Problem 47 - Fundamentals of Mathematics

Question

Determine if the pairs of fractions are equivalent. $$ \frac{108}{77}, 1 \frac{5}{13} $$

Step-by-Step Solution

Answer

The fractions are not equivalent.

Step 1: Understanding the Problem

We need to determine if the fractions \( \frac{108}{77} \) and \( 1 \frac{5}{13} \) are equivalent.

Step 2: Convert Mixed Number to Improper Fraction

Since \( 1 \frac{5}{13} \) is a mixed number, convert it to an improper fraction. Multiply the whole number (1) by the denominator (13) and add the numerator (5): \[ \text{Improper Fraction} = \frac{1 \times 13 + 5}{13} = \frac{18}{13} \]

Step 3: Compare the Two Fractions

Now, compare the two fractions \( \frac{108}{77} \) and \( \frac{18}{13} \). Two fractions are equivalent if their cross products are equal: \[ 108 \times 13 \quad ?=\quad 77 \times 18 \]

Step 4: Calculate Cross Products

Calculate the cross products:\( 108 \times 13 = 1404 \) and \( 77 \times 18 = 1386 \).

Step 5: Determine Equivalence

Since \( 1404 eq 1386 \), the fractions \( \frac{108}{77} \) and \( \frac{18}{13} \) are not equivalent.

Key Concepts

Mixed Numbers

Let's dive into what a mixed number is. When you see a mixed number, it's essentially a combination of a whole number and a fraction. For example, in the number \(1 \frac{5}{13}\), the \(1\) is the whole number, and \(\frac{5}{13}\) is the fraction part.

Mixed numbers are a convenient way of expressing quantities that include a whole part and a fractional part. They are often used in everyday situations. Picture having a whole chocolate bar and a portion of another one - that's akin to owning a mixed number!

When dealing with equations or problems, mixed numbers can be converted into improper fractions to make calculations easier. This conversion allows us to work smoothly with fractions in mathematical operations such as addition, subtraction, or comparison.

To convert a mixed number into an improper fraction, here's what you do:

• Multiply the whole number by the denominator of the fraction.
• Add that result to the numerator.
• Place the sum over the original denominator.

For example, converting \(1 \frac{5}{13}\) into an improper fraction gives us \(\frac{18}{13}\). This format helps in solving complex problems by simplifying the components involved.

Improper Fractions

Improper fractions are simply fractions where the numerator is greater than or equal to the denominator. This makes them top-heavy. An example is \(\frac{18}{13}\), where 18 (the numerator) is larger than 13 (the denominator).

Improper fractions might look complex at first, but they're actually quite straightforward. They provide a unified way to display numbers and are particularly useful in arithmetic calculations. With improper fractions, it's much easier to multiply, divide, add, or subtract.

Why do we change mixed numbers to improper fractions? It makes multiple operations more consistent and simpler. Since it's often advantageous to have a consistent fraction form during arithmetic operations, switching to improper fractions is a common practice.

Here's a quick reminder of their usefulness:

• They simplify fraction arithmetic operations.
• They help maintain consistent format across equations.
• They offer a straightforward path towards solving equations.

Improper fractions open the doors to more versatile numerical manipulations, ensuring precise and efficient calculations.

Cross Multiplication

Cross multiplication is a handy tool used to determine whether two fractions are equivalent. It involves multiplying the numerator of one fraction by the denominator of the other and vice versa.

Imagine you have two fractions, like \(\frac{108}{77}\) and \(\frac{18}{13}\). To see if they are equivalent, you would:

• Multiply the numerator of the first fraction by the denominator of the second: \(108 \times 13\).
• Multiply the numerator of the second fraction by the denominator of the first: \(77 \times 18\).

Compare these two results. If they are the same, the fractions are equivalent; otherwise, they're not.

Cross multiplication is particularly useful because it avoids the need for simplifying fractions beforehand. It's a straightforward way to travel directly to the outcome:

"Are these two fractions equivalent?" Let the cross multiplication technique quickly answer that for you.

Remember:

• It's a simple process with just a few multiplications.
• You get results without simplifying to a lowest form.
• It's a quick way to compare fractions without converting to decimals.

Cross multiplication empowers you to efficiently evaluate and compare diverse fractions, making it an indispensable tool in tackling such problems.