Making Lemonade?

Algebra Level 5

Alice and Bob are given a $2000$ mL pitcher of water and an empty bottle. First, Alice pours $L$ mL of lemon juice into the pitcher, where $L$ is a positive rational number. She mixes well, then pours $L$ mL of the resulting mixture into the bottle. Afterwards, Bob pours $1000$ mL of pure water into the pitcher, mixes well, and pours $1000$ mL of the resulting mixture into the bottle. Finally, the contents of the bottle are thoroughly mixed. If the mixture in the bottle is $20\%$ lemon juice, find the integer closest to $L$ .

Notes:

When Alice and Bob mix well, the concentration of lemon juice/water is the same throughout the mixture.

Lemon juice is $100\%$ lemon juice and water is $0\%$ lemon juice.

The answer is 667.

1 solution

Abdelhamid Saadi
Oct 19, 2016

Let the $L \space mL$ of mixture poured by Alice to be composed by $\alpha \space mL$ of water and $\beta \space mL$ of juice.

We have : $\alpha + \beta = L$ $L \times \alpha = 2000 \times \beta$

Just before pouring to the bottle for the second time, the pitcher contains $3000 \space mL$ , so that a third will be poured to the bottle.

The quantity of juice at this time is $(L - \beta) \space mL$ and the third of that will go the bottle.

The final quantity of juice in the bottle is $\beta + \frac{L - \beta}{3} = \frac { L + 2 \times \beta}{3}$ which represents $20 \%$ of the final total quantity of $1000 + L$ $3\times (1000 + L) = 5 \times (L + 2 \times \beta)$ So that : $5 \times \beta = 1500 - L \quad and \quad \ 5 \times \alpha = 6 \times L - 1500$

Then: $L \times (6 \times L - 1500) = 2000 \times(1500 - L)$

This second degree equation has two solution : $-750$ and $\dfrac {2000}{3}$

So that the solution is 667

Making Lemonade?

The answer is 667.

1 solution

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