Problem 60 - Algebra A Combined Function

Question

Perform the indicated operation. \(-\frac{1}{10} \div\left(-\frac{8}{11}\right)\)

Step-by-Step Solution

Answer

\(\frac{11}{80}\)

Step 1: Recognize the division of fractions

The operation given is the division of two fractions: \(-\frac{1}{10} \div \left(-\frac{8}{11}\right)\).To divide fractions, we multiply by the reciprocal of the second fraction.

Step 2: Find the reciprocal of the divisor

The divisor here is \(-\frac{8}{11}\),so its reciprocal is \(-\frac{11}{8}\).Notice that the negative sign remains with the fraction after finding the reciprocal.

Step 3: Multiply the fractions

Replace the division with multiplication using the reciprocal found:\(-\frac{1}{10} \times -\frac{11}{8}\).When multiplying fractions, multiply the numerators and the denominators together.

Step 4: Perform the multiplication

Multiply the numerators:\(-1 \times -11 = 11\).Multiply the denominators:\(10 \times 8 = 80\).So, the product of the multiplication is:\(\frac{11}{80}\).

Step 5: Check for simplification

The fraction \(\frac{11}{80}\)is already in its simplest form because 11 is a prime number and does not divide evenly into 80.

Step 6: Determine the sign of the result

Because we multiplied two negative numbers, the result is positive. Therefore, the final answer is \(\frac{11}{80}\).

Key Concepts

Reciprocal

When we talk about the reciprocal of a fraction, we are referring to a simple yet powerful concept in mathematics. The reciprocal of a fraction is what you get when you flip the fraction upside down. This means switching the numerator (the top number) and the denominator (the bottom number). For example, the reciprocal of \(-\frac{8}{11}\) is \(-\frac{11}{8}\). It's important to note that if the original fraction is negative, the reciprocal will also be negative.

Reciprocals are crucial in division of fractions as they allow us to transform division into multiplication. Instead of directly dividing, we multiply by the reciprocal. This step simplifies the operation significantly and is the standard method for handling division of fractions.

Simplification

Simplification in the context of fractions means reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. In our exercise, we ended with the fraction \(\frac{11}{80}\).

To simplify a fraction, you look for common factors of the numerator and denominator and divide them accordingly. This process ensures that your answer is as straightforward as possible. In this case, since 11 is a prime number, it doesn't share any common factors with 80 apart from 1, hence \(\frac{11}{80}\) is already simplified.

Remember, simplifying is always a good habit because it makes fractions easier to understand and work with. Although not always necessary, it ensures the result is presented in a clean and universally recognized form.

Multiplying Fractions

Multiplying fractions might seem tricky at first, but it's really quite straightforward. When you multiply fractions, you simply multiply the numerators together and the denominators together. This was what we applied in the exercise, where we had to multiply \(-\frac{1}{10}\) by \(-\frac{11}{8}\).

Here’s the detailed breakdown:

• Multiply the numerators: \(-1 \times -11 = 11\). Multiplying two negative numbers results in a positive number.
• Multiply the denominators: \(10 \times 8 = 80\).

Thus the fraction resulting from this operation is \(\frac{11}{80}\).

Multiplying fractions is generally easier than adding or subtracting them because you don’t need a common denominator. It's an essential skill, as it frequently appears in solving algebraic problems and real-world applications. By mastering this process, you gain a great tool for handling diverse mathematical challenges.