Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.
More Regents problems.
August 2012
Part I: Each correct answer will receive 2 credits.
6. Jason’s part-time job pays him $155 a week. If he has already saved
$375, what is the minimum number of weeks he needs to work in
order to have enough money to buy a dirt bike for $900?
1) 8
2) 9
3) 3
4) 4
Answer: 4) 4
You can set up an equation, or you can start with 375 and keeping adding 155 to it until you are over 900.
Set up an inequality in the form ax + b > c. The b represents the starting amount that he has, which is 375. This isn't repeated, so it doesn't get a variable. The a is the rate of change (the slope), which is the amount that occurs repeated, which is why it goes before the x. This is the 155.
155x + 375 > 900
155x > 525
x > 3.387...
Do NOT round down. He would have enough money after only three weeks. The first integer GREATER THAN 3.387... is 4. Jason will have money left over.
7. The expression 9a2 - 64b2 is equivalent to
1) (9a - 8b)(a + 8b)
2) (9a - 8b)(a - 8b)
3) (3a - 8b)(3a + 8b)
4) (3a - 8b)(3a - 8b)
Answer: 3) (3a - 8b)(3a + 8b)
The expression 9a2 - 64b2 is the difference of two perfect squares. Each part of it is a perfect square (which means you can take a square root of each component) and the two terms are separated by a minus sign.
When we have this, it means that the polynomial can be factored into two conjugates. Conjugates are two binomials that are identical, except that one has a plus and one has a minus separating them. When you multiply the two conjugates, you will get a zero pair with the two "middle" terms. (That is, the "OI" in "FOIL" will cancel out and become zero.)
9a2 - 64b2 = (3a - 8b)(3a + 8b). Note that the order of the two binomials doesn't matter.
Note that if you have two minus signs, then the middle terms, when combined, will not have a sum of zero.
Using this line, what is the best estimate for profit in the 18th month?
8. The scatter plot below shows the profit, by month, for a new
company for the first year of operation. Kate drew a line of best fit,
as shown in the diagram.
1) $35,000
2) $37,750
3) $42,500
4) $45,000
Answer: 3) $42,500
Interesting that there were two scatter plot questions so close together. (If you missed the previous post, question 4 was about correlation.)
Look at the line of best fit. Then go to 18 on the x-axis and move up to that line. You will see that the y-coordinate is between 40,000 and 45,000. So Choice 3 is the best estimate.
9. Which statement illustrates the additive identity property?
1) 6 + 0 = 6
2) -6 + 6 = 0
3) 4(6 + 3) = 4(6) + 4(3)
4) (4 + 6) + 3 = 4 + (6 + 3)
Answer: 1) 6 + 0 = 6
The identity property for addition states that adding 0 to a number doesn't change the number. It stays the same. (Multiplying by 1 for multiplication identity.)
Choice (2) shows an example of the Inverse Property of Addition, where adding two additive inverses yields a sum of zero. These two numbers are a zero pair.
Choice (3) is the Distributive Property of Multiplication over Addition.
Choice (4) is the Associative Property.
How far, to the nearest tenth of a mile, did he walk?
10. Peter walked 8,900 feet from home to school.
1) 0.5
2) 0.6
3) 1.6
4) 1.7
Answer: 4) 1.7
To change feet into miles, divide by 5,280.
8900 / 5200 = 1.71...
The sum of the leading coefficients 3 and -1 is 2, so eliminate Choice (4).
Since 8900 is greater than 5280, it's obvious that he walked more than a mile, so Choices (1) and (2) should be eliminated immediately.
More to come. Comments and questions welcome.
More Regents problems.
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