Algebra Problems of the Day (Integrated Algebra Regents, January 2013)

While I'm waiting for new Regents exams to come along (AND THEY ARE COMING SOON!), I'm revisiting some of the older NY Regents exams.

More Regents problems.

Administered January 2013

Part III: Each correct answer will receive 3 credits. Partial credit can be earned.


34. In a game, a player must spin each spinner shown in the diagram below once.

Draw a tree diagram or list a sample space showing all possible outcomes.

Determine the number of outcomes that consist of a prime number and a letter in the word “CAT.”

Answer:

A sample space is a list of all the possible outcomes. There are five outcomes on the first spinner and three on the second, so there are 15 possible outcomes, and you can list them in order very quickly:

1A 1B 1C 3A 3B 3C 5A 5B 5C 7A 7B 7C 9A 9B 9C

You can write those in any order, but I would suggest you put the number first. (It shouldn't matter, but I strongly suggest it.) You could've written 1A 3A 5A ... etc, and then B and C, and you would have been fine.

A tree diagram is more visual and is more helpful where there are a lot of outcomes to list, particularly when you list words (e.g., red hat, blue shirt, and striped shorts).

If you have drawn a tree diagram or listed a sample space, then you only need to supply a numerical answer to the last part of the question. The first question would be considered the "work" for the second. You can look at your first response and count the positive results. If you skipped the first portion of this question, you would have to show some kind of work, how you approached the problem and got your answer.

Obviously, only the letters C and A appear on the second spinner. There are 3 prime numbers, 3, 5, and 7, on the first spinner. According to the Counting Principle 2 * 3 = 6 outcomes.

Instead of the Counting Principle, you could have just circled the six positive outcomes you were looking for, but that wouldn't have been necessary.

35. The cost of three notebooks and four pencils is $8.50. The cost of five notebooks and eight pencils is $14.50. Determine the cost of one notebook and the cost of one pencil.

[Only an algebraic solution can receive full credit.]

Answer:

Let N = the price of one notebook and P = the price of one pencil
You could use x and y, but meaningful variables are better to use.
Then set up a system of equations to solve.

3N + 4P = 8.50
5N + 8P = 14.50
Double the first equation, and then subtract.
6N + 8P = 17.00
5N + 8P = 14.50

N = 2.50
Substitute into the original equation to find the cost of a pencil.
3(2.50) + 4P = 8.50
7.50 + 4P = 8.50
4P = 1.00
P = 0.25

Write your answers in sentences. Don't leave it as N= and P=. We made those up!

Now if you defined your variables at the top of the page, you're all set! You already wrote the sentences you need!

Let N = the price of one notebook = 2.50 and P = the price of one pencil = 0.25

36.Wendy measures the floor in her rectangular bedroom for new carpeting. Her measurements are 24 feet by 14 feet. The actual measurements are 24.2 feet by 14.1 feet.

Determine the relative error in calculating the area of her bedroom. Express your answer as a decimal to the nearest thousandth.

Answer:

Relative error and percent of error used to be questions that the Regents asked about a lot. The biggest different between the two of them is that relative error is a decimal and percent of error is a percentage, and if you gave a percentage for relative error, you would lose a point. Annoying, but it wasn't the answer that they asked for.

Relative error is the different between the actual and the estimate divided by the actual amount.

The estimated area is 24 * 14 = 336.

The acutal area is 24.2 * 14.1 = 341.22

The relative error is (341.22 - 336) / 341.22 = 0.0152..., which is 0.015 to the nearest thousandth. (It rounds down because 0 - 4 rounds down.)

Be careful that you enter it into the calculator correctly. If you got 340.235 as an answer, it was because you forgot the parentheses and the Order of Operations did you in.

End of Part III






More to come. Comments and questions welcome.

More Regents problems.