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Chapter 5: Problem 119

Simplify. $$ -\frac{3}{20}-\frac{1}{8} $$

Short Answer

Expert verified
-\frac{11}{40}

Step by step solution

01

Find a common denominator

To subtract the fractions \(-\frac{3}{20}-\frac{1}{8}\), we first need to find a common denominator. The least common multiple (LCM) of 20 and 8 is 40.
02

Convert fractions to have the same denominator

Rewrite each fraction with the common denominator of 40. \(-\frac{3}{20} = -\frac{3 \times 2}{20 \times 2} = -\frac{6}{40}\) and \(-\frac{1}{8} = -\frac{1 \times 5}{8 \times 5} = -\frac{5}{40}\).
03

Subtract the fractions

Now subtract the fractions: \(-\frac{6}{40} - \frac{5}{40}= -\frac{6 + 5}{40} = -\frac{11}{40}\).

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Key Concepts

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

least common multiple
When subtracting fractions with different denominators, the first step is to find the least common multiple (LCM) of the denominators. The LCM is the smallest number that is a multiple of both denominators.
For example, in the exercise \(-\frac{3}{20}-\frac{1}{8}\), we need to find the LCM of 20 and 8.
Think about the multiples of 20: 20, 40, 60, 80, etc. Then think about the multiples of 8: 8, 16, 24, 32, 40, etc.
The smallest common multiple is 40.
This means that 40 is the LCM of 20 and 8.
common denominator
Once we have the least common multiple (LCM), we use it as our common denominator.
This is essential for simplifying fraction subtraction.
In our example, the LCM of 20 and 8 is 40.
We convert each fraction so that they both have 40 as the denominator.

\(-\frac{3}{20}\) becomes \(-\frac{6}{40}\) because \(-\frac{3 \times 2}{20 \times 2} = -\frac{6}{40}\).
Similarly, \(-\frac{1}{8}\) becomes \(-\frac{5}{40}\) because \(-\frac{1 \times 5}{8 \times 5} = -\frac{5}{40}\).
simplifying fractions
After rewriting the fractions with a common denominator, we subtract them.
The numerator of the resulting fraction needs to be simplified if possible.
In the example, \(-\frac{6}{40} - \frac{5}{40} = -\frac{11}{40}\).
Notice that the common denominator stays the same, and you just subtract the numerators.
Since \(-\frac{11}{40}\) cannot be simplified further (11 is a prime number and does not divide by the denominator 40 in any simpler form), we leave it as is.
negative fractions
Dealing with negative fractions can be tricky, but it's straightforward if you remember a few rules.
A negative sign in front of a fraction means the entire value of the fraction is negative.
So in subtraction, \-\frac{3}{20}\ and \-\frac{1}{8}\ both stay negative.
When subtracting two negative fractions, you're effectively adding their absolute values and keeping the negative sign.
Using our example,\(-\frac{6}{40} - \frac{5}{40}\) translates to \(-\frac{6+5}{40} = -\frac{11}{40}\).
Regardless of whether the fractions are positive or negative, following the same rules will yield the correct result.

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