Question

Order each set of fractions from least to greatest. $$\frac{7}{7}, \frac{3}{4}, \frac{1}{2}, \frac{2}{5}, \frac{2}{3}, \frac{11}{8}, \frac{1}{3}$$

Short answer

\[ \frac{1}{3}, \frac{2}{5}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{7}{7}, \frac{11}{8} \]

Step-by-step solution

Convert fractions to decimals

To compare fractions, convert them to decimals. \[ \frac{7}{7} = 7 \text{ divided by } 7 = 1 \]\[ \frac{3}{4} = 3 \text{ divided by } 4 = 0.75 \]\[ \frac{1}{2} = 1 \text{ divided by } 2 = 0.5 \]\[ \frac{2}{5} = 2 \text{ divided by } 5 = 0.4 \]\[ \frac{2}{3} = 2 \text{ divided by } 3 \thickapprox 0.67 \]\[ \frac{11}{8} = 11 \text{ divided by } 8 \thickapprox 1.375 \]\[ \frac{1}{3} = 1 \text{ divided by } 3 \thickapprox 0.33 \]

List the decimal values

List the decimal equivalents of the fractions: \[ 1, 0.75, 0.5, 0.4, 0.67, 1.375, 0.33 \]

Order the decimal values from least to greatest

Order the decimals from least to greatest: \[ 0.33, 0.4, 0.5, 0.67, 0.75, 1, 1.375 \]

Map decimals back to their corresponding fractions

Match each decimal back to its fraction: \[ 0.33 \rightarrow \frac{1}{3}, 0.4 \rightarrow \frac{2}{5}, 0.5 \rightarrow \frac{1}{2}, 0.67 \rightarrow \frac{2}{3}, 0.75 \rightarrow \frac{3}{4}, 1 \rightarrow \frac{7}{7}, 1.375 \rightarrow \frac{11}{8} \]

Write the fractions in order

Order the fractions from least to greatest: \[ \frac{1}{3}, \frac{2}{5}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{7}{7}, \frac{11}{8} \]

Key concepts

Fractions

Fractions are a way to represent parts of a whole. Each fraction consists of a numerator (the top number) and a denominator (the bottom number). The numerator represents how many parts you have, while the denominator represents how many equal parts the whole is divided into. For example, in \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator, meaning you have 3 parts out of a total of 4 parts.

Fractions can represent values less than 1 (proper fractions like \(\frac{1}{2}\)) or values greater than 1 (improper fractions like \(\frac{11}{8}\)). Understanding fractions are essential for comparing their sizes easily and converting them into other forms like decimals.

Decimal Conversion

Decimal conversion is handy for comparing fractions. By converting fractions to decimals, we can use a common numerical form to easily identify which fractions are larger or smaller.

To convert a fraction to a decimal, simply divide the numerator by the denominator. For example, \(\frac{3}{4}\) converts to 0.75 because 3 divided by 4 equals 0.75. This allows us to list fractions by their decimal values and order them from least to greatest.

Let's look at another example: \(\frac{1}{3}\) converts to approximately 0.33 because 1 divided by 3 is approximately 0.33. With decimals, it's easier to see that 0.33 (\(\frac{1}{3}\)) is less than 0.4 (\(\frac{2}{5}\)). Converting fractions to decimals simplifies the comparison process.

Ordering Numbers

Ordering numbers from least to greatest helps us understand the relative size of each number. Once we have the decimal forms of fractions, we can easily arrange them in ascending order.

For example, let's list these fractions converted to decimals: \(\frac{1}{3}\) is 0.33, \(\frac{2}{5}\) is 0.4, \(\frac{1}{2}\) is 0.5, \(\frac{2}{3}\) is approximately 0.67, \(\frac{3}{4}\) is 0.75, \(\frac{7}{7}\) is 1, and \(\frac{11}{8}\) is approximately 1.375.

Arrange these decimals from least to greatest: 0.33, 0.4, 0.5, 0.67, 0.75, 1, 1.375. Once ordered, we can map these decimals back to their corresponding fractions: \(\frac{1}{3}\) (0.33), \(\frac{2}{5}\) (0.4), \(\frac{1}{2}\) (0.5), \(\frac{2}{3}\) (0.67), \(\frac{3}{4}\) (0.75), \(\frac{7}{7}\) (1), and \(\frac{11}{8}\) (1.375). This method is systematic and ensures that the fractions are ordered correctly from least to greatest.