Question

Which answer expresses the ratio 10 feet to 12 feet as a fraction in lowest terms? A. \(\frac{10}{12}\) B. \(\frac{12}{10}\) C. \(\frac{5}{6}\) D. \(\frac{6}{5}\)

Short answer

C. \( \frac{5}{6} \)

Step-by-step solution

- Write the Ratio as a Fraction

The given ratio is 10 feet to 12 feet. Express this ratio as a fraction: \ \ \ \( \frac{10}{12} \)

- Find the Greatest Common Divisor (GCD)

Identify the greatest common divisor (GCD) of the numerator (10) and the denominator (12). The GCD of 10 and 12 is 2.

- Divide Both Terms by the GCD

To simplify the fraction, divide both the numerator and the denominator by the GCD (2): \ \ \ \( \frac{10 \div 2}{12 \div 2} = \frac{5}{6} \)

- Select the Correct Option

Compare your simplified fraction \( \frac{5}{6} \) with the given options. The correct answer is C. \( \frac{5}{6} \).

Key concepts

greatest common divisor

Understanding the greatest common divisor (GCD) is a crucial aspect of simplifying fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. To find the GCD of 10 and 12, we list the factors of each number:

- Factors of 10: 1, 2, 5, 10
- Factors of 12: 1, 2, 3, 4, 6, 12

The greatest common factor here is 2, as it is the largest number that appears in both lists. By identifying the GCD, we can reduce fractions to their simplest form by dividing both the numerator and the denominator by this number. This step is essential for accurate fraction simplification.

fraction simplification

Simplifying a fraction means expressing it in the smallest possible terms. Once you have the original fraction and the GCD, the next step is to divide both the numerator and the denominator by the GCD. For the fraction \( \frac{10}{12} \), the GCD is 2. Dividing the numerator and the denominator by 2, we get:

\[ \frac{10 \div 2}{12 \div 2} = \frac{5}{6} \]

This process of reducing both parts of the fraction ensures that the fraction retains the same value but in its simplest form. Simplification is fundamental in mathematics since it makes the fraction easier to understand and work with in further calculations.

mathematical problem solving

Mathematical problem solving involves breaking down a problem into manageable steps. In our example, solving the problem of simplifying the ratio 10 feet to 12 feet involves several key steps. First, express the ratio as a fraction: \[ \frac{10}{12} \]

Next, determine the gcd of the numerator and the denominator, which is 2 in this case. Then, divide both terms by the GCD: \[ \frac{10 \div 2}{12 \div 2} = \frac{5}{6} \]

Finally, compare the simplified fraction with the provided options to select the correct answer. Following this structured problem-solving approach not only helps in understanding the individual steps but also demonstrates logical reasoning and the importance of each mathematical operation.