Which detergent is a better buy?

Task

A store has two different brands of laundry detergent. Brand A can do 80 loads of laundry and costs \$12.75. Brand B does 36 loads of laundry and costs \$6.75. Which laundry detergent costs less per load?

IM Commentary

This purpose of this task is to provide a context for comparing ratios. Many techniques can be used including making a ratio table, finding unit rates, and carefully selecting equivalent ratios which can be compared directly. The numbers selected for the task statement make it well suited for the 6th grade. If students make all of their calculations by hand then they will build skill related to the standard 6.NS.3, '' Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.'' Alternatively, they can work with decimals and mental math as is shown in the third solution.

This task is based upon problem 5 from section 7.1 of Sybilla Beckmann's Mathematics for Elementary School Teachers, Pearson, 2014. Ideas from the commentary and solution have been gathered from a fall 2013 class of prospective teachers at the University of New Mexico.

Solutions

Solution: 1 Making a table

A table can be used to solve this problem in many different ways. For example we can use a table to find the cost to do 36 loads with Detergent A:

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Washing with Detergent A

Loads of Wash	Cost
80	\$12.75
40	\$6.375
36	\$5.6375

So detergent A is cheaper since it costs less than \$6 to do 36 loads while detergent B costs \$6.75 for 36 loads.

This first approach with a table has the downside of running into complicated decimals that do not carry any meaning within the context of money which is always rounded off to the nearest cent. A second method with a table which avoids this would be to look for a common multiple of 80 and 36. The least common mulitple of these is 720. Then we can make two tables to find the cost with each detergent brand for 720 loads of wash:

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Washing with Detergent A

Loads of Wash	Cost
80	\$12.75
240	\$38.25
720	\$114.75

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Washing with Detergent B

Loads of Wash	Cost
36	\$6.75
72	\$13.50
720	\$135

We see again that detergent A costs less money per load than detergent B. In these tables, we could skip the middle line and go directly to the cost for 720 loads. An extra step was included because, particularly with the second table, each step (multiplying by 2 and then multiplying by 10) can be done mentally. For the first table, multiplying by 3 twice is probably easier to do mentally than multiplying by 9.

If the ratio $$ \text{loads of wash} : \text{cost of detergent} $$ is viewed as a fraction $\frac{\text{loads of wash}}{\text{cost of detergent}}$, the two methods outlined above correspond to comparing fractions by finding a common numerator and finding a common demoninator respectively.

Solution: 2 Finding a unit rate

We can calculate the approximate cost per load of wash with each detergent and then compare these numbers directly. With detergent A we can wash 80 loads and the cost is \$12.75. This means that the cost per load of wash with detergent A is $$ \$ 12.75 \div 80 \approx \$ 0.16. $$ The exact value is slightly less than \$0.16 but closer to \$0.16 than to \$0.15. With detergent B we can wash 36 loads and the cost is \$6.75. So the coast per load with detergent B is $$ \$ 6.75 \div 36 = \$0.1875 $$ We can see that detergent A costs less per load of wash than detergent B, by almost 3 cents per load.

Solution: 3 Mental math

We can compare costs of the two detergents mentally. With detergent B we pay \$6.75 for 36 loads. With two detergent B's we will pay 2 $\times$ \$6.75 = \$13.50 for 2 $\times$ 36 = 72 loads. This is more money for fewer loads of wash compared to detergent A: \$0.75 cents more for 4 fewer loads of wash. Since detergent A does more loads of wash for less money it costs less per load of wash than detergent B.