# Kaytee s Contest Problem

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3 Bertha s weight (in lbs.) Billy s weight (in lbs.) Total weight (in lbs.) Is total too low, too high? is 848 lbs. too low is 448 lbs. too low is 48 lbs. too low is only 8 lbs. too low just right! I know that Bertha weighs 1272 lbs. and Billy weighs 424 lbs. Method 3: Make a Table Using Billy s Weight Possibilities I know from the problem that Billy weighs 848 lbs. less than Bertha. I also know that their combined weight is 1696 lbs. If I try a weight for Billy (list in the first column of my table) I can calculate Bertha s weight (list in the second column). I can combine those two weights (list in the third column) and then compare that to 1696 lbs. to see how close I am. Here s what I tried: Billy s weight (in lbs.) Bertha s weight (in lbs.) (Billy s weight + 848) Total weight (in lbs.) (Billy s weight + Bertha s weight) Is total too low or too high? is 448 lbs. too low is 48 lbs. too low is only 8 lbs. too low just right! I know that Billy weighed 424 lbs. and Bertha weighed 1272 lbs. Method 4: Proportional Reasoning and Drawing a Diagram After reading the problem I know that: Bertha weighs 848 lbs. more than Billy. Together Bertha and Billy weigh 1696 lbs. I notice that the first weight ends in 8 and the second weight ends in 6 and that leads me to confirm that: = 1696 So the difference between their weights is half of the sum of their weights. I wonder if I can use that to solve the problem. I decided to draw a picture to help me visualize the relationships (Billy and Bertha combined: 1696 lbs) (Difference between Billy and Bertha: 848 lbs, half of the total) I wasn t sure how long the line should be for Billy, but that didn t matter because I just wanted to see the relationships (Billy s weight: unknown) (Bertha s weight: Billy s weight) (Total weight: Billy s weight + Bertha s weight) Now I know that the total of the two, 1696 pounds, is made up of Billy s weight Billy s weight Billy s weight should be 1/4 of the total weight, like this:

4 (Billy s weight + Bertha s weight) Billy 848 Billy (Total weight: 1696 lbs) 1696 Billy s weight should be 1696 / 4 = 424 lbs. That means Bertha s weight is = 1272 lbs. Check: = 1696 lbs. That must be right! Extra: Billy s weight is 1/4 of their combined weights and Bertha s weight is 3/4 of the combined weight. That means the proportion of Bertha s weight to Billy s weight is 3:1. The proportions for the butter cows would be 600:200 and the slices of toast would be 19200:6400 and the lifetimes would be 2:2/3. The butter cow Billy s size would cover 6400 slices of toast It would take 2/3 of a lifetime to eat them all. Method 5: Algebraic Reasoning I know that I need to find Billy s weight. I know that Billy s weight equals Bertha s weight I also know that Billy s weight + Bertha s weight equals 1696 lbs. Using those two thoughts together, I know that Billy s weight + Billy s weight lbs. equals 1696 lbs. That means that Two of Billy s weight will equal 848 lbs. and so Billy s weight would be 424 lbs. If Billy s weight is 424 lbs. then Bertha weighs 1272 lbs. Method 6: Algebra If I let x = Billy s weight then I know that x = Bertha s weight I know that: x + (x + 848) = Billy s and Bertha s weight (total weight) but I also know 1696 = total weight So, I can write this equation and solve for x: x + (x + 848) = x = x = 848 x = 424 If Billy weighs 424 lbs., then Bertha weighs 1272 lbs. Teaching Suggestions As is the case in many of our problems, Kaytee is a real person, though not really a farmer. She s a photographer who, like Max, got to go to the Iowa State Fair in Max was looking for pictures of some awesome rides that might be included in a future GeoPoW when he saw those cows and immediately thought they could be the subject of a guess and check problem or a proportional reasoning problem. When Kaytee gave Max this photo she told him that her caption for it would be: Rule of cuteness: any animal pictured with a smaller version of itself is automatically cute. We agree! As we talked about this problem in the office we debated whether we should use the idea of a guess the weight contest since Bertha s and Billy s weights had constraints given in the problem. Some on staff thought it would be more accurate to say calculate the weight or determine the weight. We decided, however, that using guess the weight might just hint to students the idea of using a guess

7 Extra: when I looked at it I saw that I needed to use porportion.so I made a fraction out of Billy's and Bertha's weights next to x over Bertha's butter wieght. That was my first porportion. 424 X = Then I cross-mutiplied. 424 x = * 600 = * X = 1272x Next I divided x = = X = 200 Then I did my next porportion using billy and bertha's butter weights and bertha's toast amount. 200 X = I cross-mutiplied again. 200 X = * = Next I divided...again x = X = 6400 Then I divided twice / 2 = / 9600 = 2/3 And that is how I got my answer.

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