Depth · one concept from first principles · step 2 of 4
a% of 800 g is 8a g of salt, and b% of 800 g is 8b g.
Concentrations do not add, but grams of salt do.
7.RP.A.3Organize Information In More WaysHalf of A is 400 g carrying 4a g of salt. A keeps 400 g with 4a g of salt; B now has 1200 g with 4a + 8b g of salt.
The poured half carries half of A's salt.
7.RP.A.3Identify SubproblemsHalf of B, 600 g, carries 2a + 4b g of salt. A ends with 1000 g and 6a + 4b g of salt; B keeps 600 g and 2a + 4b g.
Well mixed means the half that leaves carries exactly half the salt.
7.RP.A.3Identify SubproblemsAfter the second pour, beaker A holds g of salt and beaker B holds g.
means grams of salt in every g, so g of it holds g of salt, and beaker B starts with g.
Mixed well, the salt is spread evenly, so pouring out half the saltwater pours out half the salt: g leaves A, and later half of B's , which is , leaves B.
Evenly mixed salt keeps a fixed ratio to the saltwater it sits in, so half of the saltwater carries half of the salt.
Half of is half of plus half of .
Salt is never lost, only moved, so A ends with the g it kept plus the g poured in, and B keeps the other half of its salt.
12% of 1000 g is 120 g and 8% of 600 g is 48 g.
Each beaker gives one equation about grams of salt.
8.EE.C.8Convert To AlgebraSubtract the second equation from the first, then substitute.
The 2b terms match, so one subtraction isolates a.
8.EE.C.8Convert To AlgebraWith mixtures, follow the grams of salt step by step -- the percentages only come back at the very end.
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition. Why the shapes repeat →