Slope Of A Line - College Level

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Find the slope of the line going through the following points:

(3, 6) and (-2, -4)

The term “Slope” usually refers to an incline of a straight line. In order to visualize a straight line that is drawn at an incline, it is important to note that this line will travel a certain horizontal distance while rising up or down a certain vertical distance. The horizontal distance is called the “run” and the vertical distance is called the “rise”. The ratio between the two is called the “slope” which also represents the steepness of that line.

The formula used for calculating the slope of a straight line is as follows:

Where m represents the slope going through two points that are indicated as:

In order to calculate the slope we use the above formula as follows:

Therefore, the slope of the line = 2

© Copyright 2008.Najwa S. Hirn. All rights reserved.

Solving for a Variable - Middle school math

Solve for t in the following equation:

3t + b = 5

When a problem asks to solve for a particular variable, it means to isolate that variable on one side of the equation (equal sign) and isolate the other terms on the other side of the equal sign.

In order to isolate the t term in the above problem, we must eliminate b from the left side of the equation. This can be accomplished if b is subtracted from both sides of the equal sign as follows:

3t + b – b = 5 – b

It is important to remember that whatever is done to the right side of the equation must be done to the left side also.

At this point b is eliminated from the right side of the equal sign and the resulting problem is:

3t = 5 – b

It is necessary at this time to eliminate the number 3 from the left side of the equation. There are two ways that may be used to accomplish this. Either multiply both side of the equation by (1/3) or divide both side of the equation by 3.

Lets divide each side by 3 as follows:

(3t) / 3 = (5 - b) / 3

The 3’s on the left side of the equation cancel out and the problem is finalized as follows:

t = (5 - b) / 3

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Number Sense - Elementary School Math

There were 35 pizzas at Andrew’s birthday party and 16 of them had only pepperoni topping while the rest had a mushroom topping. How many pizza had the mushroom topping?

a. 12
b. 19
c. 21
d. 8

Answer:  (b)  19 pizzass had only a mushroom topping.

The student needs to recognize number sense and know how to add, subtract, multiply and divide mathematical expressions.

In order to solve this problem, it is important to realize that the total number of pizza’s were 35 and since 16 had only pepperoni that means that the student must subtract 16 from the total of 35 to obtain the number of pizza’s with the mushroom topping.

35 – 16 = 19 pizzas with a mushroom topping.

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Fractional Expressions - College Level Math

Multiply the following fractional expression and simplify:

Answer:

In order to solve this problem, we first must factor the numerator and denominator of each fraction. Once they are factored the problem will look like the following:

Now it is time to multiply the numerators and multiply the denominators as follows:

It is obvious from the above that several factors can be cancelled. It is noted that the(a-1) factors can be cancelled from both numerator and denominator and one (a +4) can be cancelled from the numerator. The final simplified answer will look as follows:

© Copyrightreserved 2008.Najwa S. Hirn. All rights

Volume Of A Cube - High School Math

One face of a cube has an area of 64 cm square – What is the volume of this cube?

Answer : 512 cm cubed

In order to calculate the volume of a cube, the length, height and width of the cube must be known since the formula for finding the volume is:
V = l w h
The problem does not provide information about the length, width or height; therefore they must be calculated using the information we were given about the area.

In this problem, we are given the area of one face. We know that the formula for calculating an area for any two dimensional figure is:
A = l w
Since each face of a cube is constructed from a square, then we know that the length and width of this square must be the same and by multiplying the two, the area can be found.

Lets apply what we have:
64 cm square = l w
Since we know that the length and width must be equal for a square, we can take the square root of the area in order to calculate the length of width:

square root of 64 = 8

Therefore, the length and width of each face is 8 cm. Therefore, the height of this cube will also be
8 cm. The volume can now be calculated as follows:

V = (8cm)(8cm)(8cm) = 512 cm cubed.

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Reflection Of An Object - Middle School Math

If triangle ABC shown in the figure below is reflected about the y-axis what would the coordinates of the new triangle vertices be?

a. (-3, 3) (-5, 6) (-5, 3)
b. (-3, -3) (-5, -6) (-5, -3)
c. (3, -3) (6, -5) (3, -5)
d. (-3, -3) (-6, -5) (-3, -5)


























Answer: (a) (-3, 3) (-5, 6) (-5, 3)


In order to solve this problem, it is important to understand what happens when an object is reflected about an axis. When an object is reflected about an axis the vertices of the new object will have the same distance from that axis like the original object.

