Three bottles contain mixtures of syrup and water in the ratios 2:3, 3:4, and 7:5 (syrup : water). If 10 litres from the first bottle and 21 litres from the second bottle are taken, how many litres should be taken from the third bottle so that the final mixture of syrup and water becomes 1:1?

Introduction:
This problem tests advanced mixture combination: we are taking specified amounts from different sources, each with its own syrup-to-water ratio, and then choosing an unknown amount from a third source to achieve a target final ratio. The correct method is to compute how much syrup and how much water are contributed by each selected quantity, add them, and then enforce the final target ratio (here 1:1, meaning equal syrup and water).

Given Data / Assumptions:

• Bottle 1 ratio (syrup:water) = 2:3, quantity taken = 10 L• Bottle 2 ratio (syrup:water) = 3:4, quantity taken = 21 L• Bottle 3 ratio (syrup:water) = 7:5, quantity taken = x L• Final desired ratio (syrup:water) = 1:1

Concept / Approach:
Convert each ratio into syrup fraction and water fraction. Multiply by the litres taken to get actual litres of syrup and water. Add contributions, then set total syrup = total water (because 1:1 means equal). Solve for x.

Step-by-Step Solution:
Step 1: Bottle 1 (2:3) total parts = 5.Syrup fraction = 2/5, water fraction = 3/5From 10 L: syrup = 10*(2/5)=4 L, water = 10*(3/5)=6 LStep 2: Bottle 2 (3:4) total parts = 7.Syrup fraction = 3/7, water fraction = 4/7From 21 L: syrup = 21*(3/7)=9 L, water = 21*(4/7)=12 LStep 3: Combine totals so far.Total syrup = 4 + 9 = 13 LTotal water = 6 + 12 = 18 LStep 4: Bottle 3 (7:5) total parts = 12.From x L: syrup = x*(7/12), water = x*(5/12)Step 5: Enforce final ratio 1:1, so total syrup = total water.13 + (7x/12) = 18 + (5x/12)(7x - 5x)/12 = 52x/12 = 5 => x/6 = 5 => x = 30

Verification / Alternative check:
If x = 30, then syrup added = 30*(7/12)=17.5 and water added = 30*(5/12)=12.5. New totals: syrup = 13+17.5=30.5 and water = 18+12.5=30.5. They are equal, so the final mixture ratio is exactly 1:1, confirming the answer.

Why Other Options Are Wrong:
20 L and 25 L do not add enough syrup (since bottle 3 is syrup-heavy), so water stays higher than syrup.35 L adds too much syrup, making syrup exceed water.15 L is far too small to balance the initial excess water (18 vs 13).

Common Pitfalls:
• Treating ratios as litres directly instead of converting to fractions.• Forgetting that 1:1 means total syrup equals total water, not total volume equals something.• Making fraction errors with 7/12 and 5/12.

Final Answer:
30 Litres should be taken from the third bottle.