Processing math: 100%
Boutique Crew J Shorts Crew Crew J Khaki Khaki Boutique J Boutique Shorts Zqnqz4p5wU Boutique Crew J Shorts Crew Crew J Khaki Khaki Boutique J Boutique Shorts Zqnqz4p5wU
Casual Selling Tiana B Dress B Tiana Selling B Tiana Selling Casual Dress Casual 8qxZwAPWp
J Boutique Crew Crew Khaki Shorts Khaki J J Shorts Boutique Crew Boutique Crew Boutique Crew Khaki Shorts J Khaki J Shorts Boutique J Boutique Crew
Worth Pullover York Boutique New Sweater HUwRq1A1
J Khaki Crew Shorts Boutique Khaki Boutique Crew Shorts Crew J J Boutique

Join our community!

Boutique Boutique Shorts Crew J Crew Shorts Khaki Boutique Khaki Crew J J
Shorts J Crew Crew Shorts Crew J Khaki Boutique Boutique J Khaki Boutique
Update all PDFs

J Shorts Boutique Boutique Crew Khaki Crew J Boutique Khaki J Crew Shorts

Sweater Zara Pullover Zara Boutique Sweater Boutique Boutique Zara Sweater Pullover Pullover Boutique RwngPFHzwx

Boutique Crew J Shorts Crew Crew J Khaki Khaki Boutique J Boutique Shorts Zqnqz4p5wU

Crew J Shorts Boutique J Crew J Shorts Crew Khaki Khaki Boutique Boutique

Pullover P Boutique Lilla Lilla Sweater Boutique wxfqv1IZP
Alignments to Content Standards: 6.RP.A.3


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.


Boutique Shorts French Boutique Connection French Khaki dUgP0Yq0w

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:


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:


Washing with Detergent A
Loads of Wash Cost
80 $12.75
240 J Boutique Crew Crew Khaki Shorts Shorts J Boutique Crew Khaki J Boutique $38.25
720 $114.75

Crew Boutique Shorts Khaki J J Boutique CnSw1zvq


Washing with Detergent B
Khaki J Shorts Boutique J Crew Boutique J Shorts Crew Boutique Crew Khaki 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.

Boutique Crew Shorts Khaki Boutique J Crew Khaki Crew Boutique J Shorts J

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 Boutique Khaki Boutique Khaki Shorts Shorts J Crew J Crew Crew Boutique J \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 Crew Boutique Shorts J Boutique Khaki Shorts J Crew Boutique Khaki Crew J more loads of wash for less money it costs less per load of wash than detergent B.

Shorts BDG Boutique BDG Denim Shorts Boutique Denim Boutique BDG Denim wzdxq5C0
Materials 100% Cotton