Question

Tell whether each fraction is closest to \(0, \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4},\) or \(1 .\) Explain how you decided. $$\frac{9}{10}$$

Short answer

Closest to 1. \(\frac{9}{10}\) is very close to 1.

Step-by-step solution

- Analyze the Fraction

Given the fraction \(\frac{9}{10}\). Determine where \(\frac{9}{10}\) falls between the provided benchmarks: \(\frac{0}{1}, \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \text{and} \frac{1}{1}\).

- Convert Benchmarks to Decimals

Convert the benchmarks to their decimal forms for easier comparison: \(\frac{1}{4} \approx 0.25\), \(\frac{1}{3} \approx 0.33\), \(\frac{1}{2} \approx 0.5\), \(\frac{2}{3} \approx 0.67\), \(\frac{3}{4} \approx 0.75\), and \(\frac{1}{1} = 1\).

- Compare Fraction with Benchmarks

Convert \(\frac{9}{10}\) to its decimal form: \(\frac{9}{10} = 0.9\). Compare \(\frac{9}{10}\) to the decimal values of the benchmarks. It is closest to \(\frac{1}{1} \text{or} 1\), because \(\frac{9}{10} = 0.9\) is very close to 1.

Key concepts

Benchmark Fractions

Benchmark fractions are common fractions that we often use to estimate and compare other fractions. These benchmarks include fractions like \(\frac{1}{4}\), \(\frac{1}{3}\), \(\frac{1}{2}\), \(\frac{2}{3}\), and \(\frac{3}{4}\). Using these fractions helps us quickly understand where a given fraction stands in relation to these well-known values.

To use benchmark fractions, you first identify a reasonable range in which the given fraction falls. Then, you select the benchmark fraction that is nearest to the given fraction. This method simplifies the comparison process and provides a quick way to gauge the size of the fraction.

Decimal Conversion

Converting fractions to decimals can make the process of comparing them easier. This is because decimals can be compared directly and quickly by their size. To convert a fraction to a decimal, divide the numerator (top number) by the denominator (bottom number).

For example, let's convert the fraction \(\frac{9}{10}\) to a decimal:
\(\frac{9}{10} = 0.9\).

When you have a list of benchmark fractions, convert them to decimals as well:

• \(\frac{1}{4} \approx 0.25\)
• \(\frac{1}{3} \approx 0.33\)
• \(\frac{1}{2} = 0.5\)
• \(\frac{2}{3} \approx 0.67\)
• \(\frac{3}{4} \approx 0.75\)
• \(\frac{1}{1} = 1\)

Now, it's easier to see how \(\frac{9}{10}\), which is 0.9 as a decimal, compares to these benchmarks.

Comparing Fractions

Comparing fractions involves determining which fraction is larger or smaller, or closest to a given value. This can be done using benchmark fractions and decimal conversion. After converting everything to decimals, you can directly compare the numbers:
\(\frac{9}{10} = 0.9\) is very close to 1. In this specific comparison task, you look at each benchmark fraction and see how close the given fraction is.

Steps to compare fractions:

• Identify the benchmark fractions.
• Convert the given fraction and benchmarks to decimals.
• Compare the decimal values to find the closest match.

By following these steps, you can confidently conclude that \(\frac{9}{10}\), or 0.9, is closer to 1 than any of the other benchmarks like \(\frac{3}{4} = 0.75\) or \(\frac{2}{3} \approx 0.67\). By mastering fraction comparison, you can better understand where fractions fall on the number line and use them effectively in various mathematical contexts.