Question

In each pair, tell which fraction is closer to 0.5 \(\frac{2}{5}\) or \(\frac{3}{7}\)

Short answer

The fraction \( \frac{3}{7} \) is closer to 0.5.

Step-by-step solution

- Determine the decimal equivalent of \(\frac{2}{5}\)

To compare fractions, convert them to their decimal equivalents. Calculate \( \frac{2}{5} \): \( \frac{2}{5} = 0.4 \)

- Determine the decimal equivalent of \(\frac{3}{7}\)

Now calculate \( \frac{3}{7} \): \( \frac{3}{7} \) is approximately \( 0.42857 \)

- Compare the decimals to 0.5

Subtract each decimal from 0.5 to determine which is closer: \( |0.5 - 0.4| = 0.1 \) and \( |0.5 - 0.42857| \) is approximately \( 0.07143 \)

- Determine the closer fraction

Compare the differences: \( 0.1 \) and \( 0.07143 \). Since \( 0.07143 \) is smaller, \(\frac{3}{7}\) is closer to 0.5.

Key concepts

decimal conversion

Decimal conversion can simplify fraction comparisons. Instead of dealing with fractions as they are, we convert them to a format that is often easier to work with: decimals. For example, to convert \(\frac{2}{5}\) to a decimal, you divide 2 by 5, which equals 0.4. This helps us quickly compare sizes and relationships between numbers. Think of it like transforming foreign currency to your local currency to understand its value more intuitively.

fractions

Fractions represent a part of a whole and consist of two main components: the numerator (top number) and the denominator (bottom number). The denominator tells us into how many parts the whole is divided, while the numerator indicates how many of those parts we are considering. For example, in \(\frac{3}{7}\), the whole is divided into 7 parts, and we are looking at 3 of these parts. Understanding fractions is crucial in various mathematical operations, including comparison and conversion to decimals.

proximity to 0.5

When comparing fractions or decimals to determine proximity to 0.5, we look at how close the number is to 0.5 by subtracting the given number from 0.5 and taking the absolute value of the result. Suppose we have 0.4 and 0.42857. We calculate the proximity like this:
\(|0.5 - 0.4| = 0.1\) and \(|0.5 - 0.42857| = 0.07143\)
The smaller the result, the closer the number is to 0.5. In our case, since 0.07143 is smaller than 0.1, 0.42857 is closer to 0.5. Therefore, the fraction \(\frac{3}{7}\) is closer to 0.5 than \(\frac{2}{5}\). This method is useful in various mathematical scenarios where knowing the proximity to benchmark numbers like 0.5 aids in decision-making and comparisons.