Understanding the Tennis Ball Quantity Problem

This problem asks us to find the number of green tennis balls in a sports shop. We are given information about the relationships between the quantities of red, green, and white balls, and the total number of balls.

Let's define variables for the number of balls of each color:

• Let \(R\) be the number of red balls.
• Let \(W\) be the number of white balls.
• Let \(G\) be the number of green balls.

Setting Up Equations from the Problem Statement

We can translate the given information into mathematical equations:

1. The number of white balls is 60% more than the number of red balls. This means the number of white balls is 100% of red balls plus 60% of red balls, which is 160% of red balls. $$W = 160\% \text{ of } R$$ $$W = \frac{160}{100} \times R$$ $$W = 1.60 \times R$$
2. The number of green balls is 12.5% less than the number of white balls. This means the number of green balls is 100% of white balls minus 12.5% of white balls, which is 87.5% of white balls. $$G = (100\% - 12.5\%) \text{ of } W$$ $$G = 87.5\% \text{ of } W$$ $$G = \frac{87.5}{100} \times W$$ $$G = 0.875 \times W$$
3. The total number of balls is 120. $$R + G + W = 120$$

Solving for the Number of Each Color Ball

Now we have a system of equations. We need to find the value of \(G\). We can express \(G\) and \(W\) in terms of \(R\) and substitute them into the total number equation.

We know \(W = 1.60 \times R\). Substitute this into the equation for \(G\):

$$G = 0.875 \times W$$ $$G = 0.875 \times (1.60 \times R)$$

To calculate \(0.875 \times 1.60\): \(0.875\) can be written as the fraction \(\frac{875}{1000}\), which simplifies to \(\frac{7}{8}\). \(1.60\) can be written as \(\frac{160}{100}\), which simplifies to \(\frac{16}{10}\) or \(\frac{8}{5}\).

$$G = \frac{7}{8} \times \frac{8}{5} \times R$$ $$G = \frac{7 \times 8}{8 \times 5} \times R$$ $$G = \frac{7}{5} \times R$$ $$G = 1.4 \times R$$

So, we have the relationships in terms of \(R\):

• \(R = R\)
• \(W = 1.6 \times R\)
• \(G = 1.4 \times R\)

Now substitute these into the total number equation \(R + G + W = 120\):

$$R + (1.4 \times R) + (1.6 \times R) = 120$$

Combine the terms with \(R\):

$$(1 + 1.4 + 1.6) \times R = 120$$ $$4 \times R = 120$$

Solve for \(R\):

$$R = \frac{120}{4}$$ $$R = 30$$

Now that we have the number of red balls (\(R=30\)), we can find the number of white and green balls using the relationships we found:

• White balls \(W = 1.6 \times R = 1.6 \times 30 = 48\)
• Green balls \(G = 1.4 \times R = 1.4 \times 30 = 42\)

To verify, let's check if the total number of balls is 120:

$$R + G + W = 30 + 42 + 48 = 120$$

The total matches the given information, so our calculations are correct.

The number of green balls is 42.

Summary of Ball Quantities

Revision Table - Key Calculations

Additional Information - Percentage Increase and Decrease

Understanding percentage increase and decrease is crucial for problems like this.

• Percentage Increase: If a quantity is increased by \(x\%\), the new quantity is the original quantity plus \(x\%\) of the original quantity. Mathematically, New Quantity = Original Quantity + \(\frac{x}{100} \times\) Original Quantity = Original Quantity \(\times (1 + \frac{x}{100})\). In our problem, white balls are 60% more than red balls, so \(W = R \times (1 + \frac{60}{100}) = R \times (1 + 0.60) = 1.60 \times R\).
• Percentage Decrease: If a quantity is decreased by \(y\%\), the new quantity is the original quantity minus \(y\%\) of the original quantity. Mathematically, New Quantity = Original Quantity - \(\frac{y}{100} \times\) Original Quantity = Original Quantity \(\times (1 - \frac{y}{100})\). In our problem, green balls are 12.5% less than white balls, so \(G = W \times (1 - \frac{12.5}{100}) = W \times (1 - 0.125) = 0.875 \times W\).

Being comfortable converting percentages to decimals or fractions (like 12.5% = 0.125 = \(\frac{1}{8}\), so 87.5% = \(\frac{7}{8}\)) can simplify calculations.