The reflection of triangle ABC is done about the y-axis in this problem. Therefore, the distances each ordered pair has from the y-axis will be the same in the new triangle. This means that the y-coordinate of each ordered pair will be the same in the new triangle like they are in triangle ABC. However, the reflection will mean that the x-coordinates of these points will have the opposite sign in the new object from the original one. In this case, the positive x-coordinates will change to negative x-coordinates.

To simplify this, we’ll look at the location of each ordered pair in the original triangle as follows:

Ordered pair (A) has the original points of (3, 3) therefore the reflected pair will be (-3, 3)

Ordered pair (B) has the original points of (5, 6) therefore the reflected pair will be (-5, 6)

Ordered pair (C) has the original points of (5, 3) therefore the reflected pair will be (-5, 3)

Note in all of the above points, the x-coordinate changed from a positive to a negative while the y-coordinate remained unchanged.

© Copyright 2008.Najwa S. Hirn. All rights reserved.

Elementary Math - Recognizing number sense

While visiting a historic district during a field trip, the students were told the dates that four of the historic houses were built. The chart below list those dates. According to the list, which house was built last?

a. House no. (1)
b. House no. (2)
c. House no. (3)
d. House no. (4)

Answer : ( c ) – House no. (3)

In order to solve this problem, a student must recognize which number in the table represents the highest number. That number will indicate the year the last house was built. By examining the table, it is determined that 1853 is the highest number in the table. Therefore House no. (3) was built last.

The student can list the numbers in ascending or descending order to assist in identifying the highest number.

In order to list the numbers in ascending order, they can be written as follows:

1803 - 1819 - 1836 - 1853

In order to list the numbers in descending order, the can be written as follows:

1853 - 1836 - 1819 - 1803

Both of the above cases indicates that the highest number is 1853 and the lowest number is 1803.

©Copyright 2008.Najwa S. Hirn. All rights reserved.

College Level Math – Vertex, Axis of Symmetry and Maximum/Minimum points

Find the vertex, the axis of symmetry, the maximum or minimum value of the quadratic function.

The above equation is written in the standard quadratic equation form which is:

It is important to be able to calculate the vertex, line of symmetry, and the maximum or minimum points from a quadratic equation. These crucial points can assist in identifying the characteristics of the resulting parabola and thus being able to sketch the resulting graph.

It is important to note that the sign of the (a) term will determine the orientation of the parabola. A positive (a) term – an (a) term greater than zero - will indicate that the parabola will open upwards, while a negative (a) term – an (a) term less than zero – will indicate that the parabola will open downward.

A vertex is either the highest or lowest point of a parabola depending on its orientation. Once calculated, the vertex is written in an (x ,y) ordered pair.

Therefore, it is easy to determine the following characteristics about the parabola presented in this problem. Those characteristics are:

1) This parabola opens upwards since the (a) term is positive or greater than zero.

2) This parabola will have a vertex at its lowest point.

3) This parabola will have a minimum point.

4) The axis of symmetry will be the vertical line passing through the x-coordinate of the vertex and dividing the parabola into two equal parts.

The following steps detail how the vertex is calculated for this problem:

Vertex

Axis of Symmetry

The axis of symmetry is the line that splits the parabola into two identical halves. It passes through the vertex.
Since this quadratic opens upwards, that means that the axis of symmetry is the x co-ordinate of the vertex. Therefore, the axis of symmetry is the vertical line that passes through x = 5

Maximum or Minimum points

The vertex of a parabola is usually considered as either a maximum or a Minimum value.

We determine if the point is a maximum or minimum by looking at the a co-efficient of the a-term of the quadratic equation. Since the a-term in this quadratic equation is greater than zero, then the vertex is a minimum point because the parabola opens upwards.

Therefore for this quadratic the minimum value point occurs at (5,-28)
If a > 0, the vertex is a minimum point and the minimum value of the quadratic function f is equal to the y-co-ordinate of the vertex. That means the minimum point for this quadratic = -28
This minimum value occurs at x = = -b/2a which is the x-co-ordinate of the vertex. For this quadratic the minimum value = 5



© Copyright 2008.Najwa S. Hirn. All rights reserved.

