Positive and Negative Numbers - High School Math

If -8 is divided by a number less than 0 the result will be as follows:
a. A number between –8 and +8
b. A number less than 0
c. A number greater than 0
d. A number between –8 and 0

Answer : c

Explanation:
The numbers that are less than 0 are all negative numbers. Therefore when we divide a –8 by another negative number the result will be a positive number. Therefore, since all numbers greater than 0 are positive then that will be the answer that we will obtain.
For example:
-8/-2 = +4
For additional information and some practice problems go to:
Multiplying and Dividing Using Negative Numbers

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Probability - Middle School Math

Sam’s teacher hands out pencils to each of the students taking a test. She has different colors pencils. The following shows the number pencils in each color category:

Purple = 10 pencils
Green = 20 pencils
Brown = 20 pencils
Pink = 50 pencils

Total = 100 pencils

If the teacher selects a pencil at random to give to Sam. What is the probability that the pencil will be pink?

a. 1 / 4
b. 1 / 2
c. 1 / 3
d. 1 / 5

Answer : b

Explanation:
A probability fraction is set up as follows:
Number of favorable outcome / number of total outcomes.
Therefore for this problem the number of favorable outcome is 50 pink pencils while the number of total outcome is 100 pencils so we set it up as follows:
50 pink pencils / 100 total pencils
50 / 100 = 1 / 2

For additional information on calculating probabilities go to: Probability.

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Bar Graph - Elementary Grades

An attendance graph was drawn for last week's third grade class. The graph shows the days of the week and the percentage of third grade students attending on each day of that week. From the graph can you find out what percentage of students attended last Wednesday?

a. 40
b. 80
c. 100
d. 20

Answer: b

Explanation:

80% of the third graders attended class on Wednesday. To find out how we know this we examine the above graph. The names of the week are labeled on the bottom of the graph while the percentage rates of students attending are labeled on the left hand side of the graph. The red bar represents Wednesday. By looking at the top of this bar and matching it with a percentage number from the left side we can find our answer, which is 80.


© Copyright 2007.Najwa S. Hirn. All rights reserved.

FOIL method to solve a Quadratic- College Level

Factor the following quadratic equation using the FOIL method:

Answer:

Explanation:
The FOIL method is used when we have two linear equations (otherwise referred to here as factors) and we combine them to obtain a quadratic equation. The Term FOIL stands for the following:
F irst term
O uter term
I nner term
L ast term
The standard form of a quadratic equation is:

In the above equation the a term is 1, the b term is 10 and the c term is 21. In order to factor using FOIL, we will need to come up with two terms whose sum equals the b terms and whose product equals the c term. Since both terms are positive, we don’t need to worry about signs for this problem.
Upon examining the problem we determine that the numbers 7 and 3 work since their sum equals 10 and their product equals 21. We now have our answer as:

If we multiply the above factors using the FOIL method we will obtain the equation we started with.
For additional information about various factoring techniques go to: Introduction to Quadratic Functions.

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Inequality - High

Find the solution for the following inequality:
6x + 8 < 2x + 16

Answer: x less than 2

Explanation:

An inequality is solved using the same method used in solving an equation. A solution for an inequality is usually the number that when substituted into the inequality will make the statement true.

In order to solve the above inequality we start with the same steps as if it were an equation by isolating the variable on one side and the number on the other as follows:
6x – 2x + 8 < 2x – 2x +16 (subtract 2x from each side)
4x + 8 < 16 4x + 8 – 8 < 16 – 8 (subtract 8 from each side) 4
x < 8 x < 8/4 (divide each side by 4)
x less than 2

Therefore the solutions for x are all the numbers that are less than 2 For additional information on inequalities, go to: Inequality Basic Rules

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Angles (Corresponding and Alternate) - Middle Grades

In the figure below what are the measurements for angles X , Y and Z. Note that line AB is parallel to Line CD.

