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Wishing you a world filled with blessings, happiness, health and

prosperity for the New Year.

Happy New Year from our home to yours..

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

Scales - Middle School Math

The math class were asked to calculate the distance between two cities that appeared on a map that the teacher handed out. The class measured the distance on the map and found it to be 3 inches. The scale used to draw the map said that ½ inch = 20 miles. Based on this information, what was the approximate distance that the class calculated between the two cities?

a. 110 miles
b. 115 miles
c. 120 miles
d. 125 miles

Answer: c.

Explanation:

In order to calculate the distance the students can set up a proportion as follows to determine what the approximate distance is:

0.5 inches = 20 miles
3 inches = d miles

d = [3 inches x 20 miles] / 0.5 inches
d = [60 inches miles] / 0.5 inches
d = 120 miles

Note that the unites for inches cancel out when the calculation is carried out and we are left with just miles which is what is needed.

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

Number Sense - Elementary Math

Mr. Jones gave each of his 3 children an equal number of paintbrushes to carry to the garage. How many total paintbrushes did he give them?

a. 3
b. 5
c. 9
d. 11

Answer: c.

Explanation:
In order to solve this problem, we must find out which of the answers can be divided by 3 without a remainder left over. By examining each answer we realize that we can divide 9 by 3 without a remainder. Therefore each child carried 3 brushes and all of them combined carried a total of 9 brushes.

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

Solving Systems Of Linear Equations in Three Variables - College Level

Solve the following system of equations:

x + y + z = 6 ………………….. (1)
2x – y + z = 3 ………………….. (2)
x + 2y – 3z = -4 ……………….. (3)


Answer : (1,2,3)
X = 1
Y = 2
Z = 3

Solution:
Since we have three variables, we need to eliminate one of the variables form each equation in order to reduce them to two variables, which will be easier to handle. Once that is done, we solve for those two variables first. Once we have the first ordered pair, we can substitute those in our original equations and solve for the third and final variable.
Here are the steps:

Step 1.
Add equations (1) and (2) to eliminate the y variable as follows:
x + y + z = 6 ………………….. (1)
2x – y + z = 3 ………………….. (2)
-------------------
3x + 2z = 9 ………………….. (4)

Step 2.
In order to eliminate the y variable from equation (2) and (3) we must first multiply equation (2) by 2 to make the y coefficients equal and then adding the two equations as follows:
4x – 2y + 2z = 6 (the result of multiplying equation (2) above by the number 2)
x + 2y – 3z = -4 ………………….. (3)
-----------------------
5x - z = 2 ………………….. (5)

Step 3.
Equations (4) and (5) above have only two variables. If we add the two we
can eliminate one of the variables and solve for the other one. In order to do that,
we must first multiply equation (5) by the number 2 to eliminate z as follows:
3x + 2z = 9 ………………….. (4)
10x - 2z = 4 (the result of multiplying equation (5) above by the number 2
-------------------
13x = 13
x = 1 (we solve for x by dividing both sides of the equation by 13)

Step 4.
We can now substitute the value for x in either equation (4) or (5) to find z as as follows:
(3) (1) + 2z = 9
2z = 9-3
2z = 6
z = 3

Step 5.
Now that we have the values for both x and z we can substitute in any of the original equations to find y as follows:
1+ y + 3 = 6
y = 6 – 3 –1
y = 2

Therefore the triple points that will satisfy the equation is (1,2,3). We can check the accuracy of our answer by substituting the above points in any of the original equations and finding out that the right side of the equation will equal the left side.

The above problem appears in Intermediate Algebra Text/Workbook by Charles McKeague. 3rd. Edition

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

Square root - High School

What’s equivalent to ?

Answer:

Explanation:

In order to solve this problem, we first factor out the 48 as follows:

Now we take the square root of 4 which is 2. We realize that we can not take a square root for 12 so it is left under the square root sign. Therefore our answer is:

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

Percentage - 7th. Grade.

In John’s farm 20% of the trees produce fruit. The farm has a total of 575 trees. How many produce fruits?
a. 105
b. 115
c. 125
d. 130

Answer: b

Method:
We use multiplication in order to find the total number of trees that produce fruit as follows:

0.20 x 575 = 115 trees

Note that 20% = 0.20. This is obtained by dividing 20 by 100.

