Tuesday, September 30, 2014

How do we determine whether or not an ordered pair is a solution to a given inequality? 

Suppose the inequality is 4x + 3y > 19 and the ordered pair is (2, 6)

Substitute 2 for x and 6 for y in the inequality.

4(2) + 3(6) > 19

8 + 18 > 19

26 > 19.

Since that is true, the ordered pair is a solution. If the statement was false, the ordered pair is not a solution.

If you have a graph of an inequality and want to know what portion of the graph is part of the solution, pick an ordered pair on either side of the line and substitute the value into the equation as done above. If the statement is true, that side of the line is shaded to represent the solution region. If if the statement is false, the other side of the line is shaded to represent the solution region.

Thursday, September 25, 2014

Percent increase and decrease

Suppose the cost of a particular item last year was $320 and increased to $360 this year. What is the percent increase?

First, calculate the dollar amount of the increase, which is $40. From here we divide that number by the original amount of $320. Multiply that decimal result by 100 to convert to percent. So the increase is 12.5 %

Suppose a item depreciates in value by 20% every year.  The original value of the item is $1,200. What is the value of the item after two years?

We take 20% of $1,200 which is .20 times 1,200, which is 240. Subtract 240 from the initial 1,200 to get 960.  After the first year the value is $960.

Now take 20% of 960, which is 192. Subtract 192 from 960 to get 768. The value of the item after two years is $768.
Suppose two go-carts are driving around a circular track that is 360 feet in circumference. There is a camera in both cars that are 60 degrees apart. How are apart are the two cars?

The number of degrees around a circle is 360. So if the cameras are at a 60 degree angle from each other, that is 1/6 of the entire circle.

Since the circumference of the circle is 360 feet, we take 360 divided by 6 to get the distance between the cars. Therefore, the cars are 60 feet apart.

This method holds true no matter what the angle is between the cameras.

Monday, September 22, 2014

Here's a good sample SAT math problem.

Suppose a and b are real numbers and (a + b)^2 = 56 and a^2 + b^2 = 18. What is the value of  ab?

The key to the problem is knowing that when you expand (a + b)^2, you will have a^2 + b^2 and ab as part of the expansion, which makes the problem easy.

(a + b)^2 = a^2 + 2ab + b^2

Now substitute and solve for ab.

56 = 18  + 2

38 = 2ab

19 = ab

Thursday, September 18, 2014

Suppose W,X,Y,Z are all digits with the following conditions:

X = W + Y + Z
W - 5 = Y
Z = W - 1

What is the 4-digit number XWYZ?

We can approach this by listing all the possible values starting with Y.
Y must be 0, 1, 2, 3, 4 since Y + 5 = W and the largest number that W or any of the numbers could be is 9
Therefore, we know that W has to be 5,6,7,8, or 9.

One we know that, it's easy to see that Z must be either 4,5,6,7, or 8 since it's 1 less than W.
X is the other letters added together. The only possible number X can be is 9, obtained when W = 5, Y =0 and Z = 4. Any other combination of those numbers will make X greater than 9.
The 4-digit number is 9504.

Tuesday, September 16, 2014

If 2^(abc) = 64, where a, b and c are positive integers, what is the value of a + b + c and what is 2^a = 2^b + 2^c?

The first thing we need to do is figure out what power we raise 2 to get 64.
2^2 = 4
2^3 = 8
2^4 = 16
2^5= 32
2^6 = 64

So we know abc = 6 and we know that a,b, and c are different positive integers. So a, b and c must be 1, 2 and 3 in any order since 1*2*3 = 6.

Therefore a + b + c = 6 as well and 2^a + 2^b + 2^c  = 2^1 + 2^2 + 2^3 = 2 + 4 + 8 = 14.

The problem is simply if one knows the rules for exponents and breaks the problem down into steps.

Tuesday, September 9, 2014

Here's some good math jokes, as nominated by teachers and Twitter.



Talking sheepdog gets all the sheep in the pen for his farmer. He comes back and says ‘All 40 accounted for.’ Farmer says, ‘I’ve only got 36!’ Sheepdog replies, ‘I know, but I rounded them up.'

