Rizwan has a box containing red and blue marbles. Initially, the ratio of red marbles to blue marbles is 5 : 4. After he loses 5 red marbles, the ratio becomes 10 : 9. How many marbles does he have now in total?

Introduction / Context:
This is a typical ratio word problem involving an initial ratio, a change in one quantity, and a new ratio. Rizwan has a mix of red and blue marbles with a known initial ratio. After losing some red marbles, the ratio changes, and we are asked to find the final total number of marbles. The key is to translate the ratio information into algebraic equations and solve for the original numbers of marbles, then adjust for the loss.

Given Data / Assumptions:
• Initial ratio of red to blue marbles = 5 : 4. • Rizwan loses 5 red marbles. • After losing 5 red marbles, the new ratio of red to blue marbles becomes 10 : 9. • We must find the final total number of marbles after the loss.

Concept / Approach:
We start by assigning variables based on the initial ratio. Let the number of red marbles initially be 5k and the number of blue marbles be 4k. After the loss of 5 red marbles, the red marbles become 5k − 5, while blue marbles remain 4k. We are told that the new ratio (5k − 5) : 4k is 10 : 9. From this new ratio, we can set up an equation and solve for k. Once k is known, we can compute the original and final totals.

Step-by-Step Solution:
Step 1: Let initial red marbles = 5k and blue marbles = 4k. Step 2: After losing 5 red marbles, red marbles = 5k − 5, blue marbles remain 4k. Step 3: New ratio is red : blue = 10 : 9. So (5k − 5) / (4k) = 10 / 9. Step 4: Cross multiply: 9 * (5k − 5) = 10 * 4k. Step 5: Expand left side: 9 * (5k − 5) = 45k − 45. Step 6: Right side: 10 * 4k = 40k. Step 7: Equate both sides: 45k − 45 = 40k. Step 8: Subtract 40k from both sides: 5k − 45 = 0, so 5k = 45. Step 9: Solve for k: k = 9. Step 10: Initial red marbles = 5k = 5 * 9 = 45. Step 11: Initial blue marbles = 4k = 4 * 9 = 36. Step 12: After losing 5 red marbles, red marbles = 45 − 5 = 40, blue marbles remain 36. Step 13: Final total number of marbles = 40 + 36 = 76.

Verification / Alternative Check:
The final ratio given in the problem is 10 : 9. Check the final red and blue marbles: 40 : 36 simplifies to 10 : 9 when both numbers are divided by 4. This matches the stated final ratio. Also check the initial ratio: 45 : 36 simplifies to 5 : 4 when divided by 9, which matches the given initial ratio. Both conditions are satisfied, confirming that the calculations are correct and the final total is 76.

Why Other Options Are Wrong:
• 81 and 86 are close to the totals we might get if we mis-handle the loss or miscalculate k. • 91 is another distractor that does not satisfy both given ratios when back-calculated.

Common Pitfalls:
A common mistake is to subtract 5 from the total instead of just from the red marbles. Another is to reverse the ratio and set 4k / (5k − 5) = 10 / 9, which leads to the wrong equation. Clearly defining variables and carefully writing the ratio as red : blue each time helps avoid such errors.

Final Answer:
After losing 5 red marbles, Rizwan now has a total of 76 marbles in the box.