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Chapter 11: Problem 54

Subtract. $$ -\frac{2}{3}-\frac{5}{8} $$

Short Answer

Expert verified
-\frac{31}{24}

Step by step solution

01

- Identify the common denominator

The given fractions are \(-\frac{2}{3}\) and \(-\frac{5}{8}\). To subtract these fractions, find a common denominator. The least common multiple (LCM) of 3 and 8 is 24.
02

- Convert fractions to have the common denominator

Convert \(-\frac{2}{3}\) to a fraction with a denominator of 24. \[ -\frac{2}{3} = -\frac{2 \times 8}{3 \times 8} = -\frac{16}{24} \] Convert \(-\frac{5}{8}\) to a fraction with a denominator of 24. \[ -\frac{5}{8} = -\frac{5 \times 3}{8 \times 3} = -\frac{15}{24} \]
03

- Subtract the fractions

Now subtract the fractions: \(-\frac{16}{24} - (-\frac{15}{24})\) becomes \(-\frac{16}{24} - \frac{15}{24}\). Since both fractions now have the same denominator, subtract the numerators: \[ -\frac{16}{24} - \frac{15}{24} = -\frac{16 + 15}{24} = -\frac{31}{24} \]

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Key Concepts

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

common denominator
When subtracting or adding fractions, they must have the same denominator. This is called a common denominator. It allows you to easily combine the fractions without changing their value.
For example, to subtract \(-\frac{2}{3}-\frac{5}{8}\), you first identify a common denominator. This helps us compare the fractions more directly.
Without a common denominator, the numerators (top numbers) are not comparable because they are parts of different-sized 'wholes'.
least common multiple
To find a common denominator, we look for the least common multiple (LCM) of the original denominators.
The LCM is the smallest number that is a multiple of both denominators.
For denominators 3 and 8, the LCM is 24.
Finding the LCM is crucial because it helps to minimize the size of the numbers we work with, making the calculations simpler.
Once we have the LCM, we can convert each fraction to have this common denominator.
subtracting fractions
With fractions converted to a common denominator, subtraction is straightforward.
First, convert \(-\frac{2}{3}\) and \(-\frac{5}{8}\) to \(-\frac{16}{24}\) and \(-\frac{15}{24}\), respectively.
Then subtract the numerators: \(-\frac{16}{24} - \frac{15}{24} = -\frac{16 + 15}{24} = -\frac{31}{24}\).
Remember, only the numerators are subtracted while the denominator remains the same. This is because both fractions are now parts of the same-sized whole.
negative fractions
Negative fractions can be tricky, but the concept is straightforward.
A negative fraction simply means the value is below zero.
When subtracting negative fractions, you follow the same rules as with positive fractions.
In our example, we dealt with \(-\frac{2}{3}\) and \(-\frac{5}{8}\).
After finding the common denominator and adjusting the numerators, we subtract: \(-\frac{16}{24} - \frac{15}{24} = -\frac{31}{24}\).
The result is a negative fraction, representing a value less than zero.

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