Problem 1 - Basic College Mathematics

Question

Find another name for the given number, but with the denominator indicated. Use multiplying by 1 . $$ \frac{1}{2}=\frac{?}{10} $$

Step-by-Step Solution

Answer

\frac{1}{2}=\frac{5}{10}

Step 1 - Understanding the Problem

The goal is to find an equivalent fraction to \(\frac{1}{2}\) that has a denominator of 10.

Step 2 - Write the Fraction Multiplication Form

We will multiply \(\frac{1}{2}\) by \(\frac{5}{5}\) (which is equivalent to 1, hence not changing the value of the fraction). This is because our target denominator is 10, and \(\frac{2 \times 5 = 10}\).

Step 3 - Perform the Multiplication

Multiply the numerator and the denominator by 5: \(\frac{1 \times 5}{2 \times 5} = \frac{5}{10}\)

Step 4 - Verify the Result

Verify that \(\frac{5}{10}\) is equivalent to \(\frac{1}{2}\). Indeed, \(\frac{5}{10}\) simplifies back to \(\frac{1}{2}\).

Key Concepts

fraction multiplication

To start with fraction multiplication, remember this simple idea: when you multiply a fraction by 1, you don’t change its value. Now, 1 can look different in fractions. For example, \(\frac{5}{5}\) is the same as 1. Multiplying \(\frac{1}{2}\) by \(\frac{5}{5}\) helps us convert it to \(\frac{5}{10}\). To do this, multiply the numerator (top number) by 5, and the denominator (bottom number) by 5. Here’s how it looks: \[ \frac{1}{2} \times \frac{5}{5} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10} \]. By multiplying both parts of the fraction by the same number, we get a fraction with the same value but a different form.

denominator

The denominator is the bottom number in a fraction. It tells you how many equal parts make up a whole. In our example, we wanted to change the fraction \(\frac{1}{2}\) to have a denominator of 10. So, what we did was find a number to multiply by that would convert 2 into 10. We chose 5 because \(\frac{2 \times 5 = 10}\). Changing the denominator helps to compare, add or subtract fractions but remember, whatever you do to the denominator, you must do to the numerator too!

fractions simplification

Simplifying fractions means making them as simple as possible. For \(\frac{5}{10}\), we noticed it can be reduced to \(\frac{1}{2}\). You can simplify a fraction by finding the greatest common divisor (GCD) of both the numerator and the denominator and dividing them both by that number. For \(\frac{5}{10}\), the GCD is 5: \[ \frac{5 \text{ (numerator)}}{10 \text{ (denominator)}} \] Divide both by 5: \[ \frac{5 \text{ (numerator)}/5}{10 \text{ (denominator)}/5} = \frac{1}{2} \] Simplification ensures that fractions are as straightforward as possible, making calculations easier to handle.