the original numbers of badges owned by Edwin and Janson were in the ratio of 2:3. If Janson gave 4 badges to Edwin, the ratio becomes 8:7. How many badges did each of them has originally?

Let's solve the problem step by step.

### Step 1: Set up the original ratio
Let the original number of badges owned by Edwin be \(2x\) and the original number of badges owned by Janson be \(3x\). The ratio of badges between Edwin and Janson is given as \(2:3\).

### Step 2: Set up the equation after the exchange
After Janson gives 4 badges to Edwin, the number of badges Edwin has becomes \(2x + 4\), and the number of badges Janson has becomes \(3x - 4\). The new ratio of badges between Edwin and Janson is given as \(8:7\).

We can write this as:
\[
\frac{2x + 4}{3x - 4} = \frac{8}{7}
\]

### Step 3: Solve the equation
Cross-multiply to eliminate the fractions:
\[
7(2x + 4) = 8(3x - 4)
\]

Expand both sides:
\[
14x + 28 = 24x - 32
\]

### Step 4: Simplify and solve for \(x\)
Move all terms involving \(x\) to one side:
\[
14x - 24x = -32 - 28
\]
\[
-10x = -60
\]

Divide both sides by \(-10\):
\[
x = 6
\]

### Step 5: Find the original numbers of badges
Now that we have \(x = 6\), we can find the original number of badges each person had:

- Edwin's original number of badges: \(2x = 2(6) = 12\)
- Janson's original number of badges: \(3x = 3(6) = 18\)

### Final Answer
Edwin originally had **12 badges**, and Janson originally had **18 badges**.

Would you like any further details or have any questions?

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Here are 8 related questions to explore further:
1. What would happen if Janson gave more than 4 badges to Edwin?
2. How would the original ratio change if Edwin had given badges to Janson instead?
3. What if the new ratio was 7:8 instead of 8:7?
4. Can you set up a similar problem with a different original ratio?
5. How would the problem change if the total number of badges was provided?
6. What if Edwin and Janson started with an equal number of badges?
7. How can we check if the solution is correct after finding the numbers?
8. What if Janson gave badges to someone else instead of Edwin?

**Tip:** Always cross-multiply to eliminate fractions when working with ratios in equations!

Learn how to solve a problem involving ratios and badge exchanges between Edwin and Janson. Understand the step-by-step solution where the original ratio was 2:3, changing to 8:7 after an exchange of badges. Suitable for Grades 7-9 students learning about ratios and equations.

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