Problem 67 - Prealgebra

Question

Perform the indicated operations. $$256 \div 640$$

Step-by-Step Solution

Answer

256 divided by 640 is 0.4.

Step 1: Simplify the Division

First, recognize that dividing 256 by 640 can also be expressed as a fraction: \( \frac{256}{640} \). To simplify this fraction, divide both the numerator and the denominator by their greatest common divisor (GCD), which is 64.

Step 2: Calculate the GCD

To find the GCD of 256 and 640, use the Euclidean algorithm or inspect the factors of each number. The factors of 256 are: 1, 2, 4, 8, 16, 32, 64, 128, 256, and for 640: 1, 2, 4, 5, 8, 10, 16, 32, 40, 64, 80, 160, 320, 640. The greatest common factor is 64.

Step 3: Divide by the GCD

Divide both the numerator and the denominator by 64: \( \frac{256 \div 64}{640 \div 64} = \frac{4}{10} \).

Step 4: Further Simplify the Fraction

The fraction \( \frac{4}{10} \) can still be simplified. The GCD of 4 and 10 is 2. Divide both the numerator and the denominator by 2: \( \frac{4 \div 2}{10 \div 2} = \frac{2}{5} \).

Step 5: Convert to Decimal Form

Now that we have simplified the fraction to \( \frac{2}{5} \), convert it to decimal form for clarity in division: \( \frac{2}{5} = 0.4 \).

Key Concepts

Understanding the Greatest Common Divisor (GCD)

The Greatest Common Divisor, or GCD, is the largest number that can evenly divide two or more integers. Finding it helps in simplifying fractions. To determine the GCD, you first list all of the factors of each number. Factors are integers that divide the number without leaving a remainder.
Suppose we have two numbers, like in the exercise: 256 and 640. We start by listing their factors:

• Factors of 256: 1, 2, 4, 8, 16, 32, 64, 128, 256
• Factors of 640: 1, 2, 4, 5, 8, 10, 16, 32, 40, 64, 80, 160, 320, 640

The largest number that appears in both lists is 64. So, the GCD is 64. Simplifying fractions with the GCD makes further calculations, like converting fractions to decimals, easier.

Fraction to Decimal Conversion

Converting fractions to decimals can make them easier to understand and compare. A fraction like \( \frac{2}{5} \) represents a division problem, where you divide 2 by 5. Performing this division gives 0.4.Why is this useful? Fractions give a lot of information about proportions but can be hard to visualize in calculations or when comparing different numbers. Decimals are often simpler to interpret in day-to-day situations like measuring ingredients in recipes or calculating exact change.For example, \( \frac{2}{5} = 0.4 \) shows that two-fifths is 0.4 of a whole, making it clearer in a continuous context.

Utilizing the Euclidean Algorithm

The Euclidean Algorithm is a powerful method for finding the Greatest Common Divisor (GCD) of two numbers. It works through a process of repeated division. In simple terms, you divide the larger number by the smaller number and then use the remainder to repeat the step.
Here's how it looks with our numbers, 256 and 640:

• Divide 640 by 256, which results in a remainder of 128.
• Next, divide the previous divisor (256) by this remainder (128), which gives a remainder of 0. Since the remainder is now 0, the divisor of this step, which is 128, is the GCD.

By employing the Euclidean Algorithm, you can determine the GCD efficiently, even with very large numbers.