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# Algebra

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Tanzi and Maeve each have brand new 1200 mL bottles of soap. They use soap at different constant rates. When Tanzi's bottle is completely empty, she asks Maeve to add soap from Maeve's bottle, which is still two-thirds full. How many milliliters of soap should Maeve pour from her bottle into Tanzi's so that both bottles will be completely emptied at the same time?

May 31, 2023

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Let r1 be the rate at which Tanzi uses soap, and let r2 be the rate at which Maeve uses soap. If Tanzi's bottle is completely empty when Maeve starts pouring soap, then the total amount of soap used by Tanzi is:

1200 mL

Let's say that Maeve pours x mL of soap from her bottle into Tanzi's bottle. Then the total amount of soap used by Maeve is:

(2/3) × 1200 mL - x mL

since Maeve's bottle is initially two-thirds full and she pours x mL of soap out of her bottle.

The time it takes for Tanzi's bottle to be completely emptied is:

1200 mL / r1

The time it takes for Maeve's bottle to be completely emptied is:

(2/3) × 1200 mL - x mL / r2

Since both bottles are completely emptied at the same time, these two times must be equal. Therefore, we can set the two expressions for time equal to each other and solve for x:

1200 mL / r1 = (2/3) × 1200 mL - x mL / r2

Multiplying both sides by r1 r2, we get:

1200 r2 = (2/3) × 1200 r1 r2 - x r1

Multiplying out the right-hand side and simplifying, we get:

x = (2/3) × 1200 mL - 1200 r2 / r1

Therefore, to make both bottles empty at the same time, Maeve should pour:

x = (2/3) × 1200 mL - 1200 r2 / r1

milliliters of soap from her bottle into Tanzi's bottle.

Jun 1, 2023