Problem 51 - Algebra for College Students

Question

A jar of 20 coins contains 4 pennies. What percent of the coins are pennies?

Step-by-Step Solution

Answer

20%

Step 1 - Identify Total Number of Coins

Determine the total number of coins in the jar. The problem states that there are 20 coins in total.

Step 2 - Identify Number of Pennies

Determine the number of pennies in the jar. The problem states that there are 4 pennies.

Step 3 - Set Up the Fraction

Set up the fraction of pennies over the total number of coins: \ \[ \frac{4}{20} \]

Step 4 - Simplify the Fraction

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor. \ \[ \frac{4 \div 4}{20 \div 4} = \frac{1}{5} \]

Step 5 - Convert Fraction to Percent

Multiply the simplified fraction by 100 to find the percentage. \ \[ \frac{1}{5} \times 100 = 20\% \]

Key Concepts

fractions

Fractions are a way to represent parts of a whole. They consist of two numbers: a numerator and a denominator.
The numerator is the top number and indicates how many parts we have.
The denominator is the bottom number and shows how many equal parts the whole is divided into.
In the exercise, the fraction is set up as \(\frac{4}{20}\), where 4 (pennies) is the numerator and 20 (total coins) is the denominator.
Fractions are essential because they allow us to describe proportions and make comparisons easily.

simplifying fractions

Simplifying fractions means reducing them to their simplest form. This involves finding the greatest common divisor (GCD) of both the numerator and the denominator and then dividing both by this number.
In our example, the fraction is \(\frac{4}{20}\). First, find the GCD of 4 and 20, which is 4.
Next, divide both the numerator and the denominator by the GCD:
\[ \frac{4}{20} = \frac{4 \div 4}{20 \div 4} = \frac{1}{5} \]
This means that \(\frac{4}{20}\) simplifies to \(\frac{1}{5}\), making it easier to work with.

percentages

Percentages are a way to express a number as a fraction of 100. They help us understand ratios and proportions.
To convert a simplified fraction to a percentage, you multiply it by 100. In our case, the simplified fraction is \(\frac{1}{5}\).
To find the percentage, we perform this calculation:
\[ \frac{1}{5} \times 100 = 20\% \]
So, 20% of the coins in the jar are pennies. Percentages are very useful for quickly comparing different quantities and understanding their relationships.