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

Numbers - Elementary Math

John’s dad was walking in the car rental lot to pick up their rental car. He passed several spots on the left. They were numbered as follows:

201 203 205 207

Which number would he most likely see on the next parking lot on the left?

a. 200
b. 206
c. 209
d. 212

Answer : c

Why:
The student should notice the following:
a. The parking lot numbers are all odd numbers so the next number coming should be an
odd number.
and
b. Each parking lot number is increased by 2 from the previous number.

Therefore if we add 2 to the last number which is 207 we should get 209 as follows:
207 + 2 = 209

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

Word Problem - College Level

Nancy can clean her room in 2 hours. It takes her brother Sam 3 hours to complete the same task. How long will it take them if they worked together to get the room cleaned?

Answer: 6/5 hours

Steps to solve the problem:

This problem can be solved when we assume that both Nancy and Sam are working at a constant rate which means that each of them does the same amount of work in the first hour as they do in the last hour.

To try and find the total hours it takes both of them, we must find how much work can each one of them do in 1 hour. Therefore if Nancy completes the cleaning job in 2 hours therefore in 1 hour she completes 1/2 of the work. Also, if Sam completes the job in 3 hours, then in 1 hour he completes 1/3 of the work.

If we let x = the total number of hours it takes for both of them working together, then in 1 hour, they must do 1/x of the time.

Lets put everything together as follows:

Amount of work done by Nancy = 1/2
Amount of work done by Sam = 1/3
Total amount of work = 1/x

Lets add the hours to get the total as follows:

1/2+1/3 = 1/x

The lowest common denominator is 6x. Therefore, both sides of the equation are multiplied by 6x as follows:

6x (1/2) + 6x (1/3) = 6x (1/x)
3x + 2x = 6
5x = 6
x = 6/5 = 1.2 hours

It will take them 6/5 or 1.2 hours to get the room cleaned together.

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

Square Roots - High School

Which of the following is equivalent to:

a. 36

b.

c.

d. 28

Answer: b

Why:

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

Rectangles - 7th. Grade Math

The length of a rectangle is 3 times the width. The perimeter of the rectangle is 48 feet. What are the measurements for the length and width?

Answer:
Width = 6 ft.
Length = 18 ft.

Steps:
If the width of the rectangle = x
Then the length will be 3x
The formula for the perimeter of a rectangle = 2length + 2 width
Substituting the numbers we get:
48 = 2(3x) + 2x
48 = 6x + 2x
48 = 8x
x = 48/8 = 6
Therefore the width = 6 feet
The length = 3 (6 ft) = 18 feet

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

Multiplication - Elementary Math

While visiting a chocolate factory, John was told that the factory produces 250 chocolate pieces per second. Based on this information, John’s teacher asked him to find out how many pieces of chocolate the factory can produce per minute.
Note: 1 minute = 60 seconds

a. 12,000
b. 15,000
c. 18,000
d. 21,000

Answer: b

Why:
This is a multiplication problem. If one minute = 60 seconds and we know that 250 chocolate pieces are produced per one minute then we must multiply that number by how many seconds are in one minute to get our answer as follows:

(60 seconds / 1 minute) x (250 pieces / 1 second) = 15,000 pieces / 1 minute

Don’t forget to cancel the units for seconds. At that point you are left with the number of chocolate pieces produced per one minute.

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

Word Problem - College Level

Linda is 4 years older than Allyson. Two years ago Linda’s age was three times Allyson’s age. Find the age of each girl now.

Solution: Allyson is 4 years old and Linda is 8 years old

steps:
Allysons age now = x
Linda’s age now = x + 4

Allyson’s age two years ago = x – 2
Linda’s age two years ago = x + 4 – 2 = x + 2

Also, two years ago Linda’s age was three times Allyson’s age = 3 (x-2)

In order to set up the equation, we equate the girl’s ages from two years ago and solve for x as follows:

x + 2 = 3 ( x – 2)
x + 2 = 3x – 6
Isolate the variable:
x – 3x = - 6 – 2
-2x = -8
x = -8/-2 = 4

Now we know that Allyson’s age now = 4 years old. We substitute for x to find Linda’s age:
Linda’s age = 4 + 4 = 8 yearls old.

