Depth · one concept from first principles · step 2 of 4
Beaker A holds g of salt water that is salt, and beaker B holds g of salt water that is salt. You pour g of the salt water from beaker A into beaker B and stir well. Then you pour g of the salt water from beaker B into beaker A. Write the concentration of the salt water in beaker A in terms of and .
100 g of A's water at a% carries a g of salt. B already has b g of salt in its 100 g.
a% of 100 g is exactly a g.
7.RP.A.3Analyze The UnitsB now has 200 g of water containing a+b g of salt.
B's strength is the average of the two, since equal amounts were mixed.
7.RP.A.3Analyze The Units50 g of B's mixture carries a quarter of B's salt, because 50 g is a quarter of 200 g.
Same concentration throughout B, so salt splits in proportion to weight.
7.RP.A.3Identify SubproblemsThe g poured back from beaker B carries one quarter of all the salt in B.
After stirring, every gram of B's water holds the same share of salt, so the salt in any portion is in proportion to its weight.
g is one quarter of B's g, since four groups of g make g.
B holds g of salt at that moment.
of g is g of salt, because a percent counts parts out of every hundred.
Mixing loses no salt, so B's salt is the g it had plus the g poured in.
A kept 200 g at a%, which holds 2a g of salt, and receives 50 g more.
Write 2a as 8a/4 to combine.
7.EE.A.1Convert To AlgebraDivide salt by total weight and multiply by 100.
100/(4×250)=1/10.
7.EE.A.2Convert To AlgebraPercents don't add, but grams of salt do: follow the salt.
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition. Why the shapes repeat →