Problem 99 - Elementary Algebra

Question

For exercises \(85-108\), write \(>\) or \(<\) between the numbers to make a true statement. $$ \frac{5}{8} \quad \frac{2}{3} $$

Step-by-Step Solution

Answer

\frac{5}{8} < \frac{2}{3}

Step 1 - Find a common denominator

To compare the fractions \(\frac{5}{8}\) and \(\frac{2}{3}\), find a common denominator. The least common multiple (LCM) of 8 and 3 is 24.

Step 2 - Convert the fractions

Convert each fraction to have the common denominator of 24. For \(\frac{5}{8}\), multiply both the numerator and denominator by 3 to get \(\frac{15}{24}\). For \(\frac{2}{3}\), multiply both the numerator and denominator by 8 to get \(\frac{16}{24}\).

Step 3 - Compare the numerators

Now compare the numerators of the fractions with the same denominator. Since 15 < 16, \(\frac{15}{24} < \frac{16}{24}\), which means \(\frac{5}{8} < \frac{2}{3}\).

Key Concepts

Understanding Common Denominator

When comparing or adding fractions, one important step is finding a common denominator. A common denominator is a shared multiple of the denominators of two or more fractions. This makes it easier to directly compare the fractions or perform operations on them.

To find a common denominator for \(\frac{5}{8}\) and \(\frac{2}{3}\), you need a number that both 8 and 3 can divide into without a remainder. We often use the least common multiple (LCM) for this purpose. In this case, the LCM of 8 and 3 is 24. When converting fractions to have a common denominator, you're ensuring both fractions are expressed in a comparable form.

For example:

• Multiply the numerator and denominator of \(\frac{5}{8}\) by 3. This gives \(\frac{15}{24}\).
• Multiply the numerator and denominator of \(\frac{2}{3}\) by 8. This gives \(\frac{16}{24}\).

Once both fractions have the same denominator (24), you can easily compare the numerators.

Least Common Multiple (LCM)

The least common multiple (LCM) is the smallest multiple that is exactly divisible by each one of a set of numbers. It's a shorthand for finding the smallest shared multiple. This is crucial when working with fractions because having the same denominator simplifies many operations.

To find the LCM of two numbers like 8 and 3, list out a few multiples of each:

• Multiples of 8: 8, 16, 24, 32, 40, ...
• Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, ...

The smallest multiple they share is 24. This is the LCM. When using the LCM to find a common denominator, you transform fractions so they have this same bottom number. This allows for straightforward comparison.

Role of the Numerator

The numerator is the top number in a fraction. It indicates how many parts of the whole are being considered. When fractions have the same denominator, you can easily compare them by just looking at the numerators.

In our example, once \(\frac{5}{8}\) and \(\frac{2}{3}\) were converted to \(\frac{15}{24}\) and \(\frac{16}{24}\) respectively, we can compare the numerators 15 and 16 directly. Because 15 is less than 16, we conclude that \(\frac{15}{24} < \frac{16}{24}\). Thus, \(\frac{5}{8} < \frac{2}{3}\).

Comparing numerators becomes straightforward when both fractions share a common denominator. The fraction with the larger numerator represents the larger part of the whole.