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

Variables in an Equation - High School Math

When Sandra ordered candy for her store, she received three types of boxes containing the chocolate bars. The boxes are labeled x , y and z. If box x contains 10 bars, and box y contains 20 bars and box z contains 15 bars then the total number of bars is represented in the following equation:

T = 10x + 20y + 15z

If Sandra received 70 type x boxes, 30 type y boxes and 20 type z boxes, find the total number of bars she received?

Answer: 1600

Steps:
We must find out the total number of bars received in each box type as follows:
box x = 10 (70) = 700
Box y = 20 (30) = 600
box z = 15 (20) = 300

The total number of bars received will be the addition of all the above as follows:

T = 700 + 600 + 300 = 1600 chocolate bars.

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

Algebraic Thinking - Middle grades

JoAnn was hired as a book sales rep. Her hourly wage is $8.50 per hour. In addition, she is paid $3.50 bonus for each sale she makes. Which of the following expression represents how much she makes per week if she made 5 sales and worked n number of hours per week.

a. 5n(8.50 + 3.50)
b. 5(8.50n + 3.50)
c. 8.50n + (3.50 x 5)
d. (8.50 + 3.50) x (5 + n)

Solution: c

Why:
In order to solve this problem we must follow the steps below:
1) The amount that she earned from her job per week = 8.50 x n = 8.50n
2) The amount she earned per week by making 5 sales = 3.50 x 5
3) The total amount she would earn = The total of both of the above which will be:
8.50n + (3.50 x 5). Therefore this will be the correct expression.

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

Finding the Mean - Elementary Math

The numbers below show the grade each student received on the math exam:

Sam received 38
Adam received 40
John received 43
Katie received 45
Joy received 49

What was the mean number of grades the students received?

a. 35
b. 43
c. 48
d. 50

Solution: b

The Steps:
The mean value of any set of numbers indicates the average of those numbers. The average of the numbers is calculated by adding all the numbers and dividing them by how many times they occurred. For this example it will be as follows:
38 + 40 + 43 + 45 + 49 = 215
215 / 5 = 43
(see post 08-31 and 10-04)

For additional information on this subject go to : How to Find the Mean

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

Polynomials - College Algebra

Expand the following Binomial:

The Solution:

The Steps:

The rule for taking the square of a binomial states that:
The square of a binomial is the sum of the square of the first term, twice the product of the two terms, and the square of the last term.
By using this rule, we can expand the above binomial as follows:

For additional information and examples, go to: Square Trinomials



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

Pythagorean Theorem - High School Math

In the following right angle triangle, side b = 13.5 and side c = 15.5 Calculate side a



http://www.flourmilling.net/maths.html

Answer: 7.6

Explanation:
To solve this problem we must use the Pythagorean theorem, which states:

Substituting the values for c and b in the above equation results in:

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

Angles - Grade 8

Joe’s new swimming pool is in the shape of a Pentagon such as the shape on the left. What is the value of each interior angle in this shape?
a. 108
b. 128
c. 148
d. 168

Answer: a

Explanation:
A Pentagon is a Polygon with five sides. This also means that it has five equal angles.
The formula used to find the sum of all the interior angles in any Polygon is as follows:
180 ( n-2) where n represents the number of angles in the polygon.
In this problem n will equal 5. Lets apply the formula to find the sum of all the interior angles:
180 (5-2) = 540
Since we have 5 angles, we can find the value of each angle by dividing as follows:
540 / 5 = 108 degrees is each interior angle in a Pentagon.

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

Weights and Measures - Elementary Math

Rachel weighed herself on the scale. Which one of the following measurements best describes her weight?

a. 80 inches
b. 80 pounds
c. 80 ounces
d. 80 tons

Answer: b

Explanation:
A person's weight is usually measured in pounds, so when Rachel weighed herself the scale showed her weights in pounds. Therefore, the correct answer will be 80 pounds.

