Question

Order these fractions from least to greatest: \\[ \frac{1}{3} \quad \frac{12}{30} \quad \frac{9}{28} \quad \frac{11}{30} \quad \frac{12}{29} \\]

Short answer

Order: \( \frac{1}{3}, \frac{9}{28}, \frac{11}{30}, \frac{12}{30}, \frac{12}{29} \).

Step-by-step solution

Find a common denominator

To order fractions, it's easiest if they have a common denominator. Identify a common multiple of the denominators 3, 30, 28, and 29. The least common multiple (LCM) of these numbers is 2520.

Convert each fraction

Convert each fraction to an equivalent fraction with the common denominator 2520. \[ \frac{1}{3} = \frac{1 \times 840}{3 \times 840} = \frac{840}{2520} \] \[ \frac{12}{30} = \frac{12 \times 84}{30 \times 84} = \frac{1008}{2520} \] \[ \frac{9}{28} = \frac{9 \times 90}{28 \times 90} = \frac{810}{2520} \] \[ \frac{11}{30} = \frac{11 \times 84}{30 \times 84} = \frac{924}{2520} \] \[ \frac{12}{29} = \frac{12 \times 87}{29 \times 87} = \frac{1044}{2520} \]

Compare the numerators

Compare the numerators of the fractions with the common denominator 2520. The fractions in order from least to greatest based on their numerators are: \[ \frac{840}{2520}, \frac{810}{2520}, \frac{924}{2520}, \frac{1008}{2520}, \frac{1044}{2520} \] Therefore, the order of the original fractions from least to greatest is: \[ \frac{1}{3}, \frac{9}{28}, \frac{11}{30}, \frac{12}{30}, \frac{12}{29} \]

Key concepts

least common multiple (LCM)

To order fractions effectively, it's vital to find a common denominator. This makes comparing them much simpler. We use the least common multiple (LCM) for this task.
The least common multiple of a set of numbers is the smallest number that each of the numbers divides into without leaving a remainder. For example, finding the LCM of the denominators 3, 30, 28, and 29 will help us convert all the fractions to have the same denominator.
In our exercise, we identified that the least common multiple of 3, 30, 28, and 29 is 2520. This means 2520 is the smallest number that 3, 30, 28, and 29 can all divide into exactly. Converting each fraction to have 2520 as the denominator allows us to easily compare their sizes by just looking at their numerators.

equivalent fractions

Equivalent fractions have the same value, even if they look different. We get an equivalent fraction by multiplying both the numerator and the denominator by the same number.
For example, \(\frac{1}{3}\) is equivalent to \(\frac{840}{2520}\) because we multiplied both 1 and 3 by 840. Similarly, each fraction in our exercise was converted to have a denominator of 2520:
\[ \frac{12}{30} = \frac{1008}{2520}, \frac{9}{28} = \frac{810}{2520}, \frac{11}{30} = \frac{924}{2520}, \frac{12}{29} = \frac{1044}{2520} \] Once fractions are converted to have the same denominator, they are easy to compare by just looking at their numerators. This process simplifies the comparison and ordering of fractions.

numerator comparison

Once all the fractions have the same denominator, we only need to compare the numerators to order the fractions. The numerator tells us how many parts of the whole we have, while the denominator tells us into how many parts the whole is divided.
In our case, after converting the fractions to have a common denominator of 2520, the numerators were: \[ 840, 1008, 810, 924, and 1044 \] By comparing these numerators, we can easily see which fraction is smallest and which is largest.
The numerators in ascending order are: 810, 840, 924, 1008, 1044 \[ Therefore, the original fractions ordered from least to greatest are \(\frac{9}{28}, \frac{1}{3}, \frac{11}{30}, \frac{12}{30}, \frac{12}{29}\). \] Comparing numerators is a quick and efficient way to determine the order when the denominators are the same.