Author Archives: Linda

09.11.14
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SAT – number properties

Q: Find the largest number less than 4000 that is the square of an integer.

Answer: 3,969

Explanation:  The simplest, fastest way to do this is:

On a calculator, find the square root of 4000.  It is 63.245….  Round this down to 63.  Now, square 63.  It is 3,969.

This is the way you should be thinking for SAT problems.  If you decide instead to count down from 3,999, finding the square root of each number until you find a whole number, you will have to do this 31 times until you find the right one!  (And by the way, why 31?  Doesn’t 3999 – 3969 equal 30?  Was I wrong?  No!  This is another little SAT trick.  Always add 1 to include the first or last number, whichever one wasn’t counted in the original 30.)

 

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09.10.14
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Calculus – Limits

Q: Find the limit:

Answer: -5

Explanation: Direct substitution (plugging in -1) yields 0/0, an indeterminate form. Factor and cancel:

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09.10.14
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SAT – number properties

Q: The variables a, x, and b are positive integers. If ax = 7, xb = 28, and ab = 4, what is the value of a?

Answer: 1

Explanation:

a, x, and b are all positive integers, so the only possibilities for the variables are factors of 7, 28, and 4.

Since 7 is prime, a and x must be 7 and 1, but in which order? If x = 1, then b = 28, and then we can’t find an integer value for a from the last equation. So, x must instead be 7.

It follows that a = 1 and b = 4.

The answer is 1.

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09.10.14
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Geometry – polygons

Q: A hexagon has interior angles measuring (in degrees) 137, 100, 150, 114, 90, and x. What is the value of x?
 
 Answer: 129
Explanation:
The total measure of all the interior angles in a polygon is (n-2) * 180, where n is the number of sides of the polygon.
A hexagon has 6 sides, so n = 6. The total is (6-2) * 180 = 720.
Add up the five given angle measures: 137 + 100 + 150 + 114 + 90 = 591.
The measure of the remaining angle is 720 – 591 = 129.
My TI-84 Plus program POLYGON will find this total for you.
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09.10.14
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SAT – Counting and Ordering

Q: There are 11 basketball teams in a tournament. Each team plays each other team once. How many games are played?

 Answer: 55

Explanation 1: This question is really ‘How many ways are there to choose 2 teams from a total of 11 teams, if the order of the teams is not important?’ It’s a combination question. Find the nCr function on your calculator. On the TI-84 Plus it’s in MATH-PRB-3. Type in 11 nCr 2. It’s 55. (There is a longer way to do it by hand, using the formula for nCr.)

Explanation 2: Team A plays one game against each of the 10 other teams. That’s 10 games so far. Team B plays 9 other games (its game against Team A has already been counted). Team C plays 8 other games, … all the way down to the 1 last game between teams J and K. Add: 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 55.

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09.10.14
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Algebra 2 – exponential functions

Q: In 2010, River City had a population of 500. If the population has increased by 6% each year since then, what is the population this year, in 2014?

Answer: 631

Explanation:

Alternatively,

Calcululate 6% of 500 by doing .06 * 500 = 30.

Increase 500 by 30, and you find that the population is 530 in 2011.

Now, increase 530 by 6% (530*.06, added to 530, or shortcut is to do 530*1.06).

The population is 561.8 in 2012. Don’t round this number yet – not until you get to the end of the problem.

Doing this calculation 2 more times gives you 595.508 for 2013 and 631.23… for 2014.

Now, round to the nearest whole number: 631. It’s not the most efficient way, but you get the right answer.

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09.10.14
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Precalculus – Probability

Q: In a group of 10 randomly chosen people, what is the probability that at least two people share the same birthday? (same month and day)

Answer: 12% 

Explanation: (For simplicity, this explanation doesn’t take into account people with leap-day birthdays, but it wouldn’t change the answer by much!)

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09.10.14
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Precalculus – logarithms

Q: Solve the following equation involving logarithms:

Answer: x=4

Explanation:

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09.10.14
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Geometry – volume

Q: If 20 solid gumballs with diameter 1 inch are placed inside a hollow sphere of diameter 6 inches, how much empty space is there in the larger sphere?

Answer:

Explanation:

 

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09.10.14
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Algebra 1 – Find the equation of a line

Q: If the slope of the line through (-3,6) and (1,y) is 1/2, what is the value of y?

Answer:  8

Explanation: Slope is computed by the formula (y2 – y1) / (x2 – x1) … the change in y divided by the change in x.

So,

(y – 6) / (1 – -3) = 1 / 2

(y – 6) / 4 = 1 / 2

Multiply both sides by 4 and you have:

y – 6 = 2

y = 8

BUT, if you’re taking a test and having trouble remembering the method above, try this instead: Plot the point (-3,6) on a graph and move from that point with a slope of 1/2. Move up 1, right 2, up1, right 2… and repeat this pattern until your x-value is 1. Then, read the y-value at the same point. This is not my favorite way, but it works!

And one more thing!  If this were a multiple choice question, you could use my TI-84 Plus program ‘DISTANCE’ (which finds distance, midpoint, and slope for any two ordered pairs).  Substitute one of the answer choices for x, and see if the program reports that slope is 1/2.

 

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