Simplifying Fractions: A Step-by-Step Guide to 12/70
This article provides a comprehensive guide to simplifying the fraction 12/70, explaining the concepts of fractions, greatest common divisors (GCD), and different methods for simplifying fractions. We will explore practical applications and alternative representations of the simplified fraction.
Understanding the Fraction 12/70
Fractions represent parts of a whole. The number on top is called the numerator (12 in this case), and the number on the bottom is the denominator (70). Simplifying a fraction means reducing it to its lowest terms, where the numerator and denominator have no common factors other than 1. This is done by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it.
To simplify a fraction, we find the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest number that divides both numbers without leaving a remainder. For example, let’s simplify 6/18. The GCD of 6 and 18 is 6. Dividing both the numerator and denominator by 6, we get 1/3.
Another example: Simplifying 15/25. The GCD of 15 and 25 is 5. Dividing both by 5 results in 3/5.
Finding the GCD can be done using methods like the Euclidean algorithm or prime factorization.
Finding the Greatest Common Divisor (GCD) of 12 and 70
We’ll use two methods to find the GCD of 12 and 70: the Euclidean algorithm and prime factorization.
Euclidean Algorithm: This method involves repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until the remainder is 0. The last non-zero remainder is the GCD.
- 70 ÷ 12 = 5 with a remainder of 10
- 12 ÷ 10 = 1 with a remainder of 2
- 10 ÷ 2 = 5 with a remainder of 0
The last non-zero remainder is 2, so the GCD of 12 and 70 is 2.
Prime Factorization: This method involves finding the prime factors of each number and identifying the common factors.
- Prime factorization of 12: 2 x 2 x 3
- Prime factorization of 70: 2 x 5 x 7
The only common prime factor is 2. Therefore, the GCD of 12 and 70 is 2.
Both methods yield the same result: the GCD of 12 and 70 is 2.
Simplifying the Fraction 12/70
Using the GCD of 2, we simplify 12/70 by dividing both the numerator and denominator by 2.
| Numerator | Denominator | GCD | Simplified Fraction |
|---|---|---|---|
| 12 | 70 | 2 | 6/35 |
Visual Representation: Imagine 12 out of 70 equal parts. By grouping these parts into pairs (since the GCD is 2), we end up with 6 groups of 2 out of 35 groups of 2. This visually represents the simplified fraction 6/35.
The simplified fraction is 6/35.
Practical Applications and Examples
Simplifying fractions is crucial in various real-world situations.
- Recipe scaling: If a recipe calls for 12 cups of flour and 70 cups of water, simplifying the ratio 12/70 to 6/35 makes it easier to scale the recipe down.
- Ratio analysis: In business, simplifying ratios helps in comparing performance across different periods or companies.
- Probability: Simplifying fractions helps to express probabilities in their simplest form.
Expressing fractions in their simplest form improves clarity and understanding, making calculations and comparisons simpler. It is especially important when dealing with ratios and proportions.
Alternative Representations
The simplified fraction 6/35 can be represented in several ways.
- Decimal: 6 ÷ 35 ≈ 0.1714
- Percentage: (6/35) x 100% ≈ 17.14%
Visual Representation: Imagine a rectangle divided into 35 equal parts. Shade 6 of those parts to visually represent the fraction 6/35.
Converting between fraction, decimal, and percentage forms is straightforward: divide the numerator by the denominator to get the decimal, and multiply the decimal by 100 to get the percentage. To convert from decimal or percentage back to a fraction, express the decimal or percentage as a fraction and simplify.
