Problem 73 - Elementary Algebra

Question

Divide and reduce to lowest terms. $$ 5 \div 235 $$

Step-by-Step Solution

Answer

The result is \( \frac{1}{47} \).

Step 1: Convert Division to Fraction

Start by rewriting the division of 5 by 235 as a fraction: \( \frac{5}{235} \). This makes it easier to reduce the fraction to its lowest terms.

Step 2: Find the Greatest Common Divisor (GCD)

To simplify the fraction \( \frac{5}{235} \), find the greatest common divisor (GCD) of the numerator and the denominator. Here, the GCD of 5 and 235 is 5 since it is the only prime factor that divides both numbers.

Step 3: Divide by the GCD

Divide both the numerator and the denominator of the fraction \( \frac{5}{235} \) by their GCD, which is 5. This gives you a simplified fraction of \( \frac{5 \div 5}{235 \div 5} = \frac{1}{47} \).

Step 4: Review and Confirm Solution

Ensure that the fraction \( \frac{1}{47} \) is simplified to its lowest terms. This is because 1 as the numerator means no further reduction is possible. Also, 47 is a prime number, confirming that 1 and 47 have no common factors.

Key Concepts

Understanding the Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is a crucial concept when it comes to simplifying fractions. Essentially, the GCD of two numbers is the largest number that divides both of them without leaving a remainder. Knowing how to find the GCD can significantly ease fraction simplification.

• To find the GCD, list the factors of each number.
• Identify the largest factor common to both lists.

In the example of simplifying \( \frac{5}{235} \), the GCD is 5. Since 5 is a factor of itself and the only number that divides 235 and 5, it simplifies the process. Once you have the GCD, you can simplify the fraction by dividing both the numerator and the denominator by this GCD.
Whenever you're working with fractions, starting by finding the GCD can make the process smoother and lead directly to the fraction's simplest form.

Exploring Prime Numbers

Prime numbers are fundamental in understanding fractions and the process of simplification. A prime number is any number greater than 1 that cannot be formed by multiplying two smaller natural numbers. This means that primes have only two positive divisors: 1 and the number itself.

• Common prime numbers include 2, 3, 5, 7, 11, etc.
• They play a key role in identifying GCDs because if a number is a prime factor of both the numerator and the denominator, it divides both without a remainder.

In the fraction \( \frac{1}{47} \), 47 is a prime number. This confirms that \( \frac{1}{47} \) cannot be reduced further because no number other than 1 can divide both 1 and 47. Recognizing prime numbers is a useful skill in determining when a fraction is at its simplest form.

Understanding Division as Fractions

Division and fractions are closely related concepts. In many cases, division problems are better understood and solved by expressing them as fractions. This is a valuable strategy for simplifying mathematical expressions.

• For example, a division of 5 by 235 can be expressed as the fraction \( \frac{5}{235} \).
• This form allows for a straightforward approach to simplification by utilizing the GCD, as shown previously.

Once the division is expressed as a fraction, the next step typically involves finding the GCD, if any, to reduce the fraction to its simplest form. This approach simplifies complex division problems and provides clearer visualization of the mathematical relationship between the numbers involved. By transforming division into fractions, one gains both clarity and a simpler path to solution.