Quandaries
and Queries 

Bryce, Duanne, Juliana and Sonya share a 40ft by 48 ft rectangular portion of the community garden in the neighborhood. With this space each has their own rectangular garden plot. Bryce is the only one with a square plot. its are is 1/2 the area of Duanne's The area of Bryce's plot is 2/3 the area of Julianna's plot. Their plots have one side in common. Sonya's garden has an area twice that of Julianna's garden. What are the dimensions of each person's garden plot? Your help would be greatly appreciated. I am a student at The University of Texas San Antonio taking an Math Course for Elementary Teachers and I have been racking my brain for 2 days already trying to figure it out. Thank your for any assistance that you are able to give me! Michele 

Hi Michele, The first step to solving a problem like this is to define your variables: D = the area of Duanne's plot J = the area of Juliana's plot S = the area of Sonya's plot Bryce's plot is half the area of Duanne's plot: B = ^{D}/_{2} or D = 2B Bryce's plot is twothirds the area of Juliana's plot: B = ^{2J}/_{3} or J = ^{3B}/_{2} Sonya's plot is twice the area of Juliana's plot: S = 2J or S = (2)( ^{3B}/_{2}) = 3B The total area is 40 ft by 48 ft, or 1920 sq. ft. Now we makeÊan equation from the information given and our variables: B + D + S + J = 1920 sq. ft. but now we will substitute expressions in terms of B for D, S and J B + 2B + 3B + ^{3B}/_{2} = 1920 sq. ft. (now solve for B to get the area of Bryces plot) B = 256 sq. ft. Since we know Bryce's plot is square, it must have dimensions of 16 ft by 16 ft since 16 x 16 = 256 Now we can find the areas of the other plots: Sonya's plot = 3B = 768 sq. ft. Juliana's plot = ^{3B}/_{2} = 384 sq. ft. To find the actual dimensions of the plots, we need to go back to the information given in the problem: Since Bryce's plot and Juliana's plot share a side, we find the other side of Juliana's plot by dividing her area by 16: ^{384}/_{16} = 24. Thus Juliana's plot is 16 ft. by 24 ft. To find the dimensions of Duanne's plot and Sonya's plot, draw a picture of the whole garden. Try various likely placements for Bryce's and Juliana's plots and see if what is left over makes rectangles of the correct areas for Duanne's and Sonya's plots. Keep in mind that Bryce and Juliana's plots are connected along one side. Hope this helps, Leeanne 

