Problem 16 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$5 \frac{1}{4} \text { to } 1 \frac{3}{4}$$

Step-by-Step Solution

Answer

The ratio as a fraction in lowest terms is \(3\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers into improper fractions. For \(5 \frac{1}{4}\), multiply the whole number 5 by the denominator 4 and add the numerator 1: \((5 \times 4) + 1 = 21\). The fraction is \(\frac{21}{4}\). For \(1 \frac{3}{4}\), do the same: \((1 \times 4) + 3 = 7\). The fraction is \(\frac{7}{4}\).

Step 2: Write the Ratio as a Fraction

Now that you have the improper fractions \(\frac{21}{4}\) and \(\frac{7}{4}\), express the ratio between them as a fraction. Ratios compare the first number to the second, so write the fraction as \(\frac{21}{4}\) divided by \(\frac{7}{4}\).

Step 3: Simplify the Fraction Division

To divide fractions, multiply the first fraction by the reciprocal of the second. So, \(\frac{21}{4} \div \frac{7}{4}\) becomes \(\frac{21}{4} \times \frac{4}{7}\).

Step 4: Simplify the Resulting Fraction

Multiply the numerators and the denominators: \(\frac{21 \times 4}{4 \times 7}\). The \(4\) in the numerator and denominator cancel out. This simplifies to \(\frac{21}{7}\).

Step 5: Reduce to Lowest Terms

Divide the numerator and the denominator by their greatest common divisor, which is 7. \(\frac{21}{7} = 3\). Thus, the simplified fraction is \(3\).

Key Concepts

Understanding Ratios

Ratios are a key concept in mathematics that help us compare two quantities. A ratio tells us how much of one thing there is compared to another. To express a ratio as a fraction, we simply take the first number, called the antecedent, and the second number, referred to as the consequent, and write them in fraction form: antecedent over consequent. This helps in understanding and visually seeing the relationship between two different numbers.

For example, in our case where we have the numbers related as mixed numbers: \(5 \frac{1}{4}\) to \(1 \frac{3}{4}\), translating these into improper fractions facilitates easier comparison and computation. Once converted, the ratio becomes a fraction of these improper fractions, simplifying comparisons by using consistent denominators.

Keep in mind:

• A ratio can simplify into smaller numbers, much like a fraction reduces to its lowest terms.
• Ratios can describe parts of collections, rates, or scale factors in various contexts.

Remember, ratios are a powerful way to express relationships and are used throughout mathematics and real-world applications.

The Concept of Improper Fractions

Improper fractions are fractions where the numerator (top number) is larger than the denominator (bottom number). They're not as visually apparent as mixed numbers, hence we convert to or from them when required for calculations.

In our exercise, we were tasked with converting \(5 \frac{1}{4}\) and \(1 \frac{3}{4}\) into improper fractions to ease our work with ratios. This is done by multiplying the whole number by the denominator and adding the numerator, which forms a straight-line calculation and offers a uniform path for comparison through division.

Converting mixed numbers to improper fractions using the formula:

• For \(a \frac{b}{c}\), use: \((a \times c) + b = \text{Numerator}; c = \text{Denominator}\)

In numerical form:

• \(5 \frac{1}{4} = \frac{21}{4}\)
• \(1 \frac{3}{4} = \frac{7}{4}\)

This step is crucial as it eliminates the complexity of mixed numbers, streamlining the multiplication or division processes in fractions.

Simplifying Fractions for Clarity

Simplifying fractions is a fundamental process in mathematics that helps in reducing fractions to their simplest form. This makes the numbers easier to understand and work with.In the context of our exercise, after converting mixed numbers to improper fractions and expressing them as ratios, we found ourselves with the fraction \(\frac{21}{7}\).

To simplify, you find the greatest common divisor (GCD) of the numerator and denominator. You then divide both by this number to arrive at the simplest fraction.

Steps to Simplification:

• Identify the GCD of the numerator and denominator; in this case, both 21 and 7 are divisible by 7.
• Divide both the numerator and the denominator by this GCD.
• \(\frac{21}{7} = \frac{21 \div 7}{7 \div 7} = 3\).

These steps ensure that no smaller fraction can be derived from the original number, providing clarity in mathematical operations and in understanding the pure relationship between numbers. Simplifying fractions is vital in not just arithmetic but also in algebra, statistics, and beyond!