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

Variations - College Level

If y, x and z are variables and K is a non-zero constant, write the algebraic equation that represent each of the following:

a. y varies directly with x
b. y varies inversely to x
c. y varies jointly with x and z

Answer:
a. y = Kx
b. y = K / x
c. y = Kxz

Explanation:
Variation problems usually appear in sciences such as physics and chemistry.
There are two types of variations: direct and inverse. A combination of those relationships creates other variations such as joint.
In a direct variation, when one variable such as (x) increases, that causes the other variable (y) to increase also. The same happens when one decreases thus causing the other one to decrease.
Therefore, in a direct variation, a change in one variable results in the same change in the other variable.
In an inverse proportion, an increase in one variable will result in a decrease in the other and thus the name inverse.
Therefore in an inverse proportion a change in one variable causes the opposite to happen to the other.
A combination of variations occurs also. When one variable varies directly with two variables, we say that it varies jointly.

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

Related Measures - High School Math

Janet burns 15 calories per minute on the treadmill. How many calories does Janet burn per one-hour?

a. 300 calories / hour
b. 800 calories / hour
c. 900 calories / hour
d. 1100 calories / hour

Answer: c

Explanation:
The answer is 900 calories / hour
One hour = 60 minutes.
Therefore, in order to find out how many calories Janet burns in one hour, we multiply the amount of calories she burns per minute (in this case 15) by the total minutes per hour, which are 60 as follows:

(15 calories / 1 minute) x (60 minutes / 1 hour) = 900 calories / 1 hour
Note that the units for minutes disappear and we are left with calories per hour.

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

Geometry - Middle grade

Alex’s mom allowed her to grow vegetable on a small piece of the backyard. The patch that was assigned for Alex’s garden looks like this:












What geometrical shape is outlined in Alex’s vegetable patch?

a. Trapezoid
b. Octagon
c. Hexagon
d. Pentagon

Answer: b

Explanation:
The above figure represents an Octagon. An Octagon is a geometrical figure with eight sides. When all the sides are the same length, it is called a regular Octagon. For additional information about Octagon and how to construct a regular Octagon go to:
Octagon.



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

Division Problem - Elementary Math

The third grade math teacher must lock up the math test booklets in the desk drawers overnight. Each desk drawer can hold 4 test booklets. If she needs to lock 28 test booklets, how many drawers must she use?
a. 5
b. 7
c. 9
d. 10

Answer: b

Explanation:
This is a division problem. In order to find out how many drawers are needed to hold 28 booklets when each one holds only 4 we divide as follows:
28 / 4 = 7
The math teacher will need 7 drawers.

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

Using the Distributive Property - College Level

Multiply:

Answer:

Explanation:

In order to solve the problem we must do it in three steps as follows:

1. Multiply (x-1)(x-2) by using the FOIL method (first-outer-inner-last) terms

2. Multiply the answer by (x-3)Multiply that answer by the outer 3 to get the solution.

© Copyright 2009. Najwa S. Hirn. All rights reserved.

Equations - High School Math

The number students enrolled in the elementary school is 5750 and is increasing by 250 students per year. The following equation represents this relationship:
S = 5750 + 250 y
Where S represents the number of students and y represents the number of years.
Using the above equation, find out how many years it will take for the number of enrolled students to reach 8500?

Answer: 11 years

Explanation:
The above equation is used to find the solution for this problem. Since the final number of students is to become 8500, then we must substitute that for S in the equation as follows:
S = 5750 + 250 y Original Equation
8500 = 5750 + 250 y Substituting 8500 for S
8500 – 5750 = 250 y Isolating Variables
2750 = 250 y Solve for y
2750 / 250 = y Divide both sides by 250
11 = y The number of years it will take for the number of enrolled
students to become 8500

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

Percentages - Middle School

In David’s farm there are 120 horses. 45% of those horses are female. How many female horses are on the farm?

a. 43
b. 54
c. 62
d. 70

Answer: b

Explanation:
To find the number of female horses we have to multiply the total number of horses by the percentage of female horses as follows:

120 horses x 0.45 female / horse = 54 female horses

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

Multiplication - Elementary Math

Jessica’s mom drives 3 miles for 5 days to get to work during the week. Which equation can be used to find out how many miles Jessica’s mom drives during the week?

a. 3 + 5 =
b. 3 x 5 =
c. 3 – 5 =
d. 3 / 5 =

Answer: b

Explanation:
Since Jessica’s mom drives 3 miles each day, and there are 5 days per week, then we must multiply the 3 miles driven daily by the number of days to get the total miles she drives per week. Therefore, this will be a multiplication problem.

