Drinking Juice, Variation 3

Task

Alisa had some juice in a bottle. Then she drank $\frac38$ liters of juice. If this was $\frac34$ of the juice that was originally in the bottle, how much juice was there to start?

IM Commentary

This task builds on a fifth grade fraction multiplication task, “5.NF Drinking Juice.” This task uses the identical context, but asks the corresponding “Group Size Unknown” division problem. See “Drinking Juice, Variation 2” for the “Number of Groups Unknown” version.

Solutions

Solution: Solution

From the question we know that Alisa drank $\frac38$ liters of juice, which was $\frac34$ of the amount of juice that was in the bottle.

What we really want to know is how many liters or what fraction of a liter of juice was in the bottle. Therefore we are looking for how many liters are in the whole ($\frac44$) amount of juice in the bottle. To answer this we will start by breaking the amount of juice in the bottle into three equal pieces.

Since $\frac38$ of a liter is broken into three equal pieces, each piece represents $\frac18$ of a liter.

What we are really concerned with is the total amount of juice in the bottle, and because the amount of juice in the bottle is divided up into fourths, we need four of them to make the whole.

Relabeling the picture will make it just a bit easier to see the answer.

Now it is clear that the total amount of juice in the bottle was $\frac48=\frac12$ of a liter.

Solution: A computational approach

This question is equivalent to asking, "$\frac34$ of what quantity of juice is $\frac38$ liter?" We can write this symbolically as $$\frac34 \times ? = \frac38$$ which is equivalent to the division problem $$\frac38 \div \frac34=?$$ Since $$\frac38 \div \frac34 = \frac38 \times \frac43 = \frac{12}{24} = \frac 12$$ we see we get the same answer as if we did reasoning about the context in the previous solution.

Alisa had $\frac12$ liter of juice in the bottle to start.