Answer:
Angle X = 70 degrees
Angle Y = 60 degrees
Angle Z = 60 degrees

Explanation:
In order to find out what angle Z is we must know that angles on a straight line must all equal to 180 degrees. Therefore we will add the totals for the other two angles and subtract that from 180 to find our angle as follows:
180 – (70 + 50) =
180 – (120) = 60

To complete the rest of the problem we must know these two facts:

1. The total of the three angles in a triangle must equal 180 degrees.
2. When a line is parallel to a side of a triangle then the two corresponding angles must be
equal and the two alternate angles must be equal.

Therefore the angles X, Y and 50 in the triangle must equal 180 degree.

Line CD is parallel to line AB and part of line AB forms one side of the triangle that has the other three angles. which formsin the triangle then angles 70 and X are corresponding angles so they both equal each other and equal 70 degrees.

Also angles Y and Z are equal because they are alternate angles. Since we found out that angle Z equals 60 degree that means that angle Y must also equal 60 degrees.

For additional information on corresponding angles go to: Corresponding Angles.
For additional information on alternate angles go to: Alternate Interior Angles.

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Number Sequence - Elementary School Math

When James was walking down the street, he read the numbers written on the houses that were to his left. The number of houses was as follows:

25 27 29 31

What would the number written on the fifth house be?

a. 26
b. 28
c. 33
d. 38

Answer: c

Explanation:
The houses are all numbered with odd numbers. A 2 is added to each house number to come up with the next one. Therefore the first is 25. The second will be 25 + 2 = 27 and so on. To find out what the fifth house will be add 2 to the number of the fourth house as follows:
31 + 2 = 33

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Slope - Intercept Form - College

Find the x- and y- intercepts and the slope for the following equation:
4x – 2y = 12

Answer:
x- intercept = 3
y- intercept = - 6
slope = 2

Explanation:
To find the x- intercept we must set y = 0 and solve for x as follows:
4x – 2(0) = 12
4x = 12
x = 12/4 = 3

To find the y-intercept we must set x=0 and solve for y as follows:
4(0) – 2y = 12
-2y = 12
y = 12/-2 = -6

To find the slope we must first rewrite the original equation in slope-intercept form, which is:
y = mx + b where m is the slope and b is the y intercept.

4x – 2y = 12
-2y = -4x + 12 (subtract 4x from both sides)
y = (-4/-2x) + (12/-2) (divide both sides by –2)
y = 2x – 6

Now we can identify the slope m = 2 and the y intercept is –6 which we found earlier also.

For additional information on intercepts go to: Review of Intercepts
For additional information on the Slope-Intercept form with practice problems go to:
Slope-Intercept Form.

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Calculating Percentages of Sales - High

At the new store in town, girls’ jeans are sold at ½ off what the original price. The store does a promotion on Saturdays and takes off an additional 15% the sale price. Joann bought one of the jeans on Saturday. The original price on the tag was $45. What will Joann pay for the jeans after the discounts not including the sales tax?

Answer: $19.12

Explanation:
Since the jeans are usually ½ off the original price then that price would be:
½ x $45 = $22.5
The sale on Saturday will cut an additional 15% off so we must find out what $15% off the $22.5 as follows:
0.15 x $22.5 = $3.38
The final price will be:
$22.5 - $3.38 = $19.12

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Conversions of Units - Middle

The distance from Jacob’s house to his grandmothers is 5 ½ miles. What is this distance in feet?

a. 12125 feet
b. 22500 feet
c. 29040 feet
d. 34335 feet

Answer: c

Explanation:
We must use conversions to solve this problem. There is 5280 feet in 1 mile so we set up the solution as follows:

1 mile = 5280 feet
5 ½ miles = unknown feet

To find the unknown multiply the 5 ½ miles by the number of feet per mile as follows:
5 ½ miles x (5280 feet / 1 mile) = 29040 feet
Note that the units for miles cancel out and we are left with only feet, which is what the problem is asking for.

