6. give a teacher's solution using algebra for the following
problem: a truck contains 1000 pounds of sand and concrete in the ratio 2 : 3. after x pounds of sand is added, the ratio of the sand and concrete becomes 4: 5. find x

  80 pounds

Step-by-step explanation:

The problem can be worked in two parts. For the first part, we will find the amounts of sand and concrete in the truck. Then, we will use those numbers in the second part to find the added sand.

The given relations are ...

  s + c = 1000 . . . . . . . . there are 1000 pounds total of sand and concrete

  s/c = 2/3 . . . . . . . . . . . the ratio of sand to concrete is 2:3

Multiplying the second equation by c, we get an expression we can substitute into the first equation.

  s = (2/3)c

  (2/3)c + c = 1000 . . . . . . substitute for s

  5/3c = 1000 . . . . . . . . . . collect terms

  c = 1000(3/5) = 600 . . . multiply by the inverse of the coefficient of c

  s = 2/3(600) = 400 . . . . find s using the relation above

Now, we know the truck initially contains 400 pounds of sand and 600 pounds of concrete.

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When we add x pounds of sand, the ratio of the mix changes to 4/5:

  (s+x)/c = (400 +x)/600 = 4/5

  400 +x = (4/5)(600) = 480 . . . . . . multiply by 600

  x = 80 . . . . . . . . . . . . . . . . . . . . . . . subtract 400

The amount of added sand is x = 80 pounds.

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