Additional information about multiplication can be found at: Multiplication Idea Bank

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

Exponents - College Level

Simplify:

Answer:

Explanation:

The steps involved in solving this problems are as follows:

1) Separate each inside the first parenthesis and raise each term to the second power as

shown below.

2) Do the same for the terms inside the second set of parenthesis.

3) Working with the first term, do the following steps:

• Raise the number (3) to the second power by multiplying it by itself which results in (9).
  
• The rules of exponents states that when raising a variable to an exponent, the exponents are multiplied together. Therefrore the second term becomes (x) raised to the sixth power.
  
• The same applies to (y) and we get (y) raised to the eigth power.
  
• Repeate those steps for the terms inside the second pair of parenthesis.
  
• Now combine like terms. This means the numbers can be multiplied together and the exponents on the variables are added together. Remember that the second rule of exponents states that when two similar variables are multiplied together, their exponents are added.
  
• The entire problem is worked out below.
  
  
  

For additional information go to: The laws of Exponents


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

Equations - High school Math

Sam opened a Pizza shop. The rent for the Pizza shop is $670 per month. Sam makes $5 for each Pizza that he sells. The following equation represents the profit he can make:
P = 5z – 670
How many Pizza’s must Sam sell in order to make $800 profit. P represents the profit while z represents the number of Pizzas that he must sell.

Answer: 290 Pizzas

Explanation:
P = 5z – 670 Original equation
800 = 5z – 670 Substitute 800 for the profit
800 + 650 = 5z Add 570 to both sides to isolate variable
1450 = 5z Isolate variable
z = 1450 / 5 = 290 Divide both sides by 5 to find z

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

Conversions - Middle school

During Stephanie’s graduation party, her mom mixed ½ liter of Orange juice, ¼ liter of Cranberry juice, and ¾ liter of Pineapple juice in a punch ball. How many milliliters of juice did Stephanie’s mom mix in the ball?

a. 0.0015 milliliters
b. 1.5 milliliters
c. 150 milliliters
d. 1500 milliliters

Answer: d

Explanation:
1 liter = 1000 milliliter.
Therefore to be able to solve the problem we must find out the total liters that Stephanie’s mom mixed and then convert them to milliliters as follows:
½ + ¼ + ¾ = 1.5 liters
1.5 Liters x [1000 milliliter / 1 Liter] = 1500 milliliters

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

Addition & Subtraction - Elementary Math

The park in town has 53 different rides, 7 of those rides were there when the park was built in 2000. How many rides were not available in 2000?

a. 32
b. 40
c. 43
d. 46

Answer: d

Explanation:
This is an addition and subtraction problem. Since we know that the ride has 53 rides now, we can subtract the 7 rides that were built in the year 2000 to come up with how many rides were not built in 2000 as follows:

53 – 7 = 46 rides

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

Interest Rate - College Level

Tom invested a total of $1400 in two accounts. One of the accounts pays 6% annual interest. The second account pays $10% annual interest. If Tom’s total interest for the year is $120, how much money did Tom invest in each of the accounts?

Answer: $500 at 6% and $900 at 10%

Explanation:
To solve this problem we set up two equations with two variables x and y and then we solve for each variable as follows:
Let x represent the amount invested at 6%
Let y represent the amount invested at 10%

x + y = 1400
0.06x + 0.10y = 120

In order to solve for one of the variables we must eliminate the other one. In this case if either the x or y coefficients in the first equation equaled the x and y in the second then we can subtract the equations. To accomplish this we will multiply the first equation by 0.1 and eliminate the y as follows:

0.10x + 0.10y = 140
0.06x + 0.10y = 120

When we subtract the two equations we end up with:
0.04x = 20
x = 20 / 0.04 = 500

Substituting 500 for x in the first equation we can now solve for y as follows:
500 + y = 1400
y = 1400 – 500 = 900

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

Slope - High School Math

A line on a graph passes through the points (-1,-2) and (1,4). Find the slope using the point slope formula. If y-intercept is +1, write the answer in the slope intercept form?

