Understanding the Ratio Problem

The problem describes a bag containing red and green balls, initially in a specific ratio. When the number of green balls changes, the ratio changes. We need to find the original number of red balls.

Setting up the Initial Quantities

The initial ratio of red balls to green balls is given as 4 ∶ 9. This means that for every 4 red balls, there are 9 green balls. We can represent the number of balls using a variable. Let the common multiplier be \(x\).

• Number of red balls = \(4x\)
• Number of green balls = \(9x\)

Analyzing the Change

The problem states that 6 more green balls were added to the bag. The number of red balls remains unchanged, but the number of green balls increases.

• New number of red balls = \(4x\)
• New number of green balls = \(9x + 6\)

Formulating the New Ratio Equation

After adding the green balls, the ratio of red balls to green balls becomes 1 ∶ 3. We can set up an equation based on this new ratio:

$$ \frac{\text{Number of red balls}}{\text{Number of green balls}} = \frac{1}{3} $$

Substituting the expressions for the new number of balls:

$$ \frac{4x}{9x + 6} = \frac{1}{3} $$

Solving the Equation for the Unknown

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

$$ 3 \times (4x) = 1 \times (9x + 6) $$

$$ 12x = 9x + 6 $$

Now, we isolate the term with \(x\) by subtracting \(9x\) from both sides:

$$ 12x - 9x = 6 $$

$$ 3x = 6 $$

Finally, divide by 3 to find the value of \(x\):

$$ x = \frac{6}{3} $$

$$ x = 2 $$

Calculating the Number of Red Balls

The question asks for the number of red balls in the bag. From our initial setup, the number of red balls is \(4x\). Now that we have found \(x = 2\), we can calculate the number of red balls:

Number of red balls = \(4 \times x = 4 \times 2 = 8\)

So, there are 8 red balls in the bag.

Verification of the Solution

Let's check if this number of red balls (8) gives the correct ratios:

• Initial red balls = 8
• Initial green balls (using \(x=2\)) = \(9x = 9 \times 2 = 18\)
• Initial ratio of red to green = 8 : 18. Dividing both by 2, we get 4 : 9. This matches the given initial ratio.
• After adding 6 green balls, the new number of green balls = \(18 + 6 = 24\).
• New ratio of red to green = 8 : 24. Dividing both by 8, we get 1 : 3. This matches the given new ratio.

The numbers fit the conditions given in the problem, confirming our answer is correct.

Revision Table: Ratio Problem Key Steps

Additional Information: Working with Ratios

A ratio is a comparison of two quantities. It can be written in different ways, such as \(a:b\), \(\frac{a}{b}\), or "a to b". When solving problems involving changing ratios:

• Represent the initial quantities as multiples of a variable (e.g., \(ax\) and \(bx\) for a ratio \(a:b\)).
• Adjust the quantities based on the changes described in the problem (adding or subtracting).
• Set up a new ratio using the adjusted quantities and the new given ratio.
• Solve the resulting equation (often a proportion) to find the value of the variable.
• Use the variable's value to find the actual quantities asked for in the question.

Ratio problems often involve setting up and solving linear equations or proportions, which is a fundamental skill in algebra and quantitative aptitude.