Problem 24 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$6.4 \text { to } 0.8$$

Step-by-Step Solution

Answer

The ratio 6.4 to 0.8 simplifies to 8.

Step 1: Convert to Fraction

First, write the ratio 6.4 to 0.8 as a fraction. This ratio can be expressed as \( \frac{6.4}{0.8} \).

Step 2: Eliminate Decimals

Multiply both the numerator and the denominator by 10 to get rid of the decimal points: \( \frac{6.4 \times 10}{0.8 \times 10} = \frac{64}{8} \).

Step 3: Simplify the Fraction

Divide the numerator and the denominator by their greatest common divisor (GCD). The GCD of 64 and 8 is 8, so \( \frac{64 \div 8}{8 \div 8} = \frac{8}{1} \).

Step 4: Write as Final Simplified Fraction

The fraction \( \frac{8}{1} \) simplifies further to just \( 8 \), since dividing by 1 doesn't change the value.

Key Concepts

Simplifying Fractions

Simplifying fractions is a key concept in understanding ratios and proportions. It involves reducing a fraction to its simplest form, where the numerator and the denominator have no common factors other than 1.

• To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).
• The GCD is the largest number that evenly divides both numbers.

Simplifying helps in comparing fractions more easily. For example, when the fraction \( \frac{64}{8} \) is simplified, we find the GCD, which in this case is 8. Dividing both the numerator and the denominator by 8 results in \( \frac{8}{1} \).
Fractions are considered simplest when the numerator and the denominator are coprime, which means they have no common divisors other than 1. This allows for easy calculations and comparisons within mathematical problems.

Greatest Common Divisor

Finding the greatest common divisor (GCD) is crucial when working with fractions. The GCD is the largest integer that divides both numbers without leaving a remainder.

• There are several methods to find the GCD. One common method is the Euclidean algorithm, which involves repeated division until a remainder of zero is reached.
• For smaller numbers, listing the factors of each number and identifying the largest shared factor can also work well.

Let's consider the numbers 64 and 8 from our problem. The GCD here is 8 because 8 divides both 64 and 8 without a remainder. Knowing how to calculate the GCD is essential because it allows you to simplify fractions, which makes solving problems much easier.

Converting Decimals to Fractions

Converting decimals to fractions is a foundational skill in mathematics, especially when dealing with ratios and percentages. Understanding the conversion process helps in working seamlessly between different forms of numbers.

• To convert a decimal to a fraction, identify the place value of the last digit. For example, in the decimal 0.8, the 8 is in the tenths place, so it can be rewritten as \( \frac{8}{10} \).
• Once converted, simplify the fraction to its lowest terms by dividing both the numerator and the denominator by their GCD. In this case, the GCD of 8 and 10 is 2, simplifying \( \frac{8}{10} \) to \( \frac{4}{5} \).

In the case of our original problem 6.4 to 0.8, both decimal numbers were first multiplied by 10 to eliminate the decimal point, creating the fraction \( \frac{64}{8} \). From this step, they can be simplified as described earlier. Learning how to accurately convert decimals and subsequently simplify them is an essential math skill that builds a solid foundation for more advanced concepts.