Problem 14 - Basic College Mathematics

Question

Divide and simplify. \(\frac{10}{9} \div \frac{1}{3}\)

Step-by-Step Solution

Answer

\frac{10}{3}\.

Step 1 - Understand the Problem

We need to divide the fraction \(\frac{10}{9}\) by \(\frac{1}{3}\).

Step 2 - Reciprocal of the Divisor

To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac{1}{3}\) is \(\frac{3}{1}\).

Step 3 - Multiply the Fractions

Multiply \(\frac{10}{9}\) by \(\frac{3}{1}\). This gives us: \(\frac{10}{9} \times \frac{3}{1} = \frac{10 \times 3}{9 \times 1} = \frac{30}{9}\).

Step 4 - Simplify the Result

Simplify \(\frac{30}{9}\). Both the numerator and the denominator can be divided by their greatest common divisor, which is 3. Thus, \(\frac{30}{9} = \frac{30 \div 3}{9 \div 3} = \frac{10}{3}\).

Key Concepts

reciprocal of a fraction

When you need to divide fractions, you often hear about the 'reciprocal' of a fraction. The reciprocal of a fraction is created by swapping the numerator (top number) and the denominator (bottom number). For example, the reciprocal of \(\frac{1}{3}\) is \(\frac{3}{1}\).
Think of it like flipping the fraction.
Why is this useful? When we divide by a fraction, it's the same as multiplying by its reciprocal. For example, dividing by \(\frac{1}{3}\) is the same as multiplying by \(\frac{3}{1}\).
So in our problem, \(\frac{10}{9} \div \frac{1}{3}\) becomes \(\frac{10}{9} \times \frac{3}{1}\). If you remember this trick, division of fractions becomes much easier!

simplifying fractions

Simplifying fractions makes them easier to work with and understand.
Here's how you simplify a fraction:

• First, find the greatest common divisor (GCD) of the numerator (top number) and the denominator (bottom number).
• Next, divide both the numerator and the denominator by the GCD.

For example, in our problem, we had to simplify \(\frac{30}{9}\).
First, find the GCD of 30 and 9. Both numbers are divisible by 3, so the GCD is 3.
Now, divide both 30 and 9 by 3.
This gives us: \(\frac{30 \div 3}{9 \div 3} = \frac{10}{3}\).
Simplified fractions are clearer and often more manageable, especially when working with more complex problems.

multiplying fractions

Multiplying fractions is straightforward and often simpler than other fraction operations.
To multiply fractions, follow these steps:

• Multiply the numerators (top numbers) of each fraction to get the new numerator.
• Multiply the denominators (bottom numbers) of each fraction to get the new denominator.

In our example, we had \(\frac{10}{9}\) and \(\frac{3}{1}\).
To multiply them, we did: \(\frac{10 \times 3}{9 \times 1} = \frac{30}{9}\).
After multiplying, sometimes you need to simplify the fraction to its simplest form, as we did earlier.
This final step ensures the fraction is as clear and concise as possible.
Remember:

• Always perform multiplication across the numerators and denominators.
• Don't forget to simplify if possible.

With practice, these steps become second nature!