Problem 44 - Elementary Algebra

Question

For exercises 39-46, rewrite the percent as a fraction. Simplify the fraction into lowest terms. $$ 35 \% $$

Step-by-Step Solution

Answer

\[\frac{7}{20}\].

Step 1 - Convert Percent to Fraction

To convert a percent into a fraction, first write the percent as a fraction with a denominator of 100. Therefore, 35% becomes \[\frac{35}{100}\].

Step 2 - Simplify the Fraction

Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 35 and 100 is 5. Divide both the numerator and the denominator by their GCD. \[\frac{35 \div 5}{100 \div 5} = \frac{7}{20}\].

Step 3 - Final Simplified Fraction

The fraction \[\frac{7}{20}\] is now in its simplest form. So, 35% as a simplified fraction is \[\frac{7}{20}\].

Key Concepts

simplifying fractions

Simplifying fractions is the process of reducing a fraction to its simplest form. This involves dividing both the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD). When a fraction is simplified, it becomes easier to understand and work with. For instance, the fraction \(\frac{35}{100}\) was simplified to \(\frac{7}{20}\) in the exercise. This means both 35 and 100 were divided by 5, their GCD.
Steps to simplify fractions:

• Identify the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.
• The resulting fraction is in its simplest form.

Simplifying fractions helps in comparing, adding, subtracting, multiplying, and dividing fractions easily.

greatest common divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that can evenly divide two numbers. For example, in the exercise, the GCD of 35 and 100 is 5. This was determined because 5 is the highest number that can divide both 35 and 100 without leaving a remainder.
Here's how to find the GCD of two numbers:

• List the factors of each number.
• Identify the common factors.
• The largest number in the common factors list is the GCD.

The GCD is essential in simplifying fractions because it ensures the fraction is reduced to its simplest form.

fraction fundamentals

Fractions represent a part of a whole. They are written as \(\frac{a}{b}\), where 'a' is the numerator and 'b' is the denominator. In the exercise, the fraction \(\frac{35}{100}\) was simplified to \(\frac{7}{20}\). This is crucial in understanding and representing parts more efficiently.
Key points about fractions:

• The numerator represents how many parts you have.
• The denominator represents the total number of equal parts in the whole.
• Fractions can be converted to percentages, and vice versa, for comparison and various calculations.

Mastering fractions allows for better comprehension of ratios, proportions, and helps in solving more complex mathematical problems.