Sara, a 7th grader in Algebra 1, solves a word problem

Sara was working on rates and ratios in her math class. Don had some word problems that involved ratios. He started Sara with this problem (from Dolciani):

In a dry Concrete mixture the ratio Sand to Cement is 4:1 and the ratio of Gravel to Sand is 5:4. How many pounds of each are in 1100 pounds of the Concrete?

Sara wrote the ratios as 5:4:1, then proceeded to multiply each of these by some number she chose, say 50 to get 250:200:50 then added these to get 250+200+50 = 500, which is less than the 1100 lbs. needed. Sara continued this to eventually she multiplied each by 110 to get 550+440+110 = 1100 lbs.
which was the correct total. So there was 550 lbs. of Gravel, 440 lbs. of sand, and 110 lbs. of cement in the 1100 lbs of Concrete.

Then Don showed Sara how he did it using algebra: Let x equal the multiplying number, so he got 5x+4x+1x=1100 and 10x=1100, so x=110, and the weight of each would be 5*110 + 4*110+ 1*110= 1100 lbs. of concrete.

Then Don asked Sara 'Suppose the ratio of Sand to Cement was 3:1 instead of 4:1 and the ratio of Gravel to Sand was 5:4. Instead of 5:4:1 we have 5:4 and 3:1. How would you do that?'

Immediately Sara suggested we use the lowest common multiple of 4 and 3, or 12, to get the ratios 5*3:4*3 and 3*4:1*4 and ended up with 15:12:4. Then she calculated the multiplying number to be 35+15/31 which was 1100/31. She then multiplied each of the numbers 15, 12, and 4, by 35+15/31 and added them, to get 15*(35+15/31) [lbs. of Gravel] + 12*(35+15/31) [lbs. of sand] + 4*(35+15/31) [lbs. of cement]= 1100 lbs. of concrete, which was correct!

                                                      Fine work, Sara!!!