Problem 85 - Prealgebra

Question

Write the numbers in order from smallest to largest. $$1 \frac{5}{6} \quad \frac{3}{2} \quad 1 \frac{2}{3} \quad \frac{25}{12}$$

Step-by-Step Solution

Answer

\( \frac{3}{2}, 1 \frac{2}{3}, 1 \frac{5}{6}, \frac{25}{12} \).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers into improper fractions. The first mixed number is \( 1 \frac{5}{6} \). To convert it, multiply the whole number (1) by the denominator (6) and add the numerator (5). So it becomes: \( 1 \times 6 + 5 = 11 \). Thus, \( 1 \frac{5}{6} = \frac{11}{6} \).For \( 1 \frac{2}{3} \), multiply 1 by 3 and add 2: \( 1 \times 3 + 2 = 5 \). Thus, \( 1 \frac{2}{3} = \frac{5}{3} \).

Step 2: Compare Fractions by Finding a Common Denominator

Next, we need to compare \( \frac{11}{6}, \frac{3}{2}, \frac{5}{3}, \) and \( \frac{25}{12} \). To do this, find a common denominator. The least common denominator for the fractions is 12.Convert each fraction:- \( \frac{11}{6} \) to \( \frac{22}{12} \) by multiplying the numerator and denominator by 2.- \( \frac{3}{2} \) to \( \frac{18}{12} \) by multiplying both by 6.- \( \frac{5}{3} \) to \( \frac{20}{12} \) by multiplying both by 4.- \( \frac{25}{12} \) remains as \( \frac{25}{12} \).

Step 3: Order the Fractions

Now, list the fractions from smallest to largest comparing their numerators since the denominators are the same:- \( \frac{18}{12} \), \( \frac{20}{12} \), \( \frac{22}{12} \), \( \frac{25}{12} \).The original fractions corresponding to these are:- \( \frac{3}{2} \)- \( \frac{5}{3} \)- \( \frac{11}{6} \)- \( \frac{25}{12} \).

Step 4: Write the Order in Mixed Numbers and Original Form

Convert the improper fractions back to their mixed number form if necessary:- \( \frac{3}{2} \) does not need conversion as it’s already in proper form.- \( \frac{5}{3} = 1 \frac{2}{3} \).- \( \frac{11}{6} = 1 \frac{5}{6} \).Write the order: \( \frac{3}{2}, 1 \frac{2}{3}, 1 \frac{5}{6}, \frac{25}{12} \).

Key Concepts

Understanding Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. This means that the fraction represents a value greater than or equal to one whole. Such fractions can often be converted into mixed numbers, which makes them easier to understand.To convert a mixed number into an improper fraction:

• Multiply the whole number by the denominator.
• Add the numerator to the result.
• Place this total over the original denominator.

For example, to convert the mixed number \(1 \frac{2}{3} \) into an improper fraction, you multiply 1 by 3 (resulting in 3), then add 2 to get 5, so you have \( \frac{5}{3} \). This process helps when you need to compare or perform mathematical operations with fractions.

Why Common Denominators Matter

Finding a common denominator is crucial when comparing fractions or performing operations like addition and subtraction. A common denominator is a shared multiple of the denominators of two or more fractions. This allows the fractions to be expressed with the same denominator, making it easy to compare their sizes directly by looking at the numerators.The process to find a common denominator involves:

• Identifying the least common multiple (LCM) of all denominators involved.
• Adjust each fraction so that the denominator is the LCM.
• Multiply both the numerator and the denominator by the same number to maintain the value of the fraction.

For example, for fractions like \( \frac{3}{2} \) and \( \frac{5}{3} \), you would use 6 as a common denominator: converting \( \frac{3}{2} \) to \( \frac{18}{12} \) and \( \frac{5}{3} \) to \( \frac{20}{12} \). This way, fractions can be easily ordered or compared.

Mixed Numbers and Their Conversion

Mixed numbers consist of a whole number and a fraction, making them a bit more intuitive to understand than improper fractions. They bridge the gap between whole numbers and fractions, helping to visualize the quantity.Converting from improper fractions back to mixed numbers involves:

• Dividing the numerator by the denominator to find the whole number part.
• Using the remainder as the new numerator while keeping the original denominator.

Take \( \frac{11}{6} \): dividing 11 by 6 gives 1 with a remainder of 5, so \( \frac{11}{6} \) becomes \( 1 \frac{5}{6} \). Using mixed numbers can simplify mathematical reasoning and enhance our understanding of the fractions' size relative to whole numbers.