Problem 1 - Calculus with Concepts in Calculus

Question

Determine whether \(ab\). \(a=\frac{4}{9}, b=\frac{7}{16}\)

Step-by-Step Solution

Answer

The fraction \( \frac{4}{9} \) is greater, so \( a > b \).

Step 1: Understand the Problem

We need to determine whether the fraction \( \frac{4}{9} \) is less than or greater than the fraction \( \frac{7}{16} \).

Step 2: Equalize the Denominators

To compare the fractions, let's find a common denominator. The denominators are 9 and 16. The least common denominator (LCD) of 9 and 16 is 144.

Step 3: Convert \(a\) to LCD

Convert \(a = \frac{4}{9}\) to have the denominator of 144.Calculate: \( \frac{4}{9} \times \frac{16}{16} = \frac{64}{144} \).

Step 4: Convert \(b\) to LCD

Convert \(b = \frac{7}{16}\) to have the denominator of 144.Calculate: \( \frac{7}{16} \times \frac{9}{9} = \frac{63}{144} \).

Step 5: Compare the Numerators

Now with the same denominator, compare the numerators.Since 64 (numerator of \(a\)) is greater than 63 (numerator of \(b\)), it follows that \( \frac{64}{144} > \frac{63}{144} \).

Step 6: State the Conclusion

Hence, \( \frac{4}{9} > \frac{7}{16} \), meaning \( a > b \).

Key Concepts

Common Denominator

When comparing fractions, it's essential that they share a common denominator. This means they must have the same bottom number, making comparison straightforward. Imagine two delicious pies—one cut into thirds and the other into sixths. To compare the size of one slice from each pie, you'd want both pies to be cut into the same number of pieces.

• This step allows you to directly compare the top numbers, or numerators, since the slices (or denominators) are the same size.
• Without a common denominator, direct comparison could be misleading.

Finding a common denominator means modifying the fractions, but it doesn't change their actual value—it just reframes how you look at them so they can "speak the same language."

Numerator Comparison

Once fractions have a common denominator, comparing them becomes a breeze. The numerators—the digits above the fraction line—are what you'll examine next.

• If you think about it, numerators tell you how many pieces of the pie you have.
• The larger the numerator, the more pie you own.

Therefore, after equalizing the denominators, you simply need to check which numerator is bigger to determine the larger fraction. For example, with fractions like \( \frac{64}{144} \) and \( \frac{63}{144} \), a numerator of 64 indicates more pie than a numerator of 63.

Least Common Denominator

The least common denominator (LCD) is the smallest number that both denominators can divide into evenly. This is the critical step in aligning denominators for comparison.

• For the fractions \( \frac{4}{9} \) and \( \frac{7}{16} \), the LCD is 144.
• This involves finding the least common multiple (LCM) of 9 and 16.

Multiplying the top and bottom of each fraction by necessary factors gives each the same base, allowing for an easy comparison of numerators.

Fraction Conversion

Fraction conversion is a method used to rewrite fractions so their denominators match, permitting direct comparison. To convert a fraction like \( \frac{4}{9} \) into a denominator of 144:

• Multiply both the top (numerator) and bottom (denominator) by the same number that will turn the denominator into 144.
• For instance, \( 9 \times 16 = 144 \), so multiply the fraction by \( \frac{16}{16} \) resulting in \( \frac{64}{144} \).

This technique preserves the fraction's value while recasting it into a form with a desired denominator, facilitating easy and accurate comparison.