Algebra Problems of the Day (Integrated Algebra Regents, August 2011)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones. The Integrated Algebra Regents covered most of the same material as the current Algebra Regents, with a few differences.

More Regents problems.

Integrated Algebra Regents, August 2011

Part I: Each correct answer will receive 2 credits.


21. What is the solution of the inequality −6x − 17 ≥ 8x + 25? rectangle?

1) x ≥ 3
2) x ≤ 3
3) x ≥ −3
4) x ≤ −3

Answer: 4) x ≤ −3

You could check a few numbers or you could just solve the inequality in a few steps.

−6x − 17 ≥ 8x + 25

−14x − 17 ≥ 25

−14x ≥ 42

x ≤ −3

Don't forget to flip the inequality symbol when you divide by Negative 14.

Without actually solving the problem, you could narrow down the choices by substituting numbers. If you start with x = 0, you get:

−6(0) − 17 ?≥ 8(0) + 25

− 17 ?≥ 25

This is Not True, so x = 0 is NOT a solution

Since x =/= 0, this means that Choices (2) and (3) can be eliminated. Next try x = 10, a large round number.

−6(10) − 17 ?≥ 8(10) + 25

-60 − 17 ?≥ 80 + 25

-77 ?≥ 105

Even without combining the terms, it should be obvious that this is Not True. So x = 10 is NOT a solution, which means that x ≥ 3 cannot be the answer. This leaves only Choice (4). You could check x = -10, but it isn't necessary.

22. Which set of data can be classified as qualitative?

1) scores of students in an algebra class
2) ages of students in a biology class
3) numbers of students in history classes
4) eye colors of students in an economics class

Answer: 4) eye colors of students in an economics class

Qualitative is descritpive and quantitative is numeric (countable, averagable, etc). Even without knowing that, three of these are one type, and one is not. Which is that one?

Scores, ages and numbers of students are all numeric data that you can average. Eye color is descriptive.

You could make histograms of the first three, but you could only make a bar graph of the last one.

23. Jack wants to replace the flooring in his rectangular kitchen. He calculates the area of the floor to be 12.8 square meters. The actual area of the floor is 13.5 square meters. What is the relative error in calculating the area of the floor, to the nearest thousandth?

1) 0.051
2) 0.052
3) 0.054
4) 0.055

Answer: 2) 0.052

Relative error is the difference between the actual size and the meausured size, divided by the actual size.

So it's (13.5 - 12.8) / 13.5, which is 0.05185..., and that rounds to 0.052.

24. The current student population of the Brentwood Student Center is 2,000. The enrollment at the center increases at a rate of 4% each year. To the nearest whole number, what will the student population be closest to in 3 years?

1) 2,240
2) 2,250
3) 5,488
4) 6,240

Answer: 2) 2,250

If you know anything about saving money, $2000 at 4% will NOT become $5000 or $6000 is just 3 years. Those numbers are silly.

You can use an exponential equation to solve this, or you can add 4% three times.

A = 2000(1.04)³ = 2249.728, which rounds to 2250, because you can't have .728 people.

If you want to do it in three parts, then it would be:

2000 * 0.04 = 80

2080 * 0.04 = 83.2

2163.2 * 0.04 = 86.528

2163.2 + 86.528 = 2249.728

25. Maria has a set of 10 index cards labeled with the digits 0 through 9. She puts them in a bag and selects one at random. The outcome that is most likely to occur is selecting

1) an odd number
2) a prime number
3) a number that is at most 5
4) a number divisible by 3

Answer: 3) a number that is at most 5

There are five odd numbers between 0 and 9 (1, 3, 5, 7, 9). There are four prime numbers in that range (2, 3, 5, 7). There are SIX numbers that are at most 5 (0, 1, 2, 3, 4, 5). And there are four numbers divisible by 3 (0, 3, 6, 9).

The correct answer is, therefore, 3.

More to come. Comments and questions welcome.

More Regents problems.

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