What This Problem Teaches

• Setting up equations when equal quantities are removed from different starting amounts
• Understanding how ratios change when you subtract the same value from different quantities
• Recognizing that the difference between containers stays constant during equal removal
• Translating "times as much" language into algebraic relationships
• Verifying solutions by checking both the arithmetic and the logical constraints

Solution: Method 1 — The Direct Ratio Approach

This is a classic "equal removal" problem where the key insight is that removing the same amount from different quantities changes their ratio. Let's work through this systematically.

Step 1 — Define the variable

Let x = the number of liters removed from each container.

Step 2 — Express the final amounts

After removing x liters from each container:

• Container A will have: 18.6 - x liters
• Container B will have: 10.3 - x liters

Step 3 — Set up the ratio equation

We're told that after removal, container A has 5 times as much water as container B. This gives us:

Step 4 — Solve the equation

Expand the right side:

Add 5x to both sides:

Subtract 18.6 from both sides:

Solution: Method 2 — The Constant Difference Strategy

Here's a more elegant approach that uses the fact that the difference between the containers stays the same when we remove equal amounts.

Step 1 — Find the initial difference

The difference between the containers initially is:

Step 2 — Use the final ratio to find the individual amounts

After removal, if container B has y liters, then container A has 5y liters. The difference must still be 8.3 liters:

Step 3 — Find how much was removed

Container B ends with 2.075 liters, so the amount removed was:

We can verify: Container A ends with 5 × 2.075 = 10.375 liters, so 18.6 - 10.375 = 8.225 liters was also removed from A.

Verification

Let's check our answer by substituting back into the original conditions:

After removing 8.225 liters from each container:

• Container A: 18.6 - 8.225 = 10.375 liters
• Container B: 10.3 - 8.225 = 2.075 liters

Check the ratio:

Perfect! Container A indeed has exactly 5 times as much water as container B.

Common Pitfalls

The Pattern Behind This

This problem follows the general pattern for "equal removal with a final ratio" problems. If you start with quantities A and B, remove x from each, and end with a ratio of r:1, then:

Solving this general equation:

Does This Seem Reasonable?

Let's do a sanity check on our answer of 8.225 liters removed:

Also consider: we're removing more than 80% of the original water from both containers. This makes sense because we're going from a ratio of about 1.8:1 down to 5:1 — a dramatic change requires dramatic removal.

If we'd only removed a small amount, the ratio would have stayed close to the original 1.8:1. To get to 5:1, we need to remove most of the water, leaving container B nearly empty.