There are red counters and black counters in a bag in the ratio 3:1. 9 red counters are removed and 5 black counters are added into the bag, and the ratio becomes 5:3. Work out the original number of black counters in the bag.

To solve the problem, let's set up the equation using the information provided.

### Step 1: Define Variables
- Let the original number of red counters be \( 3x \).
- Let the original number of black counters be \( x \).

### Step 2: Set Up the Initial Ratio
The initial ratio of red to black counters is given as 3:1, so:
\[
\text{Number of red counters} = 3x
\]
\[
\text{Number of black counters} = x
\]

### Step 3: Account for Changes in Counters
- 9 red counters are removed, so the new number of red counters is \( 3x - 9 \).
- 5 black counters are added, so the new number of black counters is \( x + 5 \).

### Step 4: Set Up the New Ratio
After these changes, the ratio becomes 5:3. Therefore:
\[
\frac{3x - 9}{x + 5} = \frac{5}{3}
\]

### Step 5: Solve the Equation
Cross-multiply to solve for \( x \):
\[
3(3x - 9) = 5(x + 5)
\]
Expanding both sides:
\[
9x - 27 = 5x + 25
\]
Subtract \( 5x \) from both sides:
\[
4x - 27 = 25
\]
Add 27 to both sides:
\[
4x = 52
\]
Divide by 4:
\[
x = 13
\]

### Step 6: Find the Number of Black Counters
The original number of black counters is \( x \), so the original number of black counters is 13.

### Final Answer:
The original number of black counters in the bag was **13**.

Would you like further details or have any other questions?

Here are five related questions to consider:
1. How would the problem change if only 4 red counters were removed?
2. What if instead of adding, the black counters were removed?
3. How would the solution differ if the initial ratio was 2:1 instead of 3:1?
4. How can we verify that the new ratio 5:3 is correct?
5. What if the total number of counters in the bag was given instead?

**Tip:** Always double-check your work by substituting the solution back into the original problem to ensure it satisfies all conditions.

Explore how to determine the original number of black counters in a bag using ratio and proportion concepts. This problem involves setting up and solving equations to find the initial count of black counters after a series of changes. Suitable for Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation