Question

In each pair, tell which fraction is closer to 0.5 \(\frac{3}{17}\) or \(\frac{16}{17}\)

Short answer

The fraction \(\frac{3}{17}\) is closer to 0.5.

Step-by-step solution

Understand the Problem

Determine which fraction in the pair is closer to 0.5.

Convert Fractions to Decimals

Convert each fraction to decimal form to compare their distance from 0.5. For \(\frac{3}{17}\): \[ \frac{3}{17} \approx 0.176 \] For \(\frac{16}{17}\): \[ \frac{16}{17} \approx 0.941 \]

Calculate Distance from 0.5

Find the absolute distance of each decimal from 0.5: For \(\frac{3}{17}\): \[ |0.176 - 0.5| = 0.324 \] For \(\frac{16}{17}\): \[ |0.941 - 0.5| = 0.441 \]

Compare Distances

Compare the distances calculated: \[ \frac{3}{17}\ is closer to 0.5 than \frac{16}{17}\ because \0.324\ is less than \0.441 \]

Key concepts

distance from a point

Understanding the distance from a point is crucial when comparing fractions to a specific value like 0.5. When we talk about the 'distance' of a number from another point, we're referring to how far apart these two numbers are on the number line. In this exercise, we need to measure which fraction is closer to 0.5.

To do that, we calculate the absolute value of the difference between each fraction and 0.5. The steps involve:

• Converting each fraction to its decimal form.
• Finding the absolute value of the difference between the decimal and 0.5.
• Comparing these differences to see which is smaller.

This method helps to visually and numerically determine which fraction is closer to the reference point—here, 0.5.

converting fractions to decimals

A key step in comparing fractions is converting them to decimals. This can often make it easier to see their values and their distances from another number. Converting fractions to decimals involves a simple division operation: the numerator divided by the denominator.
For example, to convert \frac{3}{17} to a decimal:
\[ \frac{3}{17} \text{ calculation steps} = 3 \text{ divided by } 17 \text{ is approximately } 0.176 \]For \frac{16}{17} to a decimal:
\[ \frac{16}{17} \text{ calculation steps} = 16 \text{ divided by } 17 \text{ is approximately } 0.941 \]This conversion can make it clearer which fractions are closer or farther from other numbers, especially when comparing to a decimal like 0.5.

absolute value

The concept of absolute value is fundamental in determining how far one number is from another, without considering the direction. It measures the 'distance' between numbers on a number line, always resulting in a non-negative number.
To find the absolute distance from 0.5 for each fraction (converted to a decimal), we take the difference and then apply the absolute value:
For \frac{3}{17}, which is approximately 0.176:
\[ |0.176 - 0.5| = 0.324 \]For \frac{16}{17}, which is approximately 0.941:
\[ |0.941 - 0.5| = 0.441 \]Since absolute value gives us the magnitude of the distance without considering direction, it helps us see clearly that 0.324 (the distance of \frac{3}{17} from 0.5) is smaller than 0.441, confirming that \frac{3}{17} is indeed closer to 0.5 than \frac{16}{17}.