Problem 62 - Elementary Algebra

Question

If 4 out of 20 shirts are flannel shirts, find the percent of the shirts that are flannel shirts.

Step-by-Step Solution

Answer

20%

Step 1 - Identify the part and the whole

Determine the number of flannel shirts (part) and the total number of shirts (whole). In this case, there are 4 flannel shirts and 20 shirts in total.

Step 2 - Set up the fraction

Create a fraction representing the part out of the whole: \( \frac{4}{20} \).

Step 3 - Simplify the fraction

Simplify \( \frac{4}{20} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 4. \( \frac{4 \div 4}{20 \div 4} = \frac{1}{5} \).

Step 4 - Convert the fraction to a percentage

To convert \( \frac{1}{5} \) to a percentage, multiply by 100: \( \frac{1}{5} \times 100 = 20\% \).

Key Concepts

Fractions

Fractions are a way of expressing parts of a whole. They consist of two numbers, the numerator and the denominator. The numerator is the number of parts you have, and the denominator is the total number of equal parts the whole is divided into. For example, in the fraction \( \frac{4}{20} \), 4 is the numerator (the part), and 20 is the denominator (the whole).
In the original exercise about shirts, the fraction \( \frac{4}{20} \) tells us that 4 out of 20 shirts are flannel.

Simplifying Fractions

Simplifying fractions is the process of making the fraction as simple as possible by reducing the numerator and denominator to their smallest whole numbers while keeping the value the same. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD), which is the biggest number that exactly divides both.
From our example, \( \frac{4}{20} \), we can simplify it by dividing both 4 and 20 by their GCD, which is 4. So, \( \frac{4}{20} = \frac{4 \div 4}{20 \div 4} = \frac{1}{5} \). This simplified fraction is easier to work with and understand.

Percentage Conversion

Converting a fraction to a percentage makes it easier to understand and compare. A percentage represents a fraction out of 100. To convert a fraction to a percentage, we multiply the fraction by 100.
In our example, we simplified \( \frac{4}{20} \) to \( \frac{1}{5} \). To convert \( \frac{1}{5} \) to a percentage, we multiply by 100:
\( \frac{1}{5} \times 100 = 20\% \).
So, 20% of the shirts are flannel.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can divide both the numerator and the denominator of a fraction without leaving a remainder. Finding the GCD is essential for simplifying fractions.
To find the GCD of two numbers, list out their factors and choose the largest one they have in common.

• The factors of 4 are 1, 2, and 4.
• The factors of 20 are 1, 2, 4, 5, and 20.

So, the greatest common factor of 4 and 20 is 4. We can then use this to simplify the fraction \( \frac{4}{20} \) to \( \frac{1}{5} \).
This step simplifies calculations and helps in understanding proportions better.