Algebra Problems of the Day (Integrated Algebra Regents, August 2010)

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 2010

Part II: Each correct answer will receive 2 credits. Partial credit is available. One computational or conceptual error will cost 1 credit. A second error will cost the other.


31. Express in simplest form:

(45a⁴b³ - 90a³b) / (15a²b)

Answer:

Use the Distributive Property in reverse to create factors. Factors can be eliminated from the numerator and the denominator.

(45a⁴b³ - 90a³b) / (15a²b)

(45a³b)(ab² - 2) / (15a²b)

(3a)(ab² - 2) / 1

(3a)(ab² - 2)

You could also have written 3a²b² - 6a, which is an equivalent expression.

32. Joseph typed a 1,200-word essay in 25 minutes. At this rate, determine how many words he can type in 45 minutes.

Answer:

Divide the number of words by the number of minutes to get the Unit Rate. Then multiply by 45.

1200 words / 25 minutes = 48 words / minute.

48 words / minute * 45 minutes = 2160 words.

You could also use a proportion, which would involve the same two operations:

1200 / 25 = x / 45

25x = (1200)(45)

x = (1200)(45) / 25 = 2160.

33. Express −3 √(48) in simplest radical form

Answer:

Find the largest perfect square that is a factor of 48, or write doen the factors of 48 and circle the pairs:

The first few perfect squares are 4, 9, and 16 -- 25 is too big to go into 48. Of these, 16 is the largest one that is a factor of 48.

So −3 √(48) = −3 √(16 * 3) = −3 √(16)√(3) = −3(4)√(3) = −12√(3).

The factor method:

48 = 2 * 24 = 2 * 2 * 12 = 2 * 2 * 2 * 6 = 2 * 2 * 2 * 2 * 3

= (2 * 2) * (2 * 2) * 3

-3 √((2 * 2) * (2 * 2) * 3) = -3 * 2 * 2 √(3) = -12 √(3)

End of Part II.

More to come. Comments and questions welcome.

More Regents problems.

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