Problem 13 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$\frac{6}{5} \text { to } \frac{6}{7}$$

Step-by-Step Solution

Answer

The ratio \( \frac{6}{5} \) to \( \frac{6}{7} \) is \( \frac{7}{5} \) in its lowest terms.

Step 1: Understand the Ratio

The given problem asks us to find the ratio between two fractions: \( \frac{6}{5} \) and \( \frac{6}{7} \). This means we need to divide the first fraction by the second fraction.

Step 2: Divide the Fractions

To find the ratio of \( \frac{6}{5} \) to \( \frac{6}{7} \), we perform the division \( \frac{6}{5} \div \frac{6}{7} \). Recall that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we rewrite this as: \[ \frac{6}{5} \times \frac{7}{6} \]

Step 3: Simplify the Multiplication

Carry out the multiplication step: \( \frac{6}{5} \times \frac{7}{6} \) is computed as \[ \frac{6 \times 7}{5 \times 6} = \frac{42}{30} \]

Step 4: Reduce to Lowest Terms

Simplify the fraction \( \frac{42}{30} \) by finding the greatest common divisor (GCD) of 42 and 30, which is 6. Divide both the numerator and the denominator by their GCD: \[ \frac{42 \div 6}{30 \div 6} = \frac{7}{5} \]

Step 5: Confirm the Final Fraction

The fraction \( \frac{7}{5} \) is already in its lowest terms, as there is no number other than 1 that can evenly divide both 7 and 5. Thus, the final answer is confirmed.

Key Concepts

Understanding Ratios

Ratios are a way to compare two quantities. They are often expressed as fractions or with a colon (e.g., 3:4). In our exercise, we are dealing with a ratio of two fractions, which requires us to divide one fraction by another. This is the critical step because converting ratios of fractions into a simpler form often involves multiplication and not just simple comparison. To find the ratio of two fractions, you divide the first fraction by the second. This was done in the solution where \( \frac{6}{5} \) was divided by \( \frac{6}{7} \), which translated mathematically to multiplication: \( \frac{6}{5} \times \frac{7}{6} \). Understanding that division can transform into multiplication by the reciprocal is key when working with fractional ratios.

Reduction to Lowest Terms

Reducing a fraction to its lowest terms means simplifying it as much as possible so that the numerator and denominator are as small as they can be while still maintaining the same value. After performing the multiplication in our exercise, we obtained \( \frac{42}{30} \). This fraction is not in its simplest form, because both the numerator and the denominator can be divided by a common factor. Finding that number, here known as the Greatest Common Divisor (GCD), is the next crucial step. Reducing fractions helps in simplifying equations and making calculations easier to manage.A fraction in lowest terms is one where the only common divisor of the numerator and denominator is 1. In this scenario, reducing \( \frac{42}{30} \) gave us \( \frac{7}{5} \), which could not be simplified further because 7 and 5 have no other common divisors.

Finding the Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest number that can divide both the numerator and the denominator of a fraction without leaving a remainder. It is critical for reducing fractions to their simplest or lowest terms. In our example with \( \frac{42}{30} \), we determined the GCD was 6. Both 42 and 30 could be divided by 6, making it the key number to simplify the fraction.To find the GCD, we can utilize several methods, such as listing out the factors or using the Euclidean algorithm. By dividing both the numerator and the denominator by their GCD, the fraction is reduced efficiently, leading to \( \frac{7}{5} \). Always check your result; if no further simplification can occur, then you've successfully found the GCD and simplified the fraction.