Problem 219 - Intermediate Algebra

Question

In the following exercises, simplify. $$ 1-\frac{3}{5} \div \frac{1}{10} $$

Step-by-Step Solution

Answer

The simplified answer is \( -5 \).

Step 1: Understand the Division of Fractions

Division of fractions is the same as multiplying by the reciprocal. So, \( \frac{3}{5} \div \frac{1}{10} \) should be converted to \( \frac{3}{5} \times \frac{10}{1} \).

Step 2: Multiply the Fractions

Now, multiply the numerators and denominators: \( \frac{3}{5} \times \frac{10}{1} = \frac{3 \times\ 10}{5 \times\ 1} = \frac{30}{5} \).

Step 3: Simplify the Fraction

Simplify \( \frac{30}{5} \) by dividing both the numerator and the denominator by their greatest common divisor: \( \frac{30}{5} = 6 \).

Step 4: Subtract from 1

Finally, subtract the result from 1: \( 1 - 6 = -5 \).

Key Concepts

Fraction Division

When we talk about dividing fractions, we use the concept of multiplying by the reciprocal. The reciprocal of a fraction is simply flipping the numerator and the denominator. For example, the reciprocal of \(\frac{1}{10}\) is \(\frac{10}{1}\).

So, when we divide \(\frac{3}{5} \) by \(\frac{1}{10}\), we actually multiply \(\frac{3}{5}\) by \(\frac{10}{1}\). Understanding this step is key to solving division problems involving fractions.

In mathematical terms, \(\frac{3}{5} \div \frac{1}{10} \) becomes \(\frac{3}{5} \times \frac{10}{1} \). This makes the calculation simpler.

Multiplication of Fractions

Multiplying fractions involves straightforward steps. You multiply the numerators together and the denominators together. Let's take our example: \(\frac{3}{5} \times \frac{10}{1}\).

First, multiply the numerators: 3 and 10: \(\frac{3 \times 10}{5 \times 1} = \frac{30}{5}.\)

Next, multiply the denominators: 5 and 1. This results in \(\frac{30}{5}\).

Remembering these steps makes the multiplication of fractions simple and quick.

Simplifying Fractions

Simplifying fractions is about reducing them to their simplest form. It involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

For example, \(\frac{30}{5}\) can be simplified because both 30 and 5 share a common factor: 5.

Divide both the numerator and the denominator by 5: \(\frac{30 \/ 5}{5 \/ 5} = 6\).

Now, \(\frac{30}{5}\) simplifies to 6. Simplifying fractions helps to make calculations easier and results more elegant.

Subtraction

Lastly, we need to perform the subtraction part of the problem. Once we have simplified the fraction division to a single number, we subtract it from 1.

In this exercise, our simplified result is 6. Therefore, we calculate: \( 1 - 6 = -5\).

Subtraction with fractions follows the same principles as whole numbers, as long as they are simplified and straightforward.

Understanding the basic operation of subtraction helps us to arrive at the correct final solution.