Rate - High School Math

Math at Hand: A Mathematics Handbook
A patient receives her nutrition by a stomach tube with the use of a feeding pump. The rate of fluid pumped is usually measured in milliliters per hour (mL/hr). The feeding pump used to infuse the formula in the tube is set to pump at a rate of 300 mL/hr. The patient is fed for only 45 minutes. How many milliliters did this patient receive?

Answer: 225 milliliter

The term milliliter per hour means a certain volume of fluid is to be pumped into a patient for a length of one hour. Therefore, in this problem the patient’s pump is set to feed the patient 300-milliliter of formula for every hour the pump is running. This also means that the rate is set for 300 mL per 60 minutes.

Since the patient received nutrition for only 45 minutes then the student must set up the problem as a ratio and proportion to be able to solve it. The steps are as follows:

[300 mL / 60 min] = [x mL / 45 min] = 225 milliliters

Note that the units for minutes disappear and the student is left with the milliliter units that are required to solve the problem.

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Measurements - Middle School Math

A long piece of wood is 8.25 feet long. If this piece were to be cut into 5 equal pieces how long will each piece be?

a. 11.25 inches
b. 19.80 inches
c. 22.75 inches
d. 27.15 inches

Answer : 19.80 inches (answer b)

The student must notice that the dimension of the original piece of wood is in feet units while the answers are in inch units. This is an important observation since all units must match in order to solve the problem. Therefore, as a starting point, we will convert the 8.25 feet into inches. In order to do that, the student must realize that each 1 foot = 12 inches. Therefore:

8.25 feet x [12 inch/1 foot] = 99 inches
(note here that the units for feet are eliminated and we are left with only the inch units we need to continue solving the problem).

Since we need to cut this piece into 5 equal pieces then we must divide the 99 inches by 5 to reach the final answer:

99 inches / 5 = 19.8 inches

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Subtraction - Elementary School Math

There are a total of 72 cars parked in the used parking lot 8 of them are blue. How many cars are not blue?

a. 58
b. 62
c. 64
d. 68

Answer: 64 (c )

This is a subtraction problem. The student must subtract the 8 blue cars from the total 72 cars in order to come up with what is remaining. The remaining number will be the number of cars that are not blue. Here is the step:

72 - 8 = 64

For additional interactive practice visit: Aplusmath.com

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Functions - College Level Math

If

f(x) = (x-3)

and

g(x) = (2x + 4)

Evaluate:

(f + g) (2)

Answer : 7

The above two equations are examples of the use of the Algebra of functions.

A function is a relation that does not repeat any first coordinates.

The problem is asking the student to evaluate (f+g)(2). This is the same as saying take the sum of functions f(2) and g(2) respectively and evaluate the result.

The steps involved in solving this problem are as follows:

• Substitute the equation of f(x) for the (f) value of the function.
  
• Substitute the equation of g(x) for the (g) value of the function.
  
• Add the two equations.
  
• Take the function with respect to (2) Evaluate the answer.
  

Here is the algebraic solution for the problem.

f(2) = ( 2 - 3) = -1

g(2) = ( 2(2) + 4 ) = ( 4 + 4 ) = 8

(f + g) (2) = ( -1 + 8 ) = 7


© Copyright 2008.Najwa S. Hirn. All rights reserved.

Parallel Lines - High School Math

The line shown on the Cartesian coordinate graph below passes by points AB. Another line needs to be drawn parallel to line AB and passes by point C shown on the graph and point D.

What coordinates should point D have that will make line CD parallel to line AB?

a. (-3, 4)
b. (-5, 2)
c. (-6, 0)
d. (-9,-1)

Answer: The coordinates for point D are (-9,-1).

In order to solve this problem the student must recall that parallel lines have the same slope. Therefore, the slope that is calculated for line AB will be the same slope for the line parallel to it that passes through point C.

Therefore, step one is to calculate the slope for line AB.
The coordinates for point A are (6,1)
The coordinates for point B are (1,0)

The slope formula states that:

Let apply the formula to find the slope for line AB:

m = (0 - 4)/(1 – 6) = (- 4)/(- 5) = 4/5

An examination of point C on the graph reveals that the coordinates for this point are:
(- 4 , 3).

The student must now use this point and check each of the above coordinates provided as a possible answer to determine which pair resulted in a slope of 4/5.

Once the coordinate pair of (-9 , -1) is checked, it reveals the following:

m = (- 1 – 3)/ (- 9 – [- 4])
m = (- 1 – 3)/ (- 9 + 4) = (- 4)/(- 5) = 4/5

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Finding Angles of a Triangle - Middle School Math

In the following triangle, if angle ABC equals 80 degrees and angle BAC equals 40 degrees, what does angle ACB equal to?