Answer:
Slope = 3
y-intercept = 1
Slope intercept form:
Y = 3x + 1

Explanation:
To calculate the slope using the point slope formula we have:
m = [4-(-2) / 1-(1)]
m= 4 + 2 / 1 + 1
m= 6 / 2 = 3
The y-intercept is +1. It is at this point the line crosses the y-axis.
Putting it all in the slope intercept form we have:
y= mx + b
y = 3x+1

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

Conversions - Middle School

John’s family bought 1½ acres of land. How many square feet is that?

a. 40,250 square feet
b. 53,370 square feet
c. 65,340 square feet
d. 71,120 square feet

Answer: c

Explanation:
We know that 1 acre = 43,560 square feet. In order to find how many square feet is in
1½ acres, we have to multiply those numbers as follows:
1½ acres x [43,560 square feet / 1 acre] = 65,340 square feet.
Note that the units for acres disappear and we are left with only square feet.

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

Telling Time - Elementary Math

Darrin went to the movies with his friends. The clock on the left shows the time the movie started. The clock on the right shows the time the movie ended. How long was the movie?

a. 1 ¼ hour
b. 1 ½ hour
c. 2 ¼ hour
d. 2 ½ hour

Answer: b

Explanation:
The move took 1 ½ hours because the first clock shows the time as 4:30 in the afternoon and the second clock shows the time as 6:00 in the afternoon. The difference between the two is

1 ½ hour.

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

Word Problem - College Level

The perimeter of a triangle is 20 inches. If the medium side is 5 inches more than the shortest side and the longest side is 3 times the shortest side, find the dimensions for all the sides?

Answer: S = 3 inches , M = 8 inches , L = 9 inches

Explanation:

To solve this problem we use the following variables:
S = shortest side
M = medium side
L = longest side
P = Perimeter

We know that the perimeter equals the sum of the three sides therefore:
P = L + M + S

Now we plug in our numbers from the problem:
P = 20
M = (5 + s)
L = 3s

20 = (5+s) + 3s + s
20 = 5 + 5s
20 – 5 = 5s
15 = 5s
s = 15/5 = 3

Now we know that the shortest side = 3 inches. We use this value to find the other sides as follows:

M = (5 + 3) = 8 inches
L = (3) (3) = 9 inches

In order to check it we add the values as follows:

P = 9 + 8 + 3 = 20 inches

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

Venn Diagram - High School Math

The Venn Diagram shows the different meals offered by two restaurants A and B.

If Restaurant A offers : {pasta, pizza, subs, chicken, hotdog, hamburger}
And
If Restaurant B offers: {steak, seafood, pasta, soup, chicken, salad}

C shows the meals offered by both restaurant A and B. What meals should appear in C?

a. Steak, Subs
b. Pasta, Chicken
c. Seafood, Chicken
d. Hotdog, hamburger

Answer : b

Explanation:
By examining the meals that are offered in each restaurant we realize that both restaurant A and B offer the Pasta and the Chicken dishes. These two dishes are common for both and therefore will appear in area C of the Venn diagram. Area C will is the intersection of areas A and B and will display the dishes that are offered at restaurant A and B.

For additional information go to: Venn Diagram

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

Inequality - Middle School

James earns $7 per hour working at the local grocery store. James needs to earn at least $63 to pay for a new computer game. Which of the following inequalities describes the number of hours (N) that James will have to work to earn at least $63?

a. N less than 9
b. N greater than 9
c. N less than or equal to 9
d. N greater than or equal to 9

Answer : c

Explanation:
To begin the solution to this problem we must first determine how many hours James must work to earn $63 at a rate of $7 per hour. We use division to do this as follows:
63 / 7 = 9

Now we know that if James worked 9 hours he will earn $63. However, since it is an inequality, we also know that to earn up to that amount ($63) he must work any number of hours (N) less than or equal to 9 hours.

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

Number recognition - Elementary Math

Jeremy lives in a neighborhood where the houses are built at different years. The list below shows the names of four of his neighbors and the years their house was built. According to the list, which of the houses was built first?

The Andersons own a house that was built in 1981
The Craigs own a house that was built in 1978
The Drakes own a house that was built in 1985
The Edwards own a house that was built in 1973

Answer : 1973

Explanation:
The problem is asking for the house that was built first meaning to find the earliest year among the four different years that are presented. The way to do this is to write down the years from smallest to largest and compare them. The number that is the smallest will represent the house that was built first. Lets try this as follows:
1973, 1978, 1981, 1985
From the above list we realize that the smallest number is 1973. Therefore the house belonging to the Edwards was built first.

