Comparing Numbers and Fractions: Finding the Largest Value

We are asked to identify the largest number from the given list: $\frac{1}{2}$, $\frac{2}{3}$, $\frac{4}{5}$, and $1$. To do this, we need to compare the values of these numbers.

Given Numbers:

• $\frac{1}{2}$
• $\frac{2}{3}$
• $\frac{4}{5}$
• $1$

There are several ways to compare numbers, especially when they include both fractions and whole numbers. A straightforward method is to convert all the numbers into a common format, such as decimals.

Step-by-Step Comparison using Decimals

Let's convert each number into its decimal equivalent:

• For $\frac{1}{2}$: Divide 1 by 2. $\frac{1}{2} = 1 \div 2 = 0.5$
• For $\frac{2}{3}$: Divide 2 by 3. $\frac{2}{3} = 2 \div 3 \approx 0.6667$ (This is a repeating decimal, so we can use an approximation for comparison)
• For $\frac{4}{5}$: Divide 4 by 5. $\frac{4}{5} = 4 \div 5 = 0.8$
• For $1$: A whole number 1 is simply $1.0$ in decimal form.

Comparing the Decimal Values

Now we have the decimal values for each number:

• $\frac{1}{2}$ is $0.5$
• $\frac{2}{3}$ is approximately $0.6667$
• $\frac{4}{5}$ is $0.8$
• $1$ is $1.0$

Let's compare these decimal values: $0.5$, $0.6667$, $0.8$, and $1.0$.

Arranging them from smallest to largest:

$0.5 < 0.6667 < 0.8 < 1.0$

Or, using the original numbers:

$\frac{1}{2} < \frac{2}{3} < \frac{4}{5} < 1$

From this comparison, it is clear that $1.0$ is the largest decimal value, which corresponds to the number $1$.

Conclusion

Comparing the decimal forms of all the given numbers, we found that $1$ is the largest value among $\frac{1}{2}$, $\frac{2}{3}$, $\frac{4}{5}$, and $1$.

Revision Table: Values and Decimals

Additional Information: Fraction Comparison Techniques

Understanding how to compare fractions is a fundamental math skill. Besides converting to decimals, here are other useful methods:

• Common Denominator Method: To compare fractions with different denominators, find a common denominator (a common multiple of the denominators). Convert each fraction to an equivalent fraction with this common denominator. Then, compare the numerators. The fraction with the larger numerator is greater. For example, to compare $\frac{2}{3}$ and $\frac{4}{5}$, the common denominator is 15. $\frac{2}{3} = \frac{10}{15}$ and $\frac{4}{5} = \frac{12}{15}$. Since $12 > 10$, $\frac{4}{5} > \frac{2}{3}$.
• Cross-Multiplication Method: To compare two fractions $\frac{a}{b}$ and $\frac{c}{d}$, compare the products $a \times d$ and $b \times c$. If $a \times d > b \times c$, then $\frac{a}{b} > \frac{c}{d}$. If $a \times d < b \times c$, then $\frac{a}{b} < \frac{c}{d}$. This method is quick for comparing two fractions at a time.
• Comparing to a Benchmark: Sometimes, comparing fractions to a simple benchmark like $\frac{1}{2}$ can help. For instance, $\frac{1}{2}$ is equal to $0.5$. $\frac{2}{3}$ is greater than $\frac{1}{2}$ because $2 \times 2 = 4$ and $3 \times 1 = 3$, and $4 > 3$. $\frac{4}{5}$ is also greater than $\frac{1}{2}$ because $4 \times 2 = 8$ and $5 \times 1 = 5$, and $8 > 5$.

When one of the numbers is a whole number greater than 1, and the fractions are proper fractions (numerator less than denominator), the whole number will generally be the largest, as proper fractions are always less than 1.