Solving the Sugar-Water Ratio Problem

We need to determine the new ratio of sugar to water in a bucket after a part of the solution is replaced with water.

Initial Quantities

• Total initial volume = 60 litres
• Initial ratio of sugar to water = 5:1
• Total parts in the ratio = 5 + 1 = 6

Calculating Initial Volumes

• Initial volume of sugar = $<60 \text{ litres} \times \frac{5}{6}> = 50 \text{ litres}$
• Initial volume of water = $<60 \text{ litres} \times \frac{1}{6}> = 10 \text{ litres}$

Solution Replacement

• 12 litres of the solution are removed. The composition of the removed solution is the same as the original (5:1 ratio).
• Sugar removed = $<12 \text{ litres} \times \frac{5}{6}> = 10 \text{ litres}$
• Water removed = $<12 \text{ litres} \times \frac{1}{6}> = 2 \text{ litres}$

Volumes After Removal

• Volume of sugar remaining = $50 \text{ litres} - 10 \text{ litres} = 40 \text{ litres}$
• Volume of water remaining = $10 \text{ litres} - 2 \text{ litres} = 8 \text{ litres}$

Adding Water

• 12 litres of water are added to the remaining solution.
• New volume of water = $8 \text{ litres} + 12 \text{ litres} = 20 \text{ litres}$
• Volume of sugar remains = $40 \text{ litres}$

Final Ratio Calculation

• New ratio of sugar to water = $40 \text{ litres} : 20 \text{ litres}$
• Simplifying the ratio by dividing both sides by 20: $2:1$

The new ratio of sugar to water is 2:1.