Problem 64 - Elementary Algebra

Question

If 6 out of 200 bills are overdue, find the percent of the bills that are overdue.

Step-by-Step Solution

Answer

3 percent of the bills are overdue.

Step 1: Identify the given values

The problem provides two values: the number of overdue bills, which is 6, and the total number of bills, which is 200.

Step 2: Set up the ratio

To find the percentage, set up the ratio of overdue bills to total bills. This ratio is \( \frac{6}{200} \).

Step 3: Convert the ratio to a fraction

The ratio \( \frac{6}{200} \) can be simplified by dividing both the numerator and the denominator by 2, resulting in \( \frac{3}{100} \).

Step 4: Convert the fraction to a percentage

To convert the fraction to a percentage, multiply it by 100: \( \frac{3}{100} \times 100 = 3\% \). This means 3 percent of the bills are overdue.

Key Concepts

Ratio

Ratios are a way to compare two quantities by showing the relative size of one quantity to the other. In our exercise, we compare the number of overdue bills to the total number of bills.
To begin, we set up a ratio to represent this comparison. The ratio of overdue bills to total bills is given by:

• Number of overdue bills: 6
• Total number of bills: 200

The ratio is written as \(\frac{6}{200}\). This simply expresses that there are 6 overdue bills out of every 200 bills.

Fraction Simplification

Simplifying fractions makes them easier to understand and work with. In fraction simplification, we find the greatest common divisor (GCD) of the numerator and the denominator.
Let's simplify \(\frac{6}{200}\):

• Find the GCD of 6 and 200, which is 2.
• Divide both the numerator and the denominator by their GCD: \(\frac{6 ÷ 2}{200 ÷ 2} = \frac{3}{100}\).

The simplified fraction is \(\frac{3}{100}\). This means 3 out of every 100 bills are overdue. By simplifying, we make the fraction easier to interpret.

Percentage Conversion

Percentages help us understand ratios and fractions in terms of 'per hundred'. For conversion, we multiply the fraction by 100.
To convert \(\frac{3}{100}\) to a percentage, follow these steps:

• Multiply the fraction by 100: \(\frac{3}{100} \times 100\).
• This simplifies as follows: \(\frac{3 \times 100}{100} = 3\text{\text{%}}\).

This means that 3% of the total bills are overdue. Converting to percentage makes the proportion easier to grasp.

Basic Arithmetic

Let's go over the essential arithmetic used in our exercise. Basic arithmetic operations include addition, subtraction, multiplication, and division.
Here, we used:

• Division to simplify the fraction: Dividing 6 by 2 and 200 by 2 to get \(\frac{3}{100}\).
• Multiplication to convert the fraction to a percentage: Multiplying \(\frac{3}{100}\) by 100 to get 3%.

These simple steps show how basic arithmetic plays a role in solving everyday problems. By practicing these operations, you can tackle similar exercises confidently.