Problem 58 - Algebra A Combined Function

Question

Perform the indicated operation. \(-\frac{5}{6}\left(-\frac{3}{10}\right)\)

Step-by-Step Solution

Answer

\(\frac{1}{4}\)

Step 1: Remove Negative Signs

First, observe that there are two negative signs in the expression \(-\frac{5}{6}\left(-\frac{3}{10}\right)\). According to the rule of multiplying integers, two negative signs cancel each other out. This means the operation becomes:\[\frac{5}{6} \times \frac{3}{10}\]

Step 2: Multiply the Numerators

Multiply the numerators of the two fractions together. The numerators are 5 and 3:\[5 \times 3 = 15\]

Step 3: Multiply the Denominators

Now, multiply the denominators of the two fractions. The denominators are 6 and 10:\[6 \times 10 = 60\]

Step 4: Form the Product of Fractions

Combine the results from Steps 2 and 3 to form the fraction:\[\frac{15}{60}\]

Step 5: Simplify the Fraction

Simplify the fraction \(\frac{15}{60}\) by finding the greatest common divisor (GCD) of 15 and 60. The GCD of 15 and 60 is 15.Divide both the numerator and denominator by 15:\[\frac{15 \div 15}{60 \div 15} = \frac{1}{4}\]

Step 6: Final Answer

After simplifying, the result of the operation is \(\frac{1}{4}\).

Key Concepts

Negative Numbers in Fractions

Fractions can sometimes have negative signs, either in the numerator, denominator, or both. When multiplying fractions with negative numbers, you apply a simple rule: two negative signs cancel each other out.
This means if you have a fraction multiplied by another negative fraction, like -\(\frac{5}{6}\left(-\frac{3}{10}\right)\), the negative signs "cancel," resulting in a positive product.
This is similar to integers, where -(-7) = 7. Therefore, while handling negative numbers, always check pairs; if both fractions are negative, the final product will be positive.
This important rule simplifies the process significantly, allowing the multiplication of the absolute values instead.

Simplifying Fractions

Simplifying fractions means reducing them to their smallest form, where the numerator and denominator have only 1 as their common divisor.
The simplified fraction is the most easy-to-understand and work-with version of the original.

• After multiplication, like in -\(\frac{15}{60}\), look for common factors.
• Use division to reduce the fraction.

If both numbers divide evenly by a number other than 1, it can be simplified further.
Keep dividing until the only common factor between the numerator and denominator is 1. A fully simplified fraction is concise, often making it clear in contexts like comparing sizes or performing further mathematical operations.

Greatest Common Divisor

Finding the greatest common divisor (GCD) is a crucial step in simplifying fractions.
The GCD of two numbers is the largest number that can divide both without leaving a remainder.

• For example, to simplify \(\frac{15}{60}\), you find the GCD of 15 and 60.
• The GCD here is 15, since both 15 and 60 can divide by 15 evenly.

Once you find the GCD, divide both the numerator and the denominator by this number to simplify the fraction.
This step is fundamental, resulting in fractions that are easier to work with and clearer in presenting data or results.

Multiplying Numerators and Denominators

Multiply the numerators and denominators separately when working with fractions.
This straightforward step helps in managing the multiplication of fractions seamlessly.

• Multiply the numerators of both fractions to get the new numerator.
• Similarly, multiply the denominators to get the new denominator.

In our example, multiplying \(5 \times 3\) gives 15 and \(6 \times 10\) gives 60.
This results in the fraction \(\frac{15}{60}\).
Keeping the multiplication of each part separate ensures clarity and reduces errors.
After obtaining these results, you can then proceed with simplifying to express the fraction in its simplest form. This method is consistent and reliable, suitable for all levels of fraction multiplication.