Problem 52 - Elementary and Intermediate Algebra

Question

Simplify. $$ \frac{140}{350} $$

Step-by-Step Solution

Answer

The simplified form is \frac{2}{5}.

Step 1: Identify the Greatest Common Divisor (GCD)

Find the GCD of 140 and 350. Identify the largest number that divides both 140 and 350 without leaving a remainder.

Step 2: Divide Both Numerator and Denominator by the GCD

Divide 140 and 350 by the GCD found in Step 1. This will simplify the fraction.

Step 3: Write the Simplified Fraction

The result from Step 2 gives the simplified form of the fraction. Write down this simplified fraction.

Key Concepts

Greatest Common Divisor (GCD)

To simplify a fraction, we first need to find the Greatest Common Divisor, or GCD, of the numerator and the denominator. The GCD is the largest number that can exactly divide both the numerator and the denominator without leaving any remainder.
For example, in the fraction \(\frac{140}{350}\), we need to find the GCD of 140 and 350.
We can list the factors of each number:

• 140: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140
• 350: 1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175, 350

The largest common factor is 70. Hence, the GCD of 140 and 350 is 70.
This step is key because we use the GCD to simplify both the numerator and the denominator.

Numerator

The numerator is the top number in a fraction. It represents how many parts of a whole are being considered.
In the fraction \(\frac{140}{350}\), 140 is the numerator.
After identifying the GCD which is 70, we divide the numerator by 70.
So, \(\frac{140}{70} = 2\).
This gives us the new, simplified numerator. Understanding the numerator helps us see how many portions we're dealing with after simplification.

Denominator

The denominator is the bottom number in a fraction and it shows the total number of equal parts the whole is divided into.
In the fraction \(\frac{140}{350}\), 350 is the denominator.
Just like with the numerator, we divide the denominator by the GCD (70).
So, \(\frac{350}{70} = 5\).
Therefore, our new denominator is 5. This shows how many equal parts the whole is divided into after simplification. It's essential to understand the role of the denominator, as it helps in comprehending the fraction better.

Fraction Simplification

Fraction simplification means reducing the fraction to its simplest form. This involves dividing both the numerator and denominator by their GCD.
In our example \(\frac{140}{350}\), we found the GCD as 70. We then divided both the numerator (140) and the denominator (350) by 70.
This gives us the simplified fraction: \(\frac{2}{5}\).
The final answer, \(\frac{2}{5}\), is the simplest form of the original fraction because 2 and 5 don't have any common divisors other than 1. Reducing a fraction to its simplest form makes it easier to work with and understand, especially when comparing fractions or solving equations.