Which fractions are greater than $\frac{1}{2}$ but less than 1?

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02-11-2026
Mathematics

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Which fractions are greater than $\frac{1}{2}$ but less than 1?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T07:27:08-05:00

To identify a fraction $\frac{a}{b}$ between $\frac{1}{2}$ and $1$, ensure it satisfies $\frac{1}{2} < \frac{a}{b} < 1$.
Convert the fraction to a decimal and check if it is between $0.5$ and $1$.
Alternatively, check if the numerator is greater than half the denominator ($2a > b$) while remaining less than the denominator ($a < b$).
For example, $\frac{2}{3} \approx 0.67$ and $\frac{4}{5} = 0.8$ both satisfy these conditions.

Explanation

Understanding the Condition for Fractions

To identify fractions that are greater than $\frac{1}{2}$ but less than $1$, we look for fractions $\frac{a}{b}$ (where $a$ and $b$ are positive integers and $a < b$) that satisfy the compound inequality:
\[\frac{1}{2} < \frac{a}{b} < 1\]
This inequality defines a specific range of values between $0.5$ and $1.0$. Any fraction whose decimal equivalent falls strictly within this interval qualifies.

Methodology for Verification

To determine if a specific fraction $\frac{a}{b}$ fits this criteria, we can use two primary methods:

Decimal conversion: Convert the fraction to a decimal by dividing the numerator by the denominator ($a \div b$). If the result is $> 0.5$ and $< 1$, the fraction is in the target range.
Cross-multiplication: We can compare the fraction to the bounds separately:

For $\frac{a}{b} > \frac{1}{2}$: Cross-multiplying gives $2a > b$.
For $\frac{a}{b} < 1$: This is naturally true whenever $a < b$.

Combining these, a fraction $\frac{a}{b}$ is between $\frac{1}{2}$ and $1$ if and only if $b < 2a < 2b$.

Testing and Identifying Examples

Let us apply this to some examples to see which ones fit:

Consider $\frac{2}{3}$: Here, $a=2, b=3$. We check $b < 2a < 2b$, which is $3 < 4 < 6$. This is true, so $\frac{2}{3}$ is between $\frac{1}{2}$ and $1$.
Consider $\frac{3}{8}$: Here, $a=3, b=8$. We check $8 < 6 < 16$. This is false because $8$ is not less than $6$.
Consider $\frac{4}{5}$: Here, $a=4, b=5$. We check $5 < 8 < 10$. This is true, so $\frac{4}{5}$ is between $\frac{1}{2}$ and $1$.

By using these criteria, you can test any set of fractions provided in a worksheet or list.

Conclusion

In summary, to find the fractions that are greater than $\frac{1}{2}$ but less than $1$, confirm that for any fraction $\frac{a}{b}$, the numerator is more than half the denominator ($a > \frac{b}{2}$) but still smaller than the denominator itself ($a < b$). These are the fractions that belong in your list.

Examples

When you are dividing a pizza or a cake among friends, identifying fractions helps ensure everyone gets a fair portion. For instance, if you have a group and decide that each person should receive more than half but less than a whole piece of the pizza to stay satisfied without overeating, you are effectively choosing portions $\frac{a}{b}$ such that $\frac{1}{2} < \frac{a}{b} < 1$. This skill of comparing fractions is essential in cooking, budgeting, and project management where resources must be allocated proportionally.