Problem 11 - Prealgebra

Question

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. $$\frac{25 m n}{65 n}$$

Step-by-Step Solution

Answer

The simplified form is \( \frac{5m}{13} \).

Step 1: Determine the Greatest Common Divisor

First, we need to find the greatest common divisor (GCD) of the numerator and the denominator, which are \(25mn\) and \(65n\). Examine the numerical coefficients 25 and 65; the gcd of 25 and 65 is 5. The variable \(n\) is present in both numerator and denominator, so it is also part of the gcd.

Step 2: Simplify the Fraction

Now divide both the numerator and denominator by their GCD. \[\frac{25mn}{65n} = \frac{25mn \div 5n}{65n \div 5n} = \frac{5m}{13}\] The variable \(n\) is cancelled out during this division.

Step 3: Confirm the Simplification

After simplifying, check if the result \( \frac{5m}{13} \) can be further reduced. The numbers 5 and 13 are both prime, and there are no common factors between them, confirming the fraction is now in its simplest form.

Key Concepts

Understanding the Greatest Common Divisor

When you want to simplify a fraction, finding the greatest common divisor (GCD) is a crucial first step. The GCD tells us the largest number that can evenly divide both the numerator and the denominator. For example, if you have an expression such as \( \frac{25mn}{65n} \), you want to determine the GCD of the numbers involved.

• Start by focusing on the numerical coefficients: In our case, 25 and 65. Break them down into their factors, and identify the greatest factor they share. Here, both can be divided by 5, which is the largest common factor.
• Consider any variables: Look at the variables, like 'n' in this example, and include them in the GCD if they appear in both numerator and denominator.

By dividing both parts of the fraction by the GCD, you effectively simplify it.

Roles of the Numerator and Denominator

In a fraction, the numerator and denominator play specific roles, much like characters in a play. The numerator is the top part of a fraction and indicates how many parts of a whole we have. The denominator, on the other hand, is the bottom part and shows into how many equal parts the whole is divided.
For instance, with the fraction \( \frac{25mn}{65n} \):

• The numerator is 25mn, which represents the parts we have.
• The denominator is 65n, representing the total number of possible parts.

Understanding this distinction helps in manipulating fractions, such as when simplifying them, because each term in the numerator can be simplified by what is in the denominator, especially when they share a common factor.

Exploring Prime Numbers

Prime numbers are the building blocks of all numbers; they are numbers greater than 1 that have no divisors other than 1 and themselves. This special property makes them very useful in simplifying fractions.
Knowing whether numbers are prime helps us understand if a fraction is fully simplified. After reducing the fraction \( \frac{25mn}{65n} \) to \( \frac{5m}{13} \), we see the numbers 5 and 13 involved.

• Check if these numbers are prime: Both 5 and 13 are indeed prime, meaning they do not share any divisors other than 1.
• Since no further division is possible, the fraction is in its simplest form.

Being aware of which numbers are prime and how they interact ensures that you can confidently state whether a fraction is simplified or needs more work.