In a mixture of three varieties of oil, the initial weights are in the ratio 4 : 5 : 8. If 5 kg of the first variety, 10 kg of the second variety and some quantity of the third variety are added, the ratio of the weights becomes 5 : 7 : 9. How many kilograms of the third variety of oil are added to the mixture?

Introduction / Context:
This question combines ratio and mixture concepts. We start with a mixture in a given ratio, then add fixed amounts of two components and an unknown amount of the third. Using the final ratio, we determine the unknown quantity.

Given Data / Assumptions:
- Initial ratio of three oil varieties = 4 : 5 : 8.
- Let their initial weights be 4k, 5k and 8k kilograms.
- Added quantities: 5 kg of first variety, 10 kg of second variety, x kg of third variety.
- Final ratio after addition = 5 : 7 : 9.
- We need to find x in kilograms.

Concept / Approach:
After additions, the new quantities are:
First variety: 4k + 5Second variety: 5k + 10Third variety: 8k + x
The final ratio states that these are in proportion to 5 : 7 : 9. We form two equations from the proportionality and solve for k and x.

Step-by-Step Solution:
Step 1: From the new ratio of first and second varieties:(4k + 5) / (5k + 10) = 5 / 7.Step 2: Cross multiply: 7(4k + 5) = 5(5k + 10).Step 3: Expand: 28k + 35 = 25k + 50.Step 4: Simplify: 3k = 15, so k = 5.Step 5: Now new quantities: First = 4 * 5 + 5 = 25 kg, Second = 5 * 5 + 10 = 35 kg.Step 6: From the ratio of second and third varieties: (5k + 10) / (8k + x) = 7 / 9.Step 7: Substitute k = 5: (25 + 10) / (40 + x) = 7 / 9, so 35 / (40 + x) = 7 / 9.Step 8: Cross multiply: 35 * 9 = 7 * (40 + x).Step 9: 315 = 280 + 7x, so 35 = 7x and x = 5.

Verification / Alternative check:
Final quantities are: first = 25 kg, second = 35 kg, third = 45 kg. The ratio 25 : 35 : 45 simplifies by dividing by 5 to 5 : 7 : 9, which matches the given final ratio, confirming that 5 kg of the third variety was added.

Why Other Options Are Wrong:
- 15 kg, 25 kg and 35 kg would produce final quantities that do not simplify to 5 : 7 : 9 when added to the initial amounts, so they fail the ratio condition.

Common Pitfalls:
- Forgetting to use two separate ratio equations, one from first to second and another from second to third.
- Mixing up which quantities include the added oil and which are the initial amounts.

Final Answer:
The quantity of the third variety of oil added is 5 kg.