Problem 48 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true sentence. $$-5 \frac{1}{3} \circ-5 \frac{3}{10}$$

Step-by-Step Solution

Answer

\(-5 \frac{1}{3} > -5 \frac{3}{10}\).

Step 1: Convert mixed numbers to improper fractions

First, we convert the mixed numbers into improper fractions. For \(-5 \frac{1}{3}\): Multiply the whole number part (-5) by the denominator (3) and then add the numerator (1): \(-5 \times 3 + 1 = -15 + 1 = -14\). So, \(-5 \frac{1}{3}\) is \(-\frac{14}{3}\).For \(-5 \frac{3}{10}\): Multiply the whole number part (-5) by the denominator (10) and then add the numerator (3): \(-5 \times 10 + 3 = -50 + 3 = -47\). So, \(-5 \frac{3}{10}\) is \(-\frac{53}{10}\).

Step 2: Find a common denominator

The fractions \(-\frac{14}{3}\) and \(-\frac{53}{10}\) need a common denominator for comparison. The least common multiple of 3 and 10 is 30. Convert \(-\frac{14}{3}\):Multiply both the numerator and denominator by 10: \(-\frac{14}{3} = -\frac{14 \times 10}{3 \times 10} = -\frac{140}{30}\).Convert \(-\frac{53}{10}\):Multiply both the numerator and denominator by 3: \(-\frac{53}{10} = -\frac{53 \times 3}{10 \times 3} = -\frac{159}{30}\).

Step 3: Compare the fractions

Now, we simply compare the converted fractions with a common denominator:\(-\frac{140}{30}\) and \(-\frac{159}{30}\).Since \(-140\) is greater than \(-159\), it follows that:\(-\frac{140}{30} > -\frac{159}{30}\).

Step 4: Replace the symbol

Now replace the symbol \(\circ\) in the sentence based on the comparison:\(-5 \frac{1}{3} > -5 \frac{3}{10}\).

Key Concepts

Improper Fractions

Improper fractions can seem a bit confusing at first. But they are quite simple once you break them down. An improper fraction is where the numerator, or the top number of the fraction, is larger than the denominator, which is the bottom part. For instance, \(-\frac{14}{3}\) is an improper fraction because 14 is bigger than 3.
When you convert a mixed number, like \(-5 \frac{1}{3}\), to an improper fraction, the process is straightforward:

• Multiply the whole number by the denominator.
• Add the numerator to the result.
• This sum becomes the new numerator, while the denominator stays the same.

This conversion helps us compare fractions more easily by having common types.

Mixed Numbers

Mixed numbers are a blend of whole numbers and fractions. They appear as such: a whole part, like \(-5\), combined with a fraction, like \(\frac{1}{3}\), to make \(-5 \frac{1}{3}\). When faced with mixed numbers, it can be easier to manage arithmetic by changing them into improper fractions. Here's how it works:

• Multiply the whole number by the denominator of the fraction.
• Add the result to the numerator of the fraction.
• Combine these to form an improper fraction.

This process turns complex problems with mixed numbers into simpler equations. It's particularly useful in math tasks like comparing fractions or adding them.

Common Denominator

When comparing fractions, having a common denominator makes the task much easier. A common denominator is the same denominator for two or more fractions. To find it, follow these steps:

• Identify the denominators of the fractions you want to compare.
• Calculate the least common multiple (LCM) of these denominators.
• Convert each fraction to an equivalent fraction by adjusting both its numerator and denominator to match the LCM.

For example, for \(-\frac{14}{3}\) and \(-\frac{53}{10}\), the LCM of 3 and 10 is 30. We adjust the fractions so their denominators become 30, converting them to equivalent forms: \(-\frac{140}{30}\) and \(-\frac{159}{30}\), respectively. This alignment makes it simple to directly compare the numerators and decide which fraction is larger or smaller.