Depth · one concept from first principles · step 2 of 4
The weight of bags of beans plus bag of rice is . The weight of bags of beans minus bag of rice is . How much do bags of beans weigh, in pounds and ounces? (Every bag of beans weighs the same.)
Adding '3 beans + 1 rice = 46' and '3 beans - 1 rice = 14' cancels the rice, leaving 6 bean bags = 46 + 14 = 60 oz.
Lining up the two facts and combining them so the rice disappears is the same idea as a two-step word problem.
3.OA.D.8Organize Information In More WaysAdding the two weight statements cancels the rice bag and shows that six bean bags together weigh 60 oz.
The first statement adds one rice bag's weight and the second takes an equal rice bag's weight away, so combining them leaves no rice behind.
Removing the same weight you just added brings you back to zero rice.
A rice weight of zero adds nothing to the pile, so the rice term simply drops out.
The three equal bean bags in the first statement and the three equal bean bags in the second join to make six bean bags.
Three equal-size groups plus three more of that same size is six of those groups.
Since 46 oz and 14 oz are the true weights of the two piles, putting the piles together must weigh 46 + 14 = 60 oz, and that 60 oz must equal the combined bean bags.
Joining two parts makes a whole equal to the two parts added, so the totals give 60 oz.
Each pile may be swapped for the amount it equals, so the combined beans equal the combined 60 oz.
Six bean bags weigh 60 oz, so one bean bag weighs 60 / 6 = 10 oz.
Splitting 60 oz equally among 6 bags is a basic within-100 division.
3.OA.A.3Organize Information In More WaysFive bags each weigh 10 oz, so together they weigh 5 x 10 = 50 oz.
Multiplying the single-bag weight by 5 is repeated addition of equal groups.
3.OA.A.3Work BackwardsSince 16 oz = 1 lb, 50 oz is 3 full pounds (48 oz) with 2 oz left over: 50 = 16 x 3 + 2.
Trading every 16 oz for 1 lb is measurement-unit sense built in Grade 3.
3.MD.A.2Work BackwardsAdd the two clues so the rice cancels - then it's just Grade 3 dividing and an ounce-to-pound trade!
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The same shape shows up later on the AMC, as Small-Domain Constrained Search. Why the shapes repeat →