Question

Order each set of fractions from least to greatest. $$\frac{3}{4}, \frac{3}{3}, \frac{3}{8}, \frac{3}{5}, \frac{3}{16}, \frac{3}{7}, \frac{3}{1}$$

Short answer

\frac{3}{16}, \frac{3}{8}, \frac{3}{7}, \frac{3}{5}, \frac{3}{4}, 1, 3

Step-by-step solution

- List the Fractions

List all the fractions provided: \(\frac{3}{4}, \frac{3}{3}, \frac{3}{8}, \frac{3}{5}, \frac{3}{16}, \frac{3}{7}, \frac{3}{1}\)

- Simplify where Possible

Simplify fractions that are not in their simplest form: \(\frac{3}{3} = 1\) and \(\frac{3}{1} = 3\)

- Compare Fractions with the Same Numerator

When fractions have the same numerator, the fraction with the larger denominator is smaller. So, compare: \(\frac{3}{16}, \frac{3}{8}, \frac{3}{7}, \frac{3}{5}, \frac{3}{4}, 1 (or \frac{3}{3}), 3 (or \frac{3}{1})\)

- Order the Fractions

Arrange the fractions in increasing order based on their denominators: \(\frac{3}{16}, \frac{3}{8}, \frac{3}{7}, \frac{3}{5}, \frac{3}{4}, 1, 3\)

Key concepts

Comparing Fractions

Comparing fractions can be tricky, but there's a simple rule you can follow when fractions have the same numerator (top number). The fraction with the smaller denominator (bottom number) is actually larger. This might seem counterintuitive, but it makes sense when you think of dividing something into fewer pieces. For example, \(\frac{3}{4}\) means 3 parts out of 4, while \(\frac{3}{8}\) is 3 parts out of 8. Since each piece is bigger when there are only 4 parts, \(\frac{3}{4}\) is larger than \(\frac{3}{8}\).

In the given exercise, all fractions have the same numerator of 3. So we can directly use the denominators to place them in order: \(\frac{3}{16}\), \(\frac{3}{8}\), \(\frac{3}{7}\), \(\frac{3}{5}\), \(\frac{3}{4}\), 1 (or \(\frac{3}{3}\)), and 3 (or \(\frac{3}{1}\)).

Now you can see, from smallest to largest, the pattern makes the fractions easier to compare based on the size of the denominators.

Simplifying Fractions

Simplifying fractions makes them easier to work with by reducing them to their simplest form. This means dividing the numerator and denominator by their greatest common divisor. For example: \(\frac{3}{3}\) simplifies to 1, and \(\frac{3}{1}\) simplifies to 3.

Here's how to simplify a fraction:

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

After simplifying, the fractions become easier to handle and compare. In the exercise example:

• \(\frac{3}{16}\) remains the same as it can't be simplified further using whole numbers.
• \(\frac{3}{3} = 1\)
• \(\frac{3}{1} = 3\)

Simplification is a crucial step to ensure fractions are in their simplest and most comparable form before proceeding to further steps like ordering or addition.

Numerators and Denominators

Understanding numerators and denominators is fundamental to working with fractions. The numerator is the top number and represents how many parts you have. The denominator is the bottom number and represents how many parts the whole is divided into.

For example, in the fraction \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator. This means you have 3 parts out of a total of 4.

The relationship between the numerator and denominator determines the value of the fraction. Here are some key points:

• When the numerators are the same, the fraction with the larger denominator is smaller.
• If the denominators are the same, the fraction with the larger numerator is larger.
• A fraction with a numerator equal to its denominator (like \(\frac{3}{3}\)) is always equal to 1.
• If the denominator is 1, the fraction (like \(\frac{3}{1}\)) equals the numerator.

By mastering numerators and denominators, you can much more easily compare, add, subtract, multiply, and divide fractions. These fundamentals are essential tools in your math toolkit.

So next time you see a fraction, remember to check out its numerator and denominator to get a grasp on its size and value.