For additional information go to: Conversions of Units

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Addition and Subtraction - Elementary

On March 2, 2005, the fourth grade class went on a field trip to visit the historical marina. An old sailboat was on display in the area. The teacher read a sign that said, “This boat was built on March 2, 1867”. The teacher asked the fourth grader to tell her how old is the sailboat on the day of their field trip? Choose the correct answer

a. 115 years
b. 138 years
c. 235 years
d. 375 years

Answer: b

Explanation:
On March 2, 2005, the sailboat is 138 years old. To find that answer we subtract the year it was built from the date of the field trip as follows:
2005 – 1867 = 138

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Completing the Square - College

Solve the following quadratic equation by completing the square:

Answer:

x = 5 or x = 1


Explanation:

First add (-5) to both sides. This is done in order to move the (-5) to the right side of the equation:

Second, add 9 to both sides. This will make the left side of the equation a perfect square:

Take the square root of both sides and solve for x :

© Copyright 2007.Najwa S. Hirn. All rights reserved.

Calculating Percentages - High

A donation of $350,000 was made for the local hospital. The hospital spent 15% of this donation to open a children’s playroom. An additional 18% was spent to buy new equipment for the cardiac center. Finally, they used 25% to open a library for the elderly ward.
How much did the hospital spend out of the original donation and what is left for them to utilize later?

a. Amount spent $147,000 Amount left $203,000
b. Amount spent $110,000 Amount left $240,000
c. Amount spent $203,000 Amount left $147,000
d. Amount spent $240,000 Amount left $110,000

Answer: a

Explanation:
To find the total amount spent add the percentages as follows:
0.15 + 0.18 + 0.25 = 0.58
Therefore 58% of the $350,000 was spent. In order to calculate that amount we must multiply as follows:
$350,000 x 0.58 = $203,000
This is the amount spent out of the original donation.

To find out what is let we subtract as follows:
$350,000 - $203,000 = $147,000
This is the amount left.

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Cartesian Coordinate and Identifying Ordered Pairs - Middle School

Identify the three pairs of points (A) (B) and (C) plotted on the following graph?

Answer:

Point (A) = (3, 0)
Point (B) = (0 , -2)
Point (C) = (-3 , -4)

Explanation:

The graph represents what is termed as the rectangular or Cartesian coordinate system. This system is constructed by drawing two perpendicular lines. The horizontal one is called the x-axis. The vertical is called the y-axis. The point were the two lines cross is called the origin which is represented as (0,0). These axes divide the plane into four quadrants numbered I to IV in a counter- clockwise direction. An ordered pair is always written in the form (x,y) where the x coordinate appears first and then the y.

Working from right to left, the first pair of points (A) is located in quadrant I. This pair of points is located on the x-axis. In this case the y will be 0. Since the x coordinate appears to the right of the origin then it is positive. Therefore, this pair is (3,0).

The second pair of points (B) is located on the y-axis. In this case x will be 0. Since the y coordinate appears below the origin then it is negative. Therefore this pair is (0, -2).

The Third pair of points (C) is located in quadrant III. In this quadrant both the x and y coordinates are negative. To find out what the pair is, start at the origin and count 3 spaces to the left and 4 spaces down. The intersection is the point (-3, -4).

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Counting Change - Elementary

Which of the following combinations should you pick in order to make $1.67?

a. 6 quarters + 1 dime + 1 nickel + 1 penny
b. 6 quarters + 1 dime + 1 nickel + 2 pennies
c. 3 quarters + 1 dime + 3 nickels + 1 penny
d. 5 quarters + 1 dime + 2 nickels + 2 pennies

Answer: b

Explanation:
A quarter = .25 cents
A dime = .10 cents
A nickel = .05 cents
A penny = .01 cent

In order to make $1.67 we’ll need to add the following:
6(.25) + .10 + .05 + 2(.01) =
1.50 + .10 + .05 + .02 = $1.67

If you like to play games, try this computer activity that can help you practice counting money:
Counting change

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Solving for a Variable - College

Solve for x: 4(2x + 1) + 8 = 5x

Answer:
x= - 4

Explanation:

