Problem 209 - Intermediate Algebra

Question

In the following exercises, simplify. $$ -\frac{3}{8} \div\left(-\frac{3}{10}\right) $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{5}{4} \).

Step 1: Rewrite the division as multiplication

To simplify the expression \(-\frac{3}{8} \, \div \, (-\frac{3}{10})\), we can rewrite the division as multiplication by the reciprocal. Remember, dividing by a fraction is the same as multiplying by its reciprocal. So: \(-\frac{3}{8} \, \times \, -\frac{10}{3}\).

Step 2: Multiply the numerators

Multiply the numerators of the fractions. \(-3 \, \times \, -10 = 30\).

Step 3: Multiply the denominators

Multiply the denominators of the fractions. \(8 \, \times \, 3 = 24\).

Step 4: Simplify the result

Now we have: \( \frac{30}{24} \). To simplify it further, find the greatest common divisor (GCD) of 30 and 24, which is 6. Divide the numerator and the denominator by 6: \( \frac{30 \, \div \, 6}{24 \, \div \, 6} = \frac{5}{4} \).

Key Concepts

multiplication by reciprocals

To solve any fraction division problem, it's helpful to understand the concept of multiplication by reciprocals.
Let's break it down: in a fraction, the reciprocal is simply flipping the numerator (top number) and the denominator (bottom number).
For example, the reciprocal of \(-\frac{3}{10}\) is \- \frac{10}{3}\.
When you divide by a fraction, you're essentially multiplying by its reciprocal.
In the given exercise, dividing \(-\frac{3}{8} \div (-\frac{3}{10})\) becomes multiplication: \(-\frac{3}{8} \times -\frac{10}{3}\) .
Once you have written it as a multiplication problem, it becomes much simpler to solve.

simplifying fractions

Simplifying fractions is a critical skill in fraction arithmetic.
It involves reducing a fraction to its simplest form, where the numerator and the denominator share no common factors other than 1.
Here’s how you simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

In the exercise, we simplified \(\frac{30}{24}\) by identifying and then dividing by their GCD of 6:
\(\frac{30 ÷ 6}{24 ÷ 6} = \frac{5}{4}\).
This process helps in obtaining the fraction in its most basic form, making arithmetic easier and results clearer.

greatest common divisor

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder.
It's a key step in simplifying fractions.
Here's how to find the GCD:

• List the factors of both numbers.
• Identify the largest common factor they share.

For 30 and 24, the factors are:

• Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

The common factors are 1, 2, 3, and 6, with the largest being 6.
We then use 6 to simplify the fraction \(\frac{30}{24}\) by dividing both the numerator and the denominator by 6.

numerators and denominators

Understanding numerators and denominators is essential when working with fractions.
The numerator is the top part of a fraction, indicating how many parts we have.
The denominator is the bottom part of the fraction, showing the total number of equal parts into which the whole is divided.
In our exercise, \(-\frac{3}{8}\) has a numerator of -3 and a denominator of 8.
When we multiply fractions like \(-\frac{3}{8} \times -\frac{10}{3}\):

• We multiply the numerators: -3 \times -10 = 30
• We multiply the denominators: 8 \times 3 = 24

The resulting fraction \(\frac{30}{24}\) needs to be simplified, bringing us back to the previous concepts of simplifying fractions and finding the GCD.Always remember, careful handling of numerators and denominators ensures accurate arithmetic operations.