Depth · one concept from first principles · step 2 of 4
Use x g of paint P and y g of paint Q. Paint P is 3/10 white and 7/10 brown; paint Q is 1/5 white and 4/5 brown.
A ratio 3:7 means 3 parts out of 10 are white.
7.RP.A.3Organize Information In More WaysWhite to brown must be 2:5, so 5 times the white equals 2 times the brown. Multiply by 10 to clear the fractions.
In any batch, paint P must be 6 times as heavy as paint Q.
8.EE.C.7Convert To AlgebraSince x = 6y, P is used six times as fast; using all of Q would need 7200 g of P. So P runs out first: x = 1200 gives y = 200.
The batch grows with y, and the most y can be is set by the scarcer ingredient.
7.RP.A.2Eliminate PossibilitiesPaint P runs out first, so the largest batch uses all g of P and g of Q.
The target ratio forces P to be times Q in any batch, so fixing one amount fixes the other.
A batch is , which is grams, so the more Q we use, the larger the batch.
Q can grow only while the grams of P stay within g, so is at most , and that limit comes well before Q's own g.
Six times reaching means is , because dividing undoes multiplying.
is at most and is less than , so the paint P limit is reached first and is the one that decides.
The largest batch uses all of paint P and 200 g of paint Q.
Total paint is 7y, so it is largest when y is.
7.RP.A.2Convert To AlgebraFirst find the recipe (how the two ingredients must compare), then let the scarcer ingredient decide how big the batch can be.
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