First we must use the distributive property in order to open the parenthesis. Therefore we multiply the 4 outside the parenthesis by each term inside as follows:
8x + 4 + 8 = 5x
Now we must collect like terms, we start by adding the numbers:
8x + 12 = 5x
In order to collect the variables we must subtract 8x from both sides:
8x – 8x + 12 = 5x – 8x
12 = -3x
To solve for x, we can multiply each side of the equation by – 1/3
(- 1/3) 12 = -3 x (-1/3)
This will cancel out the –3 and we are left with:
X = - 4

In order to check our solution, we can substitute for x in the original equation. If the left side of the equation will equal the right after the substitution, then we know that we have solved it correctly. Here goes:
4[2(-4) + 1] + 8 = 5 (-4)
4[-8+1] + 8 = -20
4 [-7] + 8 = -20
-28 + 8 = -20
-20 = -20
The solution checks out ok.

©Copyright 2007.Najwa S. Hirn. All rights reserved

Volume of a Cyliner - High school

How high should a cylindrical water tank be to hold 2034.72 cubic feet of water if its diameter is 12 feet? If one cubic foot is about 7.5 gallons, then how many gallons of water can the tank hold?

Answer:
Length = 18 feet
Volume in gallons = 15260.4 gallons

Explanation:

The volume of a cylinder is:

Since the problem is giving the diameter, we must find what the radius is before we can solve the requirement. The radius is always half the diameter so it will be 6 ft. To find the height we must substitute:

To find the volume in gallons, multiply the total volume in cubic feet by the 7.5 gallons per 1 cubic feet as follows:

Note that the cubic feet units cancel and we are left with gallons.



© Copyright 2007.Najwa S. Hirn. All rights reserved.

Scientific Notation - Middle School

Problem (1):

Write:

574,000 in scientific notation

Problem (2):

Write:

in expanded form.

Answer:

Problem (1) =

Problem (2) =

6150


Explanation:

Scientific notation is used when very large or small numbers need to be written in a way that we can manage them. It is usually expressed when a decimal number between 1 to 10 is multiplied by an integer power of 10. The power of 10 is also referred to as the exponent and it represents the number of spaces the decimal point is moved either to the left or right depending on the problem. Lets solve the above problems:

Problem (1):

574,000.
Notice the location of the decimal point at the far right of the number.
We need to count how many spaces the decimal must be moved to the left in order
To place it between the numbers 5 and 7. This turns out to be 5 spaces. This will
Represent what exponent or power of 10 to use, so the answer is:

Problem (2):

Since we know that 10 is raised to the third power, it means that this number=1000
This tells us that we must move the decimal 3 spaces to the right. By doing that we end up with 6150

For additional information go to: Scientific Notation



© Copyright 2007.Najwa S. Hirn. All rights reserved.

Additiona and Subtraction - Elementary Math

John’s mom gave him $12 to spend at the movies. He spent $3.75 to buy his movie ticket. During break time, he bought popcorn for $1.45, a drink for $1.35, and candy for $1.80.
How much money was left for John to return to his mom.

a. $1.10
b. $2.25
c. $3.65
d. $3.95

Answer : c

Explanation:

The first step is to find out how much money John spent at the movies by adding all his expenses:
$3.75 + $1.45 + $1.35 + $1.80 = $8.35

To find out how much money he had left we must subtract the total expenses from his original $12 that he started with:
$12 - $8.35 = $3.65

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Solving a Quadratic Equation - College Level

Solve the following equation by using the quadratic formula:

Answer:
x = 6 or x = -1

Explanation:

The above equation is a quadratic equation. We can identify it as such because it is written in the form:

The terms a, b and c are real numbers and a does not equal 0. This equation will have two solutions. The formula used to solve it is:

Lets solve it by substituting in the formula:

For additional information go to: Quadratic Equations: Quadratic Formula.

© Copyright 2007.Najwa S. Hirn. All rights reserved.

