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 I: Each correct answer will receive 2 credits.


26. When 8x² + 3x + 2 is subtracted from 9x² - 3x - 4, the result is

1) x² - 2
2) 17x² - 2
3) -x² + 6x + 6
4) x² - 6x - 6

Answer: 4) x² - 6x - 6

You take the first expression away from the second expresion. in other words, it's the second expression minus the first one. You may want to write it vertically.

(9x² - 3x - 4) - (8x² + 3x + 2)
9x² - 3x - 4 - 8x² - 3x - 2
9x² - 8x² - 3x - 3x - 2 - 4
x² - 6x - 6

27. Factored completely, the expression 3x³ - 33x² + 90x is equivalent to

1) 3x(x² - 33x + 90)
2) 3x(x² - 11x + 30)
3) 3x(x + 5)(x + 6)
4) 3x(x - 5)(x - 6)

Answer: 4) 3x(x - 5)(x - 6)

It should be obvious that each term is a multiple of 3 and each term has a factor of x. Of course, this is a "no-brainer" since each of the four choice includes 3x. That said, divide each term by 3x and then see if you can factor the rest.

3x³ - 33x² + 90x = 3x(x² - 11x + 30x) Eliminate Choice (1).

However, x² - 11x + 30x can be factored into (x - 5)(x - 6). Which gives you Choice (4).

28. Elizabeth is baking chocolate chip cookies. A single batch uses 3/4 teaspoon of vanilla. If Elizabeth is mixing the ingredients for five batches at the same time, how many tablespoons of vanilla will she use?

3 teaspoons = 1 tablespon

1) 1 1/4
2) 1 3/4
3) 3 3/4
4) 5 3/4

Answer: 1) 1 1/4

Find the total number of teaspoons needed, then convert that to tablespoons.

3/4 * 5 = 15/4 teaspoons * 1 tablespoon / 3 teaspoons = 15/12 tablespoons.

15/12 = 1 3/12 = 1 1/4

29. A car depreciates (loses value) at a rate of 4.5% annually. Greg purchased a car for $12,500. Which equation can be used to determine the value of the car, V, after 5 years?

1) V = 12,500(0.55)⁵
2) V = 12,500(0.955)⁵
3) V = 12,500(1.045)⁵
4) V = 12,500(1.45)⁵

Answer: 2) V = 12,500(0.955)⁵

If it is LOSING value than the rate must be less that 1.00 or 100%. Eliminate Choices (3) and (4).

You subtract the amount, 4.5% from 100%, which is 95.5% or .955. That is Choice (2).

Choice (1) would be if the appreciation was 45% instead of 4.5%.

30. The cumulative frequency table below shows the length of time that 30 students spent text messaging on a weekend.
Which 10-minute interval contains the first quartile?

1) 31 - 40
2) 41 - 50
3) 51 - 60
4) 61 - 70

Answer: 3) 51 - 60

The median would be 15.5, and the lower half of the data would be the bottom 15 numbers. The median on the lower half (which would be the lower quartile) would be the 8th student. The 8th student would be in the 31-60 interval on the cumulative frequency table, meaning that the first quartile is contained in the interval of 51-60.

More to come. Comments and questions welcome.

More Regents problems.