Problem 75 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{16}{70}$$

Step-by-Step Solution

Answer

The simplified fraction is \( \frac{8}{35} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce the fraction \( \frac{16}{70} \), we first need to find the greatest common divisor (GCD) of the numerator (16) and the denominator (70). The factors of 16 are 1, 2, 4, 8, 16 and the factors of 70 are 1, 2, 5, 7, 10, 14, 35, 70. The largest factor that 16 and 70 have in common is 2. Therefore, the GCD of 16 and 70 is 2.

Step 2: Divide by the GCD

Now that we know the GCD is 2, we divide both the numerator and the denominator by this number: \[ \frac{16}{70} = \frac{16 \div 2}{70 \div 2} = \frac{8}{35} \]

Step 3: Check for Further Reduction

After simplifying the fraction to \( \frac{8}{35} \), we check if there are any more common factors between 8 and 35 other than 1. The factors of 8 are 1, 2, 4, 8, and the factors of 35 are 1, 5, 7, 35. Since there are no common factors apart from 1, the fraction \( \frac{8}{35} \) is already in its simplest form.

Key Concepts

Greatest Common Divisor (GCD)

The Greatest Common Divisor, commonly known as the GCD, is an important concept in mathematics, especially when it comes to simplifying fractions. It refers to the largest number that divides two or more numbers without leaving a remainder.
To find the GCD of two numbers, you must first list all the factors of each number. For example, in the fraction \( \frac{16}{70} \), the factors of 16 are 1, 2, 4, 8, and 16. Meanwhile, the factors of 70 are 1, 2, 5, 7, 10, 14, 35, and 70.
Once you have these lists, you look for the highest number present in both lists. In our example, the common factors are 1 and 2, with 2 being the largest. Hence, the GCD of 16 and 70 is 2.
Identifying the GCD is a crucial step in reducing fractions, as it tells us the maximum we can divide both the numerator and the denominator to simplify the fraction.

Simplifying Fractions

Simplifying fractions is all about making them easier to understand without changing their value. This involves using the Greatest Common Divisor (GCD). Once you have the GCD, you can reduce the fraction by dividing both the numerator and the denominator by this number.
Take the fraction \( \frac{16}{70} \) for example. Once we've established that the GCD is 2, we divide 16 and 70 by this number.
- Divide 16 by 2 to get 8.
- Divide 70 by 2 to get 35.
Therefore, \( \frac{16}{70} \) simplifies to \( \frac{8}{35} \). The fraction is now in its simplest form, meaning no further division can simplify it further without changing its value.
Simplifying fractions makes them easier to work with, especially in mathematical equations and comparisons.

Factors of Numbers

Understanding factors is essential for multiple areas of math, including simplifying fractions. A factor is a number that divides another number entirely, without leaving any remainder.
To determine the factors of a number, you evaluate which numbers can be multiplied together to reach the original number. For instance, the factors of 16 include 1, 2, 4, 8, and 16, since all these numbers can completely divide 16.
Similarly, the factors of 70 are identified as 1, 2, 5, 7, 10, 14, 35, and 70. Each of these numbers divides 70 precisely.
Knowing how to find factors is vital, as it directly aids in calculating the Greatest Common Divisor (GCD), which is crucial when simplifying fractions or even in more advanced math tasks.