Algebra Problems of the Day (Integrated Algebra Regents, June 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

June 2012

Part II: Each correct answer will receive 2 credits. Partial credit is possible.


31. Solve the following system of equations algebraically for y:

2x + 2y = 9
2x - y = 3

Answer:

You only need to find y, not x. Both equations have 2x in them so they can be subtracted to eliminate the variable. Remember that you have to subtract a negative y.

2x + 2y = 9
2x - y = 3
3y = 6
y = 2

You can check your work by substituting y = 2 into each equation and seeing if you get the same value of x for both. If you don't, you made a mistake.

2x + 2(2) = 9
2x + 4 = 9
2x = 5
x = 2.5

2x - 2 = 3
2x = 5
x = 2.5

The answer checks, so y = 2.

32. Three storage bins contain colored blocks. Bin 1 contains 15 red and 14 blue blocks. Bin 2 contains 16 white and 15 blue blocks. Bin 3 contains 15 red and 15 white blocks. All of the blocks from the three bins are placed into one box.

If one block is randomly selected from the box, which color block would most likely be picked? Justify your answer.

Answer:

I remember when this question first appeared. I thought it was dumb then, and I still do. It has an unnecessary complication. My only guess is that it was supposed to be a more difficult problem, but they simplified it to use for Part II instead. That much is only a guess.

If all the contents are poured into one bin, you have to total the number of red, white and blue blocks:

Red: 15 + 0 + 15 = 30

White: 0 + 16 + 15 = 31

Blue: 14 + 15 + 0 = 29

Total: 30 + 31 + 29 = 90 (You actually don't need this because you aren't asked for an exact probability.)

You need to write something like: "A white block is most likely to be picked because there are more white blocks in the bin than any other color."

33. Students calculated the area of a playing field to be 8,100 square feet. The actual area of the field is 7,678.5 square feet. Find the relative error in the area, to the nearest thousandth.

Answer:

Reminder: relative error is NOT percent of error, even though the two are absolutely related. (One is a decimal and one is a percentage.) If you put a percentage, you will lose half credit because they didn't ask for a percentage.

Also, "to the nearest thousandth" means three decimal places, properly rounded.

Relative error is | actual - estimated | / actual. Find the difference between the two amounts and divide it by the actual amount.

| 8,100 - 7,678.5| / 7,678.5 = 0.05489..., which is 0.055 to the nearest thousandth.

End of Part II.

More to come. Comments and questions welcome.

More Regents problems.