Answer:

Angle ACB equals 60 degrees.
The measures of the three interior angles of a triangle must equal to a total of 180 degrees. Therefore we must add the two shown angles and subtract that number from 180 in order to find the correct answer. The step is as follows:

180 - (80 + 40) = 60

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Data Analysis - Elementary School Math

A shirt manufacturing company manufactures three colored shirts. The circle graph below shows the percentages of each color that is produced. Based on the graph, what percent of blue shirts are produced?


Answer:

30% Blue Shirts

In order to solve this problem, the student must add the total percent of red shirts and white shirts produced and subtract that from 100%. The answer will give the total percent of blue shirts produced. The steps are as follows:


60 % white shirts + 10% red shirts = 70 % shirts out of the total

100 % total shirts – 70 % shirts produced = 30 % blue shirts that are produced.



© Copyright 2008.Najwa S. Hirn. All rights reserved

Equation of a Circle - College Level

Find the center and radius of the circle that is determined by the following equation:

Answer :
Center = (4,-3)
Radius = 6

The solution is shown step by step below. A detailed description of the solution is provided below the equations.

The problem requires that you find the values for (h , k) and the radius. In order to do that you must first solve for x and y to be able to find the radius points.

You start the problem by making sure that the similar variables are grouped together as follows:

It is noted here that in order to move the number 11 to the other side of the equation, we must first add 11 to both sides of the equation.

You must rework the problem to bring it to the standard form of the equation of the circle. Which is:

In order to do that we must complete the square inside each bracket to find the missing values. We must complete the square for x and for y in order to come up with two quadratic equations that we can factor.

We’ll start with x :
The way to complete the square is by taking half the middle value (number) of the quadratic equation, squaring it and adding it to both sides of the equation.

In this case, the middle value is 8 so we take half of 8, which is 4, and square it so we get 16. This number (16) is added to both sides of the equation as follows:

Now we do y:
The way to complete the square is by taking half the middle value (number) of the quadratic equation, squaring it and adding it to both sides of the equation.

In this case, the middle value is 6 so we take half of 6, which is 3, and square it so we get 9. This number (9) is added to both sides of the equation. The resulting equation looks like this:

Now we have two quadratic equations that can be factored. Once they are factored we end up with the following equation:

At this point, it is feasible to solve for x and y by setting each factor to 0 and solving for the variable we come up with:
X = 4
Y = -3

Once we have solved for x and y, it means we found our h and k values which in this case comes out to be (-3,-4).

We still need to find the radius. Since we added a 9 and a 16 to the 11 on the right side of the equation that means we added a total of 36 to the right side of the equation. This 36 is the square of the radius.

If we take the square root of 36 we come up with 6, which is our radius.

Now we have solved the equation and are results are:

(4 , -3) , 6



© Copyright 2008.Najwa S. Hirn. All rights reserved.

Interpreting Line Graphs - High School Math

When a small manufacturing company started in 1950, it was producing 2500 plastic bottles as shown on the following graph:

According to the graph, what year did the company start producing five times as much as when it started?

a. 1960
b. 1965
c. 1970
d. 1975

Solution: 1970

The above is a line-graph problem. In order to solve it, we must recognize the elements that make up the graph.

The horizontal axis on the graph is labeled “Years”. This represents each year that the company has been in business. Five labeled years are shown starting with 1950 and ending with 1990. Notice that there are four equal vertical divisions between each year label (four equal division between 1950 to 1960 and so forth), therefore, each division represents 2 years. This means that the first vertical label on the right of 1950 will represent 1952, the second will be 1954, the fourth will be 1958 and the last is the 1960 shown on the graph. The same applies to the other years on this horizontal axis.

The vertical axis on the graph is labeled “Number of bottles in thousands”. This means the number of plastic bottles produced by the company as it grew and prospered. Five labeled segments are shown starting at 0 bottles and ending at 40,000 bottles. Notice, again, that there are four equal horizontal divisions between each segment (four equal divisions between 0 to 10,000 and so forth); therefore, each division represents 2500 plastic bottles. This means that the first horizontal line above the 0 mark will represent 2500; the next one up will be 5000 and so forth. The same applies to the other labels on this axis.

