Algebra Problems of the Day (Integrated Algebra Regents, June 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, June 2011

Part I: Each correct answer will receive 2 credits.


26. Michael is 25 years younger than his father. The sum of their ages is 53. What is Michael’s age?

1) 14
2) 25
3) 28
4) 39

Answer: 1) 14

If Michael is M years then his father is M + 25 and M + M + 25 = 53.

2M + 25 = 53

2M = 28

M = 14

Realistically, Choice (1) was the only one that even made sense. If Michael was 25, then his father would be 50, and that's a sum of 75. This is too big and the other numbers are even bigger.

27. What is the product of (6 × 10³), (4.6 × 10⁵), and (2 × 10^-2) expressed in scientific notation?

1) 55.2 × 10⁶
2) 5.52 × 10⁷
3) 55.2 × 10⁷
4) 5.52 × 10¹⁰

Answer: 2) 5.52 × 10⁷

Choices (1) and (3) are NOT in Scientific Notation, where the initial number must be between 0 and 10, exclusive.

When you multiply the 10s, you add the exponents, and 3 + 5 - 2 = 6. This is one off from 7, but it's obvious that 10 is totally incorrect.

Multiplying will give us 55.2 × 10⁶, which gets rewritten as 5.52 × 10⁷.

28. Which notation describes {1, 2, 3}?

1) {x | 1 ≤ x < 3, where x is an integer}
2) {x | 0 < x ≤ 3, where x is an integer}
3) {x | 1 < x < 3, where x is an integer}
4) {x | 0 ≤ x ≤ 3, where x is an integer}

Answer: 2) {x | 0 < x ≤ 3, where x is an integer}

Choice (2) is technically correct, but at the same time one of the poorer ways to write this. It would be better to write two < symbols or two ≤ symbols, and write the other numbers accordingly.

Choice (1) is incorrect because 3 is in the set, but not in the set-builder notation.

Choice (3) is incorrect because 1 is in the set, but not in the set-builder notation.

Choice (4) is incorrect because 0 is not in the set, but is included in the set-builder notation.

the intersection of the sets and Choice (3) isn't the union because it's missing 6 and 12.

29. What is 7/(12x) - y/(6x²) expressed in simplest form?

1) (7 - y)/(6x)
2) (7 - y)/(12x - 6x²)
3) -(7y)/(12x²)
4) (7x - 2y)/(12x²)

Answer: 4) (7x - 2y)/(12x²)

To combine the fractions you need a common denominator, and the least common denominator betweet 1x and 6x2 is 12x². Since the answer must be in simplest form, we know that Choice (2) can be eliminated.

7/(12x) * (x/x) - y/(6x²) * (2/2)

= 7x/(12x²) - 2y/(12x²)

= (7x - 2y) / (12x²)

That is Choice (4).

30. When 5x + 4y is subtracted from 5x − 4y, the difference is

1) 0
2) 10x
3) 8y
4) -8y

Answer: 4) -8y

Write out (5x - 4y) - (5x + 4y). This becomes 5x - 4y - 5x - 4y, which is -8y.

End of Part I.

More to come. Comments and questions welcome.

More Regents problems.

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