Order of Operations - High Grades

What is the value of the following expression:






a. 135
b. 198
c. 250
d. 301


Answer : b

Explanation:

We must use the Order of Operations when solving this problem. If you remember the following letters in the correct order then you can solve problems like these easily:

P-E-D-M-A-S

These letters stand for:

Parenthesis – Exponents – Division – Multiplication – Addition and Subtraction

Looking at the above problem and using the P-E-D-M-A-S rule, we can determine that the Exponents must be done first, then multiplication and then the addition and subtraction working left to right. Here goes:







For additional information go to: Order Of Operations.

© Copyright 2007.Najwa S. Hirn. All rights reserved.

Quadrants - Middle School Math

In the graph below, identify what quadrant are the following ordered pairs located:
(5,4) and (-6,-3)

(5,4) are in quadrant I and (-6.-3) are in quadrant III

The above graph represents what is termed as the rectangular or Cartesian coordinate system.
This system is constructed by drawing two perpendicular lines. The horizontal one is called
the x axis. The vertical is called the y axis. The point were the two lines cross is called the
origin. These axes divide the plane into four quadrants numbered I to IV in a counter-
clockwise direction.

The ordered pair (5,4) is located in quadrant I. In this quadrant both the x and y coordinates
are positive. Note that the number 5 in the ordered pair indicates the x coordinate. It
represents moving 5 units in the positive x direction (right). The number 4 in the ordered pair indicates the y coordinate. It represents moving 4 units units in the positive y direction (up).

The ordered pair (-6,-3) is located in quadrant III. In this quadrant both the x and y
coordinates are negatives. Note that the number -6 in the ordered pair indicates the x
coordinate. It represents moving -6 units in the negative x direction (left). The number -3 indicates the y coordinate. It represents moving -3 units in the negative y direction (down).

For additional information go to: The Quadrants of the Cartesian Plane.

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Simple Equations - Elementary Math

Mrs. Smith volunteered to take the neighborhood kids to the theater for a matinee. The total number of kids she chaperoned was 10. The movie’s price tickets were as follows:
Adults = $4, Children under 18 = $2.
If Mrs. Smith needed to find the total price of all the tickets, which of the following equations should she use?

a. 4 x 2 + 10 =
b. 10 + 4 + 2 =
c. 4 + (2 x 10) =
d. 10 x (4 + 6) =

Answer: c

Explanation:

The first step is to find the price of tickets for all 10 kids. Since each child’s ticket price will be $2 then the total will be (2 x 10).
In order to find the total price of all the tickets, we must add the price of Mrs. Smith’s ticket, which is $4.
Therefore the equation that will tell us the total price of all the tickets will be:
4 + (2 x 10) = Note that the question asked for the equation only and not the final answer, which comes out to be $24 for all the tickets.

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Distributive Property - Solving for a Variable - College

Solve for x : 5(x – 3) + 2 = 5(2x – 8) – 3

Answer : x = 6

Explanation:

The steps for solving this are as follows:

Apply the distributive property by multiplying the number outside of the parenthesis by each term inside the parenthesis to eliminate the parenthesis as follows:
5x – 15 + 2 = 10x – 40 – 3

Simplify both sides by adding or subtracting like terms as follows:
5x – 13 = 10x – 43

Eliminate 5x from the left side of the equation by adding a ( - 5x) to each side of the equation as follows:
5x + (-5x) – 13 = 10x + (-5x) – 43
-13 = 5x – 43

Add 43 to both sides as follows:
-13 + 43 = 5x – 43 + 43
30 = 5x

Multiply both sides by 1/5 as follows:
30 (1/5) = 5x (1/5)
6 = x

For additional information about the go to: Distributive property Page

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Proportions - High School

While in Art class, Alex drew the door to his classroom to be 5 inches. The actual height of this door is 6 ft. 8 inches high. His teacher asked him to use this same scale factor to draw the classroom wall. If the actual height of the wall is 8 ft. how high should Alex draw the wall on his paper?

Answer: 6 inches

Explanation:

This is a proportion problem, however, in order to solve it we must first make sure that the measurements all use the same units. We notice that the problem uses units in inches and in feet. We must convert one into the other. Once the units match then the problem can be solved.
I have chosen to convert the feet into inches taking into consideration that there are 12 inches in one foot:

To convert the actual height of the door: [(6 ft) x (12 inches / 1 ft)] + 8 inches = 80 inches
Notice that the feet units cancel out and we are left with inches.