Two cats called '1,2,3' & 'un,deux,trois' had swim race across channel.
1,2,3 cat won because un, deux, trois cat sank!

Followed by seven brilliant runners-up:

Hired an odd-job man to do 8 jobs for me. When I got back, he'd only done jobs 1,3,5, and 7


Have you heard about the mathematical plant? It has square roots


After careful investigation, it was found that aliens' heights were paranormally distributed


I hit someone with a scientific calculator - I used the cosh button


What kind of tree could a maths teacher climb? = Geometry

What do you get if you cross a maths teacher and a clock? Arithma-ticks!


Last night I dreamed that I was weightless! I was like, 0mg

Friday, September 5, 2014

Suppose a rectangular field used to play soccer has an area of 7,800 square yards. Is the length is 70 yards longer than the width, what are the dimensions of the field?

Area of a rectangle is length times width. Let x = the width of the soccer field. That means the length of the field is x + 70.

Now take the formula for area to get A = x(x + 70)

We know area is 7,800 therefore,
7,800 = x(x + 60)
7,800 = x^2 + 70x
0 = x^2 + 70x - 7800

Factoring gives us
0 = (x + 130)(x - 60)
x = -130 or x = 60

Since we are talking about a length, x = -130 makes no sense, so the dimensions of the field are 60 by 130.
If you draw a straight line from home plate to first base and continue around the bases in such a fashion, you form a square with area of 8,100 square feet. What is the distance from home plate to second base rounded to the nearest tenth of a foot?

First you need to know that the area of a square is side times side. So we know side times side is 8,100. Since both sides are the same length, solve by taking square root of 8,100, which is 90.

Now the distance from home plate to second base is the length of the diagonal of a square. The formula for that is square root of 2 time the side. Multiply 90 by square root of 2 and you get approximately 127.3 feet.

Tuesday, September 2, 2014

Suppose you want to fence off two rectangular pieces of land that have the same area. You have 300 feet of fencing. The total length of both pieces of land are 2x feet (each x feet in length). The width of the land is y feet (2 y's on the outside representing the width of the overall and and y feet in width down the middle separating the two pieces of land)

What is the area of the land in terms of x?  What length and width will give the maximum area of the land?

You know that area of a rectangular is length times width.  So the overall area is x(y). But we want the entire area in terms of x.

Well we know the total amount of fencing is 300 feet, so x + x + y + y + y = 300. (just took all the dimensions and added them).

2x + 3y = 300

Solve for y to get 3y = 300 - 2x

and y = 100 - (2/3)x

So now we can get the area all in terms of x..

Area = x(100 - (2/3)x)

100x - (2/3)x^2

To find the value of x which maximizes area, you can use calculus or simply take -b/2a, where b is the number in front of the x and a is the number in front of the x^2.

So -b/2a = -100/[2(-2/3)] = 75

The length is 75. To find the length, substitute 75 in for x in the equation 2x + 3y = 300. Therefore, y = 50.

The width of the land is 50 feet for a total area of 3,750 square feet.

What is the equation of the line tangent line to the circle with center (3,4) at the point (6,0)?

The tangent line is the line the touches the circle and one point, in this case the line will touch at (6,0).  If you draw a segment from the center (3,4) to the point (6,0) the tangent line will be perpendicular to that segment. Perpendicular lines have slopes that are negative reciprocals of each other, so the slopes will multiply to equal -1.

So the slope of the segment from (3,4) to (6,0) is (0-4)/(6-3) = -4/3.  Therefore the slope of the line perpendicular is 3/4. 

3/4 times -4/3 = -1, which is what the slopes of perpendicular lines must multiply to.

Now take the equation y = mx + b where m is the slope and b is the y-intercept. Remember the y-intercept is where the line crosses the y-axis.

We know m = 3/4, we know x = 6 and y = 0.

So put the numbers in and solve for b.

0 = (3/4)(6) + b

0 = 4.5 + b

b = -4.5 or - 9/2

So the equation of the line is y = (3/4)x - (9/2)