Problem 45 - Prealgebra

Question

Order the fractions from least to greatest. $$-\frac{23}{30},-\frac{4}{5},-\frac{2}{3},-\frac{7}{10}$$

Step-by-Step Solution

Answer

Order: \(-\frac{4}{5}, -\frac{23}{30}, -\frac{7}{10}, -\frac{2}{3}\).

Step 1: Find a Common Denominator

To compare fractions, we first need a common denominator. The denominators given are 30, 5, 3, and 10. The least common multiple of these numbers is 30.

Step 2: Convert Fractions to Have the Same Denominator

Convert each fraction to have a denominator of 30: - \(-\frac{4}{5} = -\frac{4 \times 6}{5 \times 6} = -\frac{24}{30}\)- \(-\frac{2}{3} = -\frac{2 \times 10}{3 \times 10} = -\frac{20}{30}\)- \(-\frac{7}{10} = -\frac{7 \times 3}{10 \times 3} = -\frac{21}{30}\)

Step 3: List the Fractions with Common Denominators

We have the fractions: \(-\frac{23}{30}, -\frac{24}{30}, -\frac{20}{30}, -\frac{21}{30}\).

Step 4: Order the Fractions

Since all fractions are negative, the fraction with the smallest numerator (in absolute value) is the greatest. Ordering from least (most negative) to greatest, we have: \(-\frac{24}{30}, -\frac{23}{30}, -\frac{21}{30}, -\frac{20}{30}\).

Step 5: Convert Back to Original Form

Returning to their original forms: \(-\frac{24}{30} = -\frac{4}{5}\),\(-\frac{23}{30}\),\(-\frac{21}{30} = -\frac{7}{10}\),\(-\frac{20}{30} = -\frac{2}{3}\).

Key Concepts

Understanding Common Denominators

When comparing fractions, the first step is to find a common denominator. The common denominator is essentially a way to bring different fractions onto the same playing field so we can easily compare them. To do this, we look for the least common multiple (LCM) of the denominators of all the fractions involved.

The LCM of a set of numbers is the smallest number that each of the numbers divides into without leaving a remainder. For example, if our denominators are 5, 3, and 2, we find that the LCM is 30 because both 5, 3, and 2 divide evenly into 30.

Once you've found the common denominator, convert each fraction using equivalent fractions that all share this denominator. This makes the fractions easy to compare and order, as you're effectively just comparing the numerators now.

Finding the Least Common Multiple (LCM)

The key to finding a common denominator is determining the least common multiple of the fractions' denominators. Here’s a simple way to do it:

**List the multiples** of each denominator:

• For 5: 5, 10, 15, 20, 25, 30...
• For 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30…
• For 10: 10, 20, 30, 40...
• For 30: 30, 60...

**Identify the smallest common multiple**: Looking at the lists, 30 appears as a common multiple, and it's the smallest one, so 30 is the LCM.

Once you've found the LCM, use it to rewrite each fraction with this common denominator. This way, you can focus solely on the numerators when comparing the fractions. It simplifies the process and helps avoid errors.

Dealing with Negative Fractions

Negative fractions can be a bit tricky but understanding them is crucial when ordering fractions. With negative numbers, the order is reversed compared to positive numbers. The closer a fraction's value is to zero, the greater it is when dealing with negatives.

This means that when you have fractions like \(-rac{23}{30}\) or \(-rac{4}{5}\), the fraction with the smallest absolute value of its numerator will be the largest in value.

For any negative fraction, think of the numbers as being on a number line to the left of zero. So, remember:

• More negative numbers are smaller.
• Less negative numbers are larger.

Keeping this in mind will help you correctly order negative fractions from least to greatest.