Understanding the Constant Ratio of Balls

The problem states that the ratio of the number of blue balls to the number of red balls in a bag is constant. This means that no matter how the total number of balls changes, the relationship between the quantities of blue and red balls remains the same. We are given information about two different scenarios and asked to find an unknown quantity in the second scenario.

Calculating the Initial Ratio

In the first scenario, we are told there are 44 red balls and 36 blue balls. The ratio of blue balls to red balls is given by:

\(\text{Ratio} = \frac{\text{Number of Blue Balls}}{\text{Number of Red Balls}}\)

Substituting the given values:

\(\text{Ratio} = \frac{36}{44}\)

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

\(\text{Ratio} = \frac{36 \div 4}{44 \div 4} = \frac{9}{11}\)

So, the constant ratio of blue balls to red balls is \(9:11\).

Setting up the Proportion for the Second Scenario

In the second scenario, we are told that the number of blue balls is 54. We need to find the number of red balls, let's call this \(R\). Since the ratio of blue balls to red balls is constant and is equal to \(9/11\), we can set up a proportion:

\(\frac{\text{Number of Blue Balls}}{\text{Number of Red Balls}} = \frac{9}{11}\)

\(\frac{54}{R} = \frac{9}{11}\)

Solving the Proportion to Find the Number of Red Balls

To solve for \(R\), we can cross-multiply:

\(54 \times 11 = 9 \times R\)

\(594 = 9R\)

Now, we isolate \(R\) by dividing both sides of the equation by 9:

\(R = \frac{594}{9}\)

Performing the division:

\(594 \div 9\)

• Divide 59 by 9: \(59 \div 9 = 6\) with a remainder of \(5\).
• Bring down the 4 to get 54.
• Divide 54 by 9: \(54 \div 9 = 6\).

So, \(R = 66\).

Therefore, when there are 54 blue balls, there will be 66 red balls in the bag, maintaining the constant ratio of \(9:11\).

Checking the Ratio

If there are 54 blue balls and 66 red balls, the ratio is:

\(\frac{54}{66}\)

Dividing both by their greatest common divisor, 6:

\(\frac{54 \div 6}{66 \div 6} = \frac{9}{11}\)

This confirms that the ratio is indeed constant.

Summary of Steps

1. Identify the given quantities for the initial scenario: 36 blue balls, 44 red balls.
2. Calculate and simplify the initial ratio of blue to red balls: \(36/44 = 9/11\).
3. Set up a proportion using the constant ratio and the given quantity in the second scenario (54 blue balls), with the unknown quantity \(R\) for red balls: \(54/R = 9/11\).
4. Solve the proportion for \(R\) using cross-multiplication: \(54 \times 11 = 9R\), which leads to \(594 = 9R\).
5. Calculate \(R\) by dividing: \(R = 594/9 = 66\).

The number of red balls will be 66.

Revision Table: Key Concepts

Additional Information: Proportional Relationships

This problem illustrates a proportional relationship. When the ratio between two quantities is constant, they are directly proportional. If the number of blue balls increases, the number of red balls must also increase proportionally to maintain the same ratio. In this case, the number of blue balls increased from 36 to 54. The scaling factor is \(54/36\).

\(\text{Scaling Factor} = \frac{54}{36} = \frac{3 \times 18}{2 \times 18} = \frac{3}{2}\)

Since the number of blue balls was multiplied by \(3/2\), the number of red balls must also be multiplied by the same factor to keep the ratio constant:

\(\text{New Red Balls} = \text{Original Red Balls} \times \text{Scaling Factor}\)

\(\text{New Red Balls} = 44 \times \frac{3}{2}\)

\(\text{New Red Balls} = \frac{44 \times 3}{2} = \frac{132}{2} = 66\)

This alternative method using the scaling factor confirms the result obtained by setting up the proportion. Both methods demonstrate the fundamental principle of constant ratios and proportional relationships.