Problem 66 - Algebra 1: Concepts and Skills

Question

Write the numbers in order from least to greatest. $$ \frac{1}{2}, \frac{2}{3}, \frac{5}{12} $$

Step-by-Step Solution

Answer

The numbers in order from least to greatest are \(\frac{5}{12}\), \(\frac{1}{2}\), \(\frac{2}{3}\).

Step 1: Find the Least Common Multiple (LCM) of the denominators

First, check for the least common multiple (also known as the least common denominator) of the numbers 2,3 and 12. By inspection, the LCM is 12.

Step 2: Express the fractions with common denominator

Now express each fraction with 12 as the denominator. The first fraction becomes \(\frac{1}{2} = \frac{6}{12}\), the second fraction becomes \(\frac{2}{3} = \frac{8}{12}\), and the third fraction remains \(\frac{5}{12}\). Now the fractions have common denominators, making it easy to compare them.

Step 3: Arrange the Fractions in Ascending Order

Next, arrange the fractions in increasing order. This can be done by simply comparing the numerators since the denominators are the same. This results in \(\frac{5}{12}\), \(\frac{6}{12}\), \(\frac{8}{12}\). When these fractions are converted back into their original form, the numbers are \(\frac{5}{12}\), \(\frac{1}{2}\), \(\frac{2}{3}\).

Key Concepts

Least Common Multiple

To solve problems involving ordering fractions, it's crucial to find the Least Common Multiple (LCM) of the given denominators. The LCM is the smallest multiple that two or more numbers have in common. Finding the LCM allows you to transform fractions with different denominators into equivalent ones with the same denominator. This simplifies the comparison process between different fractions. When determining the LCM of the numbers 2, 3, and 12, you should first list the multiples of these numbers. For example:

• Multiples of 2: 2, 4, 6, 8, 10, 12, 14...
• Multiples of 3: 3, 6, 9, 12, 15...
• Multiples of 12: 12, 24, 36...

From this, you can see that 12 is the smallest number that appears in all these lists, making it the LCM of the three denominators. By using the LCM in fraction ordering, you ensure they share a common scale, facilitating a straightforward comparison by numerators alone.

Comparing Fractions

Once you've determined a common denominator using the LCM, comparing fractions becomes a simple task. All you have to do is look at the numerators of each fraction, as the denominators are already aligned. This eliminates any complexity in the comparison process.For example, after determining the LCM, the fractions \( \frac{1}{2} \), \( \frac{2}{3} \), and \( \frac{5}{12} \) are converted into \( \frac{6}{12} \), \( \frac{8}{12} \), and \( \frac{5}{12} \) respectively. Each fraction is now expressed with a denominator of 12, and ordering them is as easy as arranging the numerators: 5, 6, and 8.This method works because when denominators are the same, larger numerators correspond to larger fractions, and the comparison is direct. Thus, we can list these fractions from smallest to largest as \( \frac{5}{12} \), \( \frac{6}{12} \), and \( \frac{8}{12} \). Reverse conversion to original terms gives us \( \frac{5}{12} \), \( \frac{1}{2} \), and \( \frac{2}{3} \).

Least Common Denominator

The Least Common Denominator (LCD) is essentially the same as the LCM, but in the context of fractions. Finding the LCD is critical when you want to add, subtract, or compare fractions. Once you've identified the LCD, you can convert each fraction to an equivalent one that shares this common denominator.In our example, by finding the LCM of 2, 3, and 12, which is 12, we established it as the LCD. You adjust each fraction accordingly:

• \( \frac{1}{2} \) becomes \( \frac{6}{12} \) by multiplying both numerator and denominator by 6.
• \( \frac{2}{3} \) becomes \( \frac{8}{12} \) by multiplying both numerator and denominator by 4.
• \( \frac{5}{12} \) is already expressed with 12 as a denominator.

To convert fractions, multiply the numerator and the denominator of each fraction by whatever number necessary to make the original denominator equal to the LCD. This allows you to compare or operate on these fractions with ease since they all live in the same fractional 'universe'.