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Chapter 3: Problem 57

Simplify. Do not use negative exponents in the answer. $$ -\frac{1}{2}-\frac{3}{10} $$

Short Answer

Expert verified
\(-\frac{4}{5}\)

Step by step solution

01

- Find a common denominator

Identify the least common multiple (LCM) of the denominators 2 and 10. The LCM of 2 and 10 is 10.
02

- Rewrite fractions with a common denominator

Convert \(-\frac{1}{2}\) to\(-\frac{5}{10}\), because \(-\frac{1}{2} * \frac{5}{5} = -\frac{5}{10}\).\(-\frac{3}{10}\) already has the denominator of 10.
03

- Combine the fractions

Add the two fractions: \(-\frac{5}{10} - \frac{3}{10} = -\frac{8}{10}\).
04

- Simplify the fraction

Simplify \(-\frac{8}{10}\) by dividing the numerator and the denominator by their greatest common divisor, which is 2: \(-\frac{8}{10} = -\frac{4}{5}\).

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Key Concepts

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

Least Common Multiple
When working with fractions, finding a common denominator is crucial. The first step in simplifying a fraction problem often involves identifying the least common multiple (LCM) of the denominators.
The LCM is the smallest number that both denominators can divide into without leaving a remainder. For example, in our exercise, the denominators are 2 and 10. The smallest number both 2 and 10 can divide into evenly is 10. Thus, the LCM of 2 and 10 is 10.
Finding the LCM helps us rewrite fractions with a common denominator, allowing for easier addition or subtraction.
Rewriting Fractions
Once you have the LCM, the next step is to rewrite each fraction to have this common denominator. This involves adjusting the numerators accordingly.
For instance, let鈥檚 look at \[-\frac{1}{2}\]. To rewrite it with the denominator 10, we multiply both the numerator and the denominator by 5. This gives us \-\frac{1 x 5}{2 x 5} = -\frac{5}{10}\.
Meanwhile, the fraction \[-\frac{3}{10}\] already has the denominator 10, so it remains unchanged. Now both fractions share the same denominator, making them ready for the next step.
Combining Fractions
With both fractions rewritten to share the same denominator, combining them becomes straightforward.
When adding or subtracting fractions, only the numerators are combined while the denominator remains the same. In our example, we combine \-\frac{5}{10}\ and \[-\frac{3}{10}\].
Adding these, we get:
\[-\frac{5}{10} - \frac{3}{10} = -\frac{8}{10}\].
Now, we have a single fraction that can be simplified.
Simplifying Fractions
The final step in our problem is to simplify the resultant fraction. Simplification involves reducing the fraction to its smallest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).
In our example, the fraction is \[-\frac{8}{10}\]. The GCD of 8 and 10 is 2.
So, we divide both 8 and 10 by 2 to get:
\-\frac{8 \div 2}{10 \div 2} = -\frac{4}{5}\.
The simplified form of the fraction is \[-\frac{4}{5}\]. And this is our final answer.
Simplifying fractions makes them easier to understand and work with in further calculations.

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