Mathnasium Wishes Everyone a Merry Christmas!

Ho, Ho, Ho! For this Mathnasium Word Problem, we've prepared a very special word problem for those of you who are gearing up for Christmas celebrations this weekend! Solve our word problem and check after the video for the answer.

Emma the Christmas elf can wrap 8 presents in 3 minutes. How long will it take Emma to wrap 28 presents?
This word problem evokes the cliche of busy elves assembling wooden toys. I think this video is a perfect blend of traditional toy making magic and modern methods. If you do watch it, I want you to listen for the variants of the word "product" <:-).

Did you hear the "product" word variants "process", "produce", "production"? Ever wondered why the result of multiplication is the "product"? Hopefully after that toy production video, it makes sense that the result of work is a product. We try our best to relate mathematical concepts back to their spoken language origin. Mathematics is only mysterious when it's shrouded in formalized presentation. This word problem is about work, and most students at this point instinctively understand that work is something to be avoided. Let's call work play, and it's viscerally felt that the amount of play is directly related to the amount of time at play. More time, more play. Ah, but not only time, but how fast you can play. Play play fast, more play. So, more play is produced by playing fast and more time playing, i.e., play is the product of speed and time. Formally, that is written as work = rate × time.

Since we know that work = rate × time, that means time = work / rate. The work in this case is 28 wrapped presents that Emma performs at a rate of 8 presents per 3 minutes (8 presents/3 minutes since "per" means divide). Hence the time = 28 / (8/3) = 28 × 3 / 8 = 10½ minutes.

Alternatively, we can approach this from the proportions perspective by reasoning that since...
8 presents are wrapped in 3 minutes.
Hence, 1 present is wrapped in 3/8 minutes.
Hence, 28 presents are wrapped in 28 × 3/8 = 10½ minutes; and with that, you'll note that we've implicitly derived the time = work / rate relationship.

Finally, we can solve this problem as a comparison of ratios. We know that Emma wraps 8 presents in (per) 3 minutes. That means we can extrapolate and equate that to 28 presents in (per) T minutes; or in English, 8 is to 3 as 28 is to T:

8 / 3 = 28 / T ... and commuting the equation (just to locate the unknown T in the familiar left-side position)
28 / T = 8 / 3 ... and taking the reciprocal
T / 28 = 3 / 8 ... and solving for T minutes
T = 28 × 3 / 8 = 10½ minutes; and we've implicitly derived time = work / rate again!

At Mathnasium, we'll approach a problem in as many perspectives it takes to make sense. That's our slogan, "We make math make sense"!

Ho, Ho, Ho! Merry Christmas!

Contact:
Ruby Yao and Benedict Zoe, Mathnasium of Fort Lee
201-969-6284 (WOW-MATH), fortlee@mathnasium.com
246 Main St. #A
Fort Lee, NJ 07024
Happily serving communities of Cliffside Park, Edgewater, Fort Lee, Leonia, and Palisades Park.
Photo credits: Debucher Baguette, aka Emma the GBGV at http://mygbgvlife.com/2015/12/21/christmas-gift-wrapping/