Problem 66 - Intermediate Algebra

Question

Simplify each numerical expression. $$ -\frac{4}{5}-\frac{1}{2}\left(-\frac{3}{5}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{11}{10}\).

Step 1: Handle Parentheses

Start by solving the expression inside the parentheses. Multiply \(-\frac{1}{2}\) by \(\left(-\frac{3}{5}\right)\) which results in a positive product. Recall the rule that multiplying two negative numbers yields a positive result: \(-\frac{1}{2} \times -\frac{3}{5} = \frac{3}{10}\).

Step 2: Combine Fractions

Now substitute back into the expression: \[-\frac{4}{5} - \frac{3}{10}\]To add or subtract fractions, they must have a common denominator. The least common multiple of 5 and 10 is 10, so convert \(-\frac{4}{5}\) to a fraction with the denominator 10: \[-\frac{4}{5} = -\frac{8}{10}\].

Step 3: Perform Subtraction

With a common denominator, subtract the fractions: \[-\frac{8}{10} - \frac{3}{10} = -\left(\frac{8 + 3}{10}\right) = -\frac{11}{10}\].

Key Concepts

Fractions

Fractions represent parts of a whole. They consist of a numerator and a denominator. The numerator is the top number and shows how many parts you have, while the denominator is the bottom number and indicates into how many parts the whole is divided.

• For example, in the fraction \( \frac{4}{5} \), 4 is the numerator, and 5 is the denominator.
• Understanding fractions is key to solving many math problems, such as the one in our example.

Fractions can be added, subtracted, multiplied, or divided, but certain rules apply. To add or subtract fractions, you need a common denominator. For multiplication, you multiply the numerators together and the denominators together.
Subtraction, as in our example, involves changing everything into a common denominator first, before performing the arithmetic.

Multiplying Negatives

One important rule in mathematics is about multiplying negative numbers. The rule is: a negative number multiplied by another negative number results in a positive number.

• This might sound tricky at first, but it's a fundamental concept.
• In the exercise, multiplying \(-\frac{1}{2}\) by \(-\frac{3}{5}\) results in the positive \(\frac{3}{10}\).

Remember this rule:

• Negative times negative equals positive.
• Positive times negative equals negative.

Try visualizing it as flipping directions. A negative direction flipped twice (multiply by two negatives) gets you back on track in the positive direction again!

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions. It ensures that you are dealing with the same units or divisions of whole numbers.

• To combine \(-\frac{4}{5}\) and \(\frac{3}{10}\), we first need the denominators to be the same.
• The rule is called finding a "common denominator," which is usually the least common multiple of the original denominators.

In our example, we changed \(-\frac{4}{5}\) into \(-\frac{8}{10}\) since 10 is the least common multiple of 5 and 10.

• This allows you to perform the operation: \(-\frac{8}{10} - \frac{3}{10}\) = \(-\frac{11}{10}\).

Remember: always convert fractions so that the denominators match before adding or subtracting! This step ensures you correctly manage parts of the whole.