Problem 46 - Fundamentals of Mathematics

Question

Determine if the pairs of fractions are equivalent. $$ 2 \frac{3}{8}, \frac{38}{16} $$

Step-by-Step Solution

Answer

The fractions are equivalent.

Step 1: Identify the Fractions

The given fractions to check for equivalence are mixed fraction \(2 \frac{3}{8}\) and improper fraction \(\frac{38}{16}\).

Step 2: Convert the Mixed Fraction to an Improper Fraction

To convert the mixed fraction \(2 \frac{3}{8}\) to an improper fraction, calculate as follows: \(2 \times 8 + 3 = 16 + 3 = 19\). So the improper fraction is \(\frac{19}{8}\).

Step 3: Find a Common Denominator

To compare fractions, we need a common denominator. However, here we can simplify to check equivalency. Let's simplify \(\frac{38}{16}\): divide both numerator and denominator by 2 to get \(\frac{19}{8}\).

Step 4: Compare the Improper Fractions

Now, compare the two improper fractions: \(\frac{19}{8}\) and \(\frac{19}{8}\). Since they are exactly the same, the fractions are equivalent.

Key Concepts

Mixed Fractions

A mixed fraction is a way of expressing a quantity that combines both a whole number and a fraction. For instance, the mixed fraction \(2 \frac{3}{8}\) represents 2 whole parts plus 3 parts of an 8-part whole. Breaking it down:

• The whole number is \(2\).
• The fractional part is \(\frac{3}{8}\), which means 3 out of 8 pieces.

Mixed fractions are useful in making sense of quantities that are not complete wholes, and they are often seen in everyday measurements, like cooking or dividing items. However, when performing mathematic operations, they are typically converted into improper fractions to simplify calculations.

Improper Fractions

An improper fraction is a fraction where the numerator (the top number) is larger than or equal to the denominator (the bottom number). This means the value of the fraction is equal to or greater than one whole. For example, the improper fraction \(\frac{38}{16}\) in the exercise is greater than one whole because 38, the numerator, is greater than 16, the denominator.

Converting a mixed fraction to an improper fraction involves some straightforward steps:

• Multiply the whole number by the denominator.
• Add the result to the numerator from the fractional part.
• Use this sum as the new numerator, over the original denominator.

For \(2 \frac{3}{8}\), this process gives us \(\frac{19}{8}\). Improper fractions are quite handy in algebra and advanced calculations because they simplify the arithmetic operations compared to using mixed numbers.

Fraction Simplification

Fraction simplification involves reducing a fraction to its simplest form where the numerator and the denominator have no common divisors other than one. Doing so makes it easier to work with fractions in addition, subtraction, and comparing equivalency. In the example, simplifying \(\frac{38}{16}\) requires looking at the factors of both the numerator and the denominator.

Here’s how you simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

For \(\frac{38}{16}\), the GCD is 2. Dividing the numerator and the denominator by 2 gives \(\frac{19}{8}\). At this stage, the fraction is fully simplified and more manageable for comparison purposes. Simplifying fractions is especially important when determining if two different fractions are equivalent since it reveals their simplest form for easier comparison.