# 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.

Concentration Amount in Bottle Price of Bottle
A 1.04% 64 fl oz $12.99 B 18.00% 32 fl oz$22.99
C 41.00% 32 fl oz $39.99 D 1.04% 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 Bottle Price of bottle Cost 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.