Problem 70 - Elementary Algebra

Question

\(-\frac{5}{6}\left(15 b-\frac{1}{4}\right)\)

Step-by-Step Solution

Answer

-12.5b + \frac{5}{24}

Step 1 - Distribute the Fraction

Distribute \(-\frac{5}{6}\) to each term inside the parentheses. This means multiplying \(-\frac{5}{6}\) by \(15b\) and \(-\frac{5}{6}\) by \(-\frac{1}{4}\).

Step 2 - Multiply the First Term

Multiply \(-\frac{5}{6}\) by \15b\: \[ -\frac{5}{6} \times 15b = -\frac{5 \times 15}{6}b = -\frac{75}{6}b = -12.5b \]

Step 3 - Multiply the Second Term

Multiply \(-\frac{5}{6}\) by \(-\frac{1}{4}\): \[ -\frac{5}{6} \times -\frac{1}{4} = \frac{5}{24} \]

Step 4 - Combine Results

Combine the results from Steps 2 and 3: \[ -12.5b + \frac{5}{24} \]

Key Concepts

Multiplying Fractions

Understanding how to multiply fractions is a key algebra skill. When you multiply fractions, you multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. For example, in the exercise, we had to multiply \(-\frac{5}{6}\) by \(-\frac{1}{4}\). The process looks like this:

- Numerator: -5 times -1
- Denominator: 6 times 4

This gives us a fraction of \(\frac{5}{24}\).

Always remember to simplify the fraction if possible, although in this case \(\frac{5}{24}\) is already in its simplest form. Negative times negative results in a positive number, so keep an eye on the signs when multiplying fractions.

Combining Terms

Once you've distributed and multiplied the fractions, the next step is to combine the results. This typically involves simplifying or combining like terms. Like terms are terms that have the same variables raised to the same power. In the exercise, after distributing and multiplying, we had \(-12.5b\) and \(\frac{5}{24}\).

Here are some tips for combining terms:

• Make sure to combine only the terms that have the same variable (or no variable).
• Simplify fractions and keep an eye out for common denominators.

So, combining the terms from our problem, we get: \(-12.5b + \frac{5}{24}\). This is the simplest form and shows how distribution and multiplication of fractions lead to the combination of terms.

Negative Numbers

Managing negative numbers is essential in algebra. Whether you're multiplying, dividing, adding, or subtracting, the sign of the number matters significantly. In the exercise, we had to distribute \(-\frac{5}{6}\) to \(15b\) and \(-\frac{1}{4}\). Here’s how you can handle negative numbers:

- When multiplying or dividing two negative numbers, the result is a positive number.
- When multiplying or dividing a negative number by a positive number, the result is a negative number.


In our exercise, multiplying \(-\frac{5}{6}\) by \(15b\) gave us \(-12.5b\), and multiplying two negatives, \(-\frac{5}{6} \times -\frac{1}{4}\), gave us \(\frac{5}{24}\). Keeping track of the signs helps ensure accuracy in your final answer, which is crucial in algebra.