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# Weed killer

Alignments to Content Standards: N-Q.A.3 N-Q.A.1 N-Q.A.2

A liquid weed-killer comes in four different bottles, all with the same active ingredient. The accompanying table gives information about the concentration of active ingredient in the bottles, the size of the bottles, and the price of the bottles. Each bottle's contents is made up of active ingredient and water.

ConcentrationAmount in BottlePrice of Bottle
A1.04%64 fl oz$12.99 B18.00%32 fl oz$22.99
C41.00%32 fl oz$39.99 D1.04%24 fl oz$5.99

1. You need to appy a 1% solution of the weed killer to your lawn. Rank the four bottles in order of best to worst buy. How did you decide what made a bottle a better buy than another?
2. The size of your lawn requires a total of 14 ﬂ oz of active ingredient. Approximately how much would you need to spend if you bought only the A bottles? Only the B bottles? Only the C bottles? Only the D bottles?

Supposing you can only buy one type of bottle, which type should you buy so that the total cost to you is the least for this particular application of weed killer?

## IM Commentary

The principal purpose of the task is to explore a real-world application problem with algebra, working with units and maintaining reasonable levels of accuracy throughout. Of particular interest is that the optimal solution for long-term purchasing of the active ingredient is achieved by purchasing bottle C, whereas minimizing total cost for a particular application comes from purchasing bottle B. Students might need the instructor's aid to see that this is just the observation that buying in bulk may not be a better deal if the extra bulk will go unused.

## Solution

1. All of the bottles have the same active ingredient, and all can be diluted down to a 1% solution, so all that matters in determining value is the cost per fl oz of active ingredient. We estimate this in the following table:
Amount active in BottlePrice of bottleCost per ounce
A$1.04\%\times 64 \approx 0.64$ fl oz $\$12.99\approx\$13$$\frac{13}{0.64} \approx \$20$per fl oz B$18.00\%\times 32 \approx 6$fl oz$ \$22.99 \approx \$23\frac{23}{ 6}\approx \$4$ per fl oz
C $41.00\% \times 32 \approx 13$ fl oz$\$39.99 \approx \$40$ $\frac{40}{ 13}\approx \$3 $per fl oz D$1.04\% \times 24 \approx 0.24$fl oz$\$5.99\approx \$6\frac{6}{ 0.24}\approx \$24$ per fl oz
If we assume that receiving more active ingredient per dollar is a better buy than less active ingredient per dollar, the ranking in order of best-to-worst buy is C,B,A,D.
2. The A bottles have about $0.64$ fl oz of active ingredient per bottle so to get $14$ fl oz we need $\frac{\mbox{14 fl oz}}{\mbox{0.64 fl oz /bottle}} \approx 22$ bottles. Purchasing $22$ A bottles at about $\$13$each will cost about$\$286$.

The B bottles have a little less than $6$ fl oz of active ingredient per bottle so to get $14$ fl oz we need $3$ bottles. Purchasing $3$ B bottles at about $\$23$each will cost about$\$69$.

The C bottles have a little more than $13$ fl oz of active ingredient per bottle, so we need $2$ bottles. Purchasing $2$ C bottles at about $\$40$each will cost about$\$80$

The D bottles have only $0.24$ fl oz of active ingredient per bottle so to get $14$ fl oz we need $\frac{\mbox{14 fl oz}}{\mbox{0.24 fl oz/bottle}} \approx 58$ bottles. Purchasing $58$ D bottles at about $\$6$each will cost about$\$348$.

Thus, although the C bottle is the cheapest when measured in dollars/fl oz, the B bottles are the best deal for this job because there is too much unused when you buy C bottles.

#### stickler says:

over 6 years

An argument could be made for saying that the C-type bottles are the ones to buy. The solution shows that buying B-type bottles results in getting the job done for about 69 dollars instead of 80 dollars, and suggests that is why B-type bottles should be chosen even though they are not the cheapest by active ingredient (as seen in part a of the problem). But the problem did not ask which type of bottle results in the least out-of-pocket expense.

Indeed, one might argue that buying the cheapest of a consumable product by volume is always the best idea, especially with a product that you are likely to need to use again. So, the result from part a of the problem could be extended to say that C-type bottles, because they are the best value, should be used. Weed killer, like laundry detergent, is something that is likely to be useful in the future. Obviously, if the user is going to throw away the unused amount, then B-type bottles would be cheaper, but if the user is going to save the remainder and use it in the future, then C-type bottles are a more economical approach. If the problem setting were about some item that did not have shelf life (or needed to be thrown away after a certain time), then the answer originally given would make more sense.

This task has a slight "trick problem" feel to it because of this. The fact that the number of bottles you have to buy is discrete creates the discrepancy between choosing B-type and C-type bottles. This seems a bit unfair as a task unless it is more clearly specified what "best" option means. If instead of asking which type of bottle the user "should" buy, the task were reworded to ask "Supposing you can only buy one type of bottle, which type should you buy so that the total cost to you is the least for this particular application of weed killer?" then I would withdraw my objection.

If I were a student who reasoned that "C-type" bottles are a better (more economical) value in the long run, I would be extremely upset to be marked wrong on this task by a scorer who is told that "B-type" bottles is "the" answer. Illustrative problems should either have an unambiguous answer by careful phrasings of the prompts, or they should allow a rubric that explains under what conditions various answers would be acceptable -- perhaps including that a suitable argument for why C-type bottles might be preferred could be an acceptable answer.

#### Cam says:

over 6 years

Agreed entirely. The intent was for this distinction between the continuous and discrete cases to be interesting, and not a "trick." I've adopted your wording for the second question, and made the commentary more clearly reflect the intent of the problem. Thanks for the careful reading and comments.