Problem 80 - Prealgebra

Question

Which of the following fractions is closest to the number 1? a. \(\frac{1}{2}\) b. \(\frac{1}{3}\) c. \(\frac{1}{4}\) d. \(\frac{1}{5}\)

Step-by-Step Solution

Answer

\(\frac{1}{2}\) is closest to 1.

Step 1: Understand the Problem

To determine which fraction is closest to 1, we need to pick the fraction that has the largest value. All the given fractions are less than 1 but approach 1 from below.

Step 2: Convert the Fractions to Decimals

Convert each fraction to its decimal form: - \(\frac{1}{2} = 0.5\)- \(\frac{1}{3} \approx 0.333\)- \(\frac{1}{4} = 0.25\)- \(\frac{1}{5} = 0.2\)

Step 3: Determine the Largest Decimal

Compare the decimal values obtained in the previous step. The largest decimal value among 0.5, 0.333, 0.25, and 0.2 is 0.5.

Step 4: Identify the Closest Fraction

The fraction corresponding to the largest decimal value, which is 0.5, is \(\frac{1}{2}\). Thus, \(\frac{1}{2}\) is the fraction closest to 1.

Key Concepts

Fraction to Decimal Conversion

Understanding fractions and their decimal equivalents is a vital math skill that simplifies comparisons. Fractions represent a part of a whole and are written in the form \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator. Converting a fraction to a decimal helps us to visually see how it compares to whole numbers. To perform this conversion, divide the numerator by the denominator. For example, \(\frac{1}{2}\) converts to 0.5, since dividing 1 by 2 gives 0.5.
- \(\frac{1}{3}\) becomes approximately 0.333 because dividing 1 by 3 results in a repeating decimal.- \(\frac{1}{4}\) easily converts to 0.25 as dividing 1 by 4 results in 0.25 exactly.
- \(\frac{1}{5}\) transitions to 0.2, given that dividing 1 by 5 results in 0.2.Having these decimal forms allows for easy comparisons to determine which is closest to 1.

Identifying Largest Value

Once you have the decimal forms of the fractions, the next step involves identifying the largest value. This is because the largest decimal is the closest to 1, provided all fractions are smaller than 1. Here, comparison becomes straightforward:- 0.5 is larger than 0.333, 0.25, and 0.2.- It's important to recognize that the highest decimal indicates the fraction with the greatest value among the given fractions.
This step is crucial when you aim to determine proximity to a specific number, like 1, making it easier to spot \(\frac{1}{2}\) as the largest and therefore the closest fraction.

Proximity to Whole Number

Finding how closely a fraction approximates a whole number, like 1, is a common problem-solving strategy. "Proximity" refers to which fraction reaches nearest to 1 but doesn’t exceed it. Since all considered fractions are less than 1, the task is to pinpoint the one barely under 1.
- \(\frac{1}{2}\) at 0.5 is only 0.5 units away from 1, closer than its alternatives.- \(\frac{1}{3}\) at approximately 0.333 is 0.667 units away from 1.- \(\frac{1}{4}\), at 0.25 is 0.75 units from 1.- \(\frac{1}{5}\), reaching just 0.2, stays 0.8 units distant from 1.Clearly, fractions with larger values come closer to our target number, reinforcing that \(\frac{1}{2}\) best approaches the whole number 1. Understanding these distances helps in quickly discerning which fraction is nearest to a specific goal.