Problem 77 - Algebra 1: Concepts and Skills

Question

Write the fractions in order from least to greatest. $$ \frac{4}{6}, \frac{5}{2}, \frac{3}{4}, \frac{2}{6}, \frac{2}{2} $$

Step-by-Step Solution

Answer

The fractions in order from least to greatest are: \( \frac{2}{6} , \frac{4}{6} , \frac{3}{4} , \frac{2}{2} , \frac{5}{2} \)

Step 1: Convert all fractions to have like denominators

Since the fractions have different denominators, it's difficult to compare them directly. Therefore, convert them to have the same denominator. It will be easier if we choose the least common multiple (LCM) of 6, 2, 4, and 6 as the common denominator, which is 12. \[ \frac{4}{6} = \frac{4 * 2}{6 * 2} = \frac{8}{12} \] \[ \frac{5}{2} = \frac{5 * 6}{2 * 6} = \frac{30}{12} \] \[ \frac{3}{4} = \frac{3 * 3}{4 * 3} = \frac{9}{12} \] \[ \frac{2}{6} = \frac{2 * 2}{6 * 2} = \frac{4}{12} \] \[ \frac{2}{2} = \frac{2 * 6}{2 * 6} = \frac{12}{12} \]

Step 2: Compare the numerators

Since all fractions now have the same denominator, it's easy to compare them by comparing their numerators. If the numerator is smaller, the value of the fraction is smaller. Now the order from least to greatest is: \( \frac{4}{12} , \frac{8}{12} , \frac{9}{12} , \frac{12}{12} , \frac{30}{12} \)

Step 3: Convert back to original denominators

Now, convert the fractions, still in order from least to greatest, back to their original denominators. This will give the sorted order of the original fractions: \( \frac{2}{6} , \frac{4}{6} , \frac{3}{4} , \frac{2}{2} , \frac{5}{2} \)

Key Concepts

Least Common Multiple

When comparing fractions with different denominators, finding a like denominator makes the process much easier. A great tool for achieving this is the 'least common multiple' or LCM. The LCM of a group of numbers is the smallest number that each of the numbers divides into evenly. For example, when we wanted to compare the fractions \( \frac{4}{6}, \frac{5}{2}, \frac{3}{4}, \frac{2}{6}, \frac{2}{2} \), we needed to find a number that 6, 2, and 4 all divide into neatly.
The steps to finding the LCM are simple but crucial:

• List the multiples of each denominator.
• Identify the smallest multiple common to all denominators.

For the denominators 6, 2, and 4, their LCM is 12. This choice enables us to streamline comparison by converting all fractions to have this common denominator. Once fractions are expressed with the same denominator, comparing their sizes becomes much more direct and intuitive.

Fraction Comparison

To determine the order of fractions from least to greatest, comparing them directly can be challenging when they have different denominators. By converting them to equivalent fractions with a common denominator, we simplify the comparison.
Here's why we use the common denominator:

• Fractions are easy to compare when they have the same bottom number.
• We only need to look at the numerators to order the fractions.

In our example, after converting all fractions to have a denominator of 12:

• \(\frac{4}{12}\)
• \(\frac{8}{12}\)
• \(\frac{9}{12}\)
• \(\frac{12}{12}\)
• \(\frac{30}{12}\)

We compare the top numbers, or numerators. The one with the smallest numerator, \(\frac{4}{12}\), is the smallest fraction, and the one with the largest, \(\frac{30}{12}\), is the greatest. The process becomes much simpler, as now we are merely comparing whole numbers.

Converting Fractions

Converting fractions is a handy skill that allows comparisons to be made easily between different fractions. This process involves changing each fraction into an equivalent fraction with a common denominator. Let's walk through this process with an example:

• Start with the original fraction, like \(\frac{4}{6}\).
• Find the new equivalent fraction with the LCM, \(12\), as the denominator.
• Multiply both the numerator and the denominator by the same number to retain the fraction's value.

For instance, \(\frac{4}{6}\) is converted into \(\frac{8}{12}\) by multiplying both the numerator and denominator by 2.
Converting these fractions ensures that they have the same base for easy comparison, transforming complex comparisons into simple integer evaluations. Once the fractions are ordered, they can revert back to their original format, providing clarity not just in solving exercises but in understanding fractions conceptually.