To convert the actual height of the wall: (8 ft) x (12 inches / 1 ft) = 96 inches
Notice that the feet units cancel out and we are left with inches.

Now we can set up the proportion:
Actual door height / drawn height = Actual wall height / X

80 inches / 5 inches = 96 inches / X

X = [(5 inches) (96 inches)] / 80 inches = 6 inches

Therefore, Alex should draw the wall 6 inches to have it proportional to the other measurements.

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Area and Perimeter - Middle Grades

Find the area and the perimeter for the rectangle shown below when its width is 5 inches and its length is 14 inches:

Answer :

Area = 70 inches square
Perimeter = 38 inches

Explanation :
The formula to find the area of a rectangle is the width multiplied by the length. Therefore
A = (W) (L) = (5 in) (14 in) = 70 inches square

The formula to find the perimeter of a rectangle is twice the width added to twice the length. Therefore:
P = (2 W) + (2 L)
P = (2)(5 in) + (2)(14 in)
P = 10 in + 28 in = 38 inches




For additional information on this topic go to: Area and Perimeter

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Addition and Subtraction - Elementary Math

On Monday Samantha bought 25 red shirts for school. On Tuesday, she went back to the store and bought an additional 15 red shirts. On Wednesday she washed 12 of them. How many more red shirts does she have to wash?

Answer : 28

Explanation:

This is an addition and subtraction problem. The goal is to find the number of red shirts that still have to be washed. In order to do that, first we must find the total number of shirts that she bought and then subtract the number of shirts that were washed. The final answer will give the number of shirts that are still left to be washed. Here goes:

Total number of shirts bought : 25 + 15 = 40 red shirts
Number of shirts left to be washed : 40 – 12 = 28 red shirts

For additional information on this subject go to the “ What is Addition and Subtraction” page.

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Greatest Common Factor (GCF) - College

Find the greatest common factor for the following polynomial:

Answer :

In order to solve the above, we must first define that the greatest common factor is the largest monomial that is a factor of each term of the polynomial. When we examine the numbers we find that the largest factor of each of the above numbers is 7. Now we must factor the variables. This is done by finding the variable with the highest exponent (power) that is a factor of :

This is found out to be:

Therefore by combining the two the answer will be:

For additional information go to: Lesson page for common factors

© Copyright 2007.Najwa S. Hirn. All rights reserved.

Volume of a Cylinder - High School Math

Find the volume of the cylinder shown given that L = 30 inches and r = 3 inches

Answer & Explanation:

A cylinder is a three dimensional geometric figure. The bases of the cylinder are two parallel and similar circles. The perpendicular line separating the two circles represents the height or length of the cylinder.

The formula for the volume of the cylinder is as follows where L is the height of the cylinder and r is the radius of the base of the cylinder :

With the given numbers and using 3.14 for pi the volume is calculated by multiplying the known values as follows:
V = (3.14)(3 in x 3 in)(30 in) = 847.8 inches cubed

©Copyright 2007.Najwa S. Hirn. All rights reserved.

Number Line - Middle Level

What is the value of point B on the number line :

a. –6

b. 9

c. –12

d. 15

Answer : c

Explanation :
In Math, Integers are placed on a number line in marked, evenly spaced sections. A Zero divides this line into two segments. The positive integers appear on the right hand of the zero while the negative segments appear on the left hand of the zero. Each segment goes on to infinity therefore only the needed segment of the line is shown for each different math problem. The number line is a good tool to use when learning addition and subtraction of positive and negative integers. This will be covered in a different problem.

The above problem is a graphing problem showing the number line and requesting the identification of the value of a certain variable (in this case B) which is located on this line. A thorough examination of the line determines that the point B is located just above the integer
(-12). Therefore, the answer is ( c ) which is –12.

For a very detailed pictorial representation of the number line and additional description go to: Number line

©Copyright 2007.Najwa S. Hirn. All rights reserved.