The donuts that appear as intersections for the vertical and horizontal axes represents how many bottles are produced in what year.

The first donut represents 2500 bottles produced in 1950 because the donut appears as the intersection of the 2500 horizontal line with the vertical line labeled 1950.

The question is asking what year the company produced five times as much bottles as when it started. The student can calculate five times 2500 as follows:
5 x 2500 = 12500
Therefore, the problem wants us to find at what year the company was producing 12500 bottles.
Examining the graph reveals that 12500 appear on the horizontal line that is one increment above the 10000 mark. (Since each equal division is 2500, that is added to 10000 to come up with 12500). When the student examines the vertical line that this donut appears at, it is noted that the year is 1970.

Therefore in 1970 the manufacturing company produced five times as much plastic bottles as when it first started in 1950

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Similar Objects - Middle School Math

The following two figures ABCDE and FGHIJ are similar. Figure ABCDE is twice as large as figure FGHIJ. Therefore the ratio of their corresponding sides is 2:1. If side BC is 12 units long, what is the length of side GH in units?

a. 3.5 units
b. 5 units
c. 6 units
d. 7.5 units

Answer: 6 units ( c )

The above two geometric figures are similar because they both have the same shape. Figure FGHIJ was constructed by scaling down (shrinking) figure ABCDE. Similarity is defined as one shape that can be obtained from another by uniformly stretching one.

In above problem, the ratio of 2:1 means that each side in figure FGHIJ is half the length of the sides in figure ABCDE which means it was stretched down by half. Therefore, since side BC is 12 units long, then side GH will be half of that which will be 6 units.

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Adding and Subtracting Fractions - Elementary Math

1) What is the answer if you add 3/4 + 2/4:

a. 5/4
b. 6/4
c. 7/4
d. 8/4

2) What is the answer if you subtract 10/3 – 6/3:

a. 1/3
b. 4/3
c. 5/3
d. 7/3

Answer : 1) 5/4 (a)
2) 4/3 (b)

A fraction consists of two parts, an upper number called numerator and a lower number called a denominator. When adding or subtracting fractions that have the same denominators then the student should leave the denominator alone and just add or subtract the numerators.

The above problems are solved as follows:

3/4 + 2/4 = 5/4
(the 3 is added to the 2 to get a result of 5. The 4’s at the bottom are left untouched)

10/3 – 6/3 = 4/3
(the 6 is subtracted from the 10 to get a result of 4. The 3’s at the bottom are left untouched)

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Solving equations for indicated variables - College Level

Solve each of the following formula for the indicated variable:

1. y = mx + b for x
2. 4x – 3y = 12 for y

Solution:

It is important to remember that when asked to solve an equation for an indicated variable that usually means isolate that variable on one side of the equal sign and move everything else to the other side.

1) y = mx + b
In order to solve for x we must try to isolate it by itself on one side of the equation. The first
In this process is to subtract b from both sides of the equation as follows:
y b = mx + b – b
y– b = mx
Now it is time to eliminate m from the right hand side of the equation by dividing both sides
By m as follows:
(y-b)/m = (mx)/m
(y-b)/m = x
we have successfully solved this equation for x.

2) 4x-3y = 12
We must isolate the y on one side to be able to solve the equation. We start by adding y to
Each side of the equation as follows:
4x – 3y + 3y = 12 + 3y
4x = 12 + 3y
Now we must subtract 12 from each side of the equation as follows:
4x – 12 = 12 – 12 +3y
4x – 12 = 3y
Now we must divide each side of the equation by 3 as follows:
(4x-12)/3 = (3y)/3
(4x-12)/3 = y
We can simplify this expression as follows:
(4x)/3 – 12/3 = y
4/3 x – 4 = y

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Measures and rates - High School Math

Allison burns 10 calories per minute when she works out on the treadmill. How many calories will she burn after she has used it for three-quarter of an hour?

a. 175
b. 225
c. 450
d. 540

Answer : c

In order to solve this problem the student must realize that three-quarters of an hour is equivalent to 45 minutes. Therefore, if Allison burns 10 calories per minute and the questions is asking how many calories she can burn in 45 minutes, then we must multiply the 10 calories that she burns per minute by the 45 minutes as follows:

10 calories/minute x 45 minutes = 450 calories (It is important to note at this time that the units for minutes disappear and we are left with only calories which is what is required)

©Copyright 2008.Najwa S. Hirn. All rights reserved.