John and Mary had some money in the ratio 11:8. After each of them spent $136 on a bicycle, the ratio of John's money to Mary's money became 3:2. How much money did John have at first?

To solve this problem, we need to set up a system of equations based on the information given about the initial and final ratios of money between John and Mary.

Let's denote:
- $ J $ as the initial amount of money John had.
- $ M $ as the initial amount of money Mary had.

### Step 1: Initial Ratio
According to the problem, the initial ratio of John's money to Mary's money is 11:8. Therefore, we can express this relationship as:
\[
\frac{J}{M} = \frac{11}{8}
\]
This implies:
\[
J = \frac{11}{8}M
\]

### Step 2: After Spending Money
After both John and Mary spent $136, the new ratio of John's money to Mary's money became 3:2. We can express the remaining money for John and Mary as:
- John's remaining money: $ J - 136 $
- Mary's remaining money: $ M - 136 $

Thus, the new ratio can be expressed as:
\[
\frac{J - 136}{M - 136} = \frac{3}{2}
\]

### Step 3: Solving the Equations
Now, we have the two equations:
1. $ J = \frac{11}{8}M $
2. $ \frac{J - 136}{M - 136} = \frac{3}{2} $

We can substitute $ J $ from the first equation into the second equation:
\[
\frac{\left(\frac{11}{8}M\right) - 136}{M - 136} = \frac{3}{2}
\]

Cross-multiplying to eliminate the fractions gives:
\[
2 \left(\frac{11}{8}M - 136\right) = 3(M - 136)
\]

Simplifying both sides:
\[
2 \times \frac{11}{8}M - 272 = 3M - 408
\]

Convert the terms to a common denominator or multiply through to eliminate fractions:
\[
\frac{22}{8}M - 272 = 3M - 408
\]

Converting $\frac{22}{8}$ to $2.75$ for simplicity:
\[
2.75M - 272 = 3M - 408
\]

### Step 4: Isolating $M$
Rearranging the equation to isolate $M$:
\[
2.75M - 3M = -408 + 272
\]

Simplifying:
\[
-0.25M = -136
\]

Solving for $M$:
\[
M = \frac{-136}{-0.25} = 544
\]

### Step 5: Calculating $J$
Now that we have $M$, we can substitute back to find $J$:
\[
J = \frac{11}{8} \times 544
\]

Calculating:
\[
J = 748
\]

Therefore, John had $748 initially.

### Conclusion
John initially had **$748**.

If you have any further questions or need a detailed explanation, feel free to ask! 

Here are some related questions to consider:
1. What would be the ratio if John and Mary spent different amounts?
2. How would the problem change if they saved money instead of spending it?
3. What if the initial ratio was different, say 2:1?
4. How would this scenario work if Mary spent more than John?
5. Can you derive a general formula for such ratio problems?
6. What happens if the ratio of remaining money becomes 1:1?
7. What if they borrowed money instead of spending it?
8. How would the calculations differ if the amounts spent were percentages?

**Tip:** When solving ratio problems, expressing variables in terms of one another often simplifies the calculation.

Learn how to solve a ratio problem involving John and Mary's initial money ratio, where the ratio changes after expenditures. Detailed step-by-step solution provided, covering ratio and proportion concepts and linear equations. Suitable for students in Grades 7-9.

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