already in its simplest form.
Always check your final fraction to see if it can be simplified for a neater answer.
equivalent fractions
Equivalent fractions represent the same value, even if their numerators and denominators differ.
For example, \(\frac{1}{2}\) can be written as \(\frac{2}{4}\) or \(\frac{4}{8}\). They all equal 0.5.
In our exercise, we turn \(\frac{1}{2}\) into \(\frac{4}{8}\) so we can easily add it to \(\frac{7}{8}\).
This step of converting fractions to equivalent ones using the common denominator is crucial in fraction addition and subtraction.
It ensures you are working with the same size parts, simplifying the process of combining fractions.

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