Problem 17 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$2 \frac{2}{3} \text { to } \frac{5}{3}$$

Step-by-Step Solution

Answer

\(\frac{8}{5}\)

Step 1: Convert Mixed Number to Improper Fraction

The given mixed number is \(2 \frac{2}{3}\). We convert it to an improper fraction. To do this, multiply the whole number part (2) by the denominator of the fractional part (3), add the numerator of the fractional part (2), and place that sum over the original denominator: \(2 \times 3 + 2 = 6 + 2 = 8\). This gives us \(\frac{8}{3}\).

Step 2: Write the Ratio as a Fraction

Write the ratio as a fraction by placing the improper fractions from Step 1 and the given fraction on top and bottom of the fraction, respectively: \(\frac{\frac{8}{3}}{\frac{5}{3}}\). This compares \(\frac{8}{3}\) to \(\frac{5}{3}\).

Step 3: Simplify the Complex Fraction

Simplify \(\frac{\frac{8}{3}}{\frac{5}{3}}\) by multiplying the numerator by the reciprocal of the denominator: \(\frac{8}{3} \times \frac{3}{5}\). This simplifies to \(\frac{8 \times 3}{3 \times 5} = \frac{24}{15}\).

Step 4: Reduce to Lowest Terms

Factor both the numerator (24) and the denominator (15) to find their greatest common divisor (GCD). The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24; the factors of 15 are 1, 3, 5, and 15. The GCD is 3. Divide both the numerator and the denominator by 3: \(\frac{24}{15} = \frac{24 \div 3}{15 \div 3} = \frac{8}{5}\).

Step 5: Conclude with the Simplified Fraction

State that the ratio of \(2 \frac{2}{3}\) to \(\frac{5}{3}\) simplified as a fraction in lowest terms is \(\frac{8}{5}\).

Key Concepts

Mixed Numbers

Mixed numbers combine a whole number and a fraction into one expression. For example, in the mixed number \(2 \frac{2}{3}\), 2 is the whole number, and \(\frac{2}{3}\) is the fractional part. To work with mixed numbers in calculations, it's often useful to convert them into improper fractions.
This makes addition, subtraction, and comparison more straightforward. Turning a mixed number into an improper fraction involves a few steps:

• Multiply the whole number by the denominator of the fractional part
• Add the result to the numerator of the fractional part
• Write that sum over the original denominator

In our example, converting \(2 \frac{2}{3}\) gives us \(\frac{8}{3}\). Converting to improper fractions allows for easier manipulation in mathematical operations.

Improper Fractions

An improper fraction is a type of fraction where the numerator is greater than or equal to the denominator. This means that the value of the fraction is equal to or greater than one.
For instance, \(\frac{8}{3}\) is an improper fraction because 8 (numerator) is greater than 3 (denominator). When dealing with mixed numbers and ratios, it's essential to convert to improper fractions to simplify calculations.
By using improper fractions, we can easily perform operations like addition, subtraction, and multiplication. When combining fractions for a ratio, using improper fractions helps streamline the process, making it straightforward to compare and simplify further. Improper fractions play a key role in accurately performing mathematical operations.

Simplifying Fractions

Simplifying fractions involves reducing a fraction to its lowest terms, which means the numerator and denominator have no common divisors other than 1. This process ensures fractions are as simple as possible, making them easier to understand and work with.
Here's how you simplify a fraction:

• Identify the greatest common divisor (GCD) for the numerator and denominator
• Divide both the numerator and denominator by the GCD
• Express the fraction in its simplest form

In our example, we simplified \(\frac{24}{15}\) to \(\frac{8}{5}\) by dividing both the numerator and the denominator by their GCD, which is 3. Simplifying fractions is a crucial step in ensuring mathematical accuracy and clarity.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is the largest positive integer that evenly divides two or more integers without leaving a remainder. It's essential to finding the simplest form of a fraction.
To find the GCD of two numbers:

• List out the factors of each number
• Identify the largest factor common to both lists

In our scenario, the GCD for 24 and 15 was 3, since it's the highest number that divides both without leftovers.
Using the GCD, you can reduce fractions like \(\frac{24}{15}\) down to their simplest form, \(\frac{8}{5}\). Mastering the concept of the GCD aids in efficient and accurate fraction simplification, crucial for any mathematical task involving ratios and comparisons.