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

Complex numbers - College Level

Today’s Problem – College –Complex nos.

Find x and y if (2x – 4) + 5i = 3 + (6y – 1) i

Answer: x = 7/2 and y = 1

Explanation:
The real parts are 2x – 4 and 3. The imaginary parts are 5 and 6y – 1

2x – 4 = 3
2x = 3 + 4
2x = 7
x = 7/2

and

5 = 6y – 1
5+1 = 6y
6 = 6y
y = 1

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

Scientific Notation - High School Math

A weight of a substance in kilogram is :




write this in expanded form.

Answer: 0.00004015

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 problem.
Since 10 is raised to a negative number that means that the decimal will have to be moved 5 spaces to the left hand side. We count the 4 as the first space and four additional zeroes to get:
0.00004015

For additional information go to: Scientific Notation

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

Equations - Middle School

Mrs. Smith went shopping to the farmers market. She bought 3 pounds of Tomatoes at $1.57 per pound and 3 pounds of Carrots at $1.15 per pound. Which of the following represents the total cost of Mrs. Smith’s purchases?

a. 3(1.57 x 1.15)
b. 3 + (1.57 x 1.15)
c. 3(1.57 + 1.15)
d. (3 + 1.57) x (3 + 1.15)

Answer: c

Explanation:
The answer will be 3(1.57 + 1.15).
The total money that Mr. Smith spent per pound is: $1.57 + $1.15 = $2.72 per pound
Since she bought 3 pounds of each then we multiply that by the price per pound to get the correct expression: 3 ($2.72) = $8.16

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

Word Problems - Elementary Math

Austin walks to school every day. If (m) represents the number of miles that Austin walks one way, which of the following represents how many miles he walks in a round trip from his house to school and back?

a. 2 + m
b. 2m
c. 2-m
d. 2/m

Answer: b

Explanation:
This is a word problem that is asking us to find the total miles walked in a round trip. The term round-trip means that Austin will have done the trip twice, once from his house to the school and back from the school to his house. Twice anything means two times the thing that is asked. In this case we are asked for miles so for the round trip the answer will be 2m.

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

Rational Expressions - College Level

Multiply:




Answer:



Explanation:
The above problem is called a rational expression because it is a polynomial that contains rational exponents. The steps taken to solve this problem are as follows:
1. Apply distributive property.
2. Add exponents
3. Simplify
Here is the solution:









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

Volume - High School Math

A sand box is in the shape of a cube measuring 2 feet on each side. If the sand box weighs 12 pounds per cubic foot, find the total weight of the sand box in pounds?

Answer: 96 pounds

Explanation:
In order to find the total weight in pounds we must first find the volume of the cube.
The formula for volume is as follows:

V = lwh
V = 2 x 2 x 2 = 8 cubic feet

Since there is 12 pounds per each cubic foot, we set up the relationship as follows:
Weight = 8 cubic feet x 12 pounds/cubic feet = 96 pounds
Note that the units for cubic feet cancel out and we are left with only the units for pounds which is what the problems is asking for.

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

Scales and conversions - Middle grade

Adam was admiring the new boat that his father bought. The boat measures 20 feet in length. Adam wanted to draw the boat using a scale of ½ inch = 2 feet. What will the length of the boat in inches once Adam draws it on the paper?

Answer: 5 inches

Explanation:
To solve this problem we must set it up as a proportion as follows:

Notice that the units for feet cancel when we multiply the 2 feet by the 20 feet and we are lefts with only inches, which is what the problem requires.

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

Additiona & Subtraction - Elementary Math

Amanda and Marcy went shopping for T-shirts. Marcy bought 3 less T-shirts than Amanda. If the letter (T) represents the number of T-shirts that Amanda bought. Which of these statements can represent the number of T-shirts that Marcy bought?
a. T + 3
b. T x 3
c. T – 3
d. T / 3

Answer : c


Explanation:
The answer is T – 3
Since Amanda bought T number of shirts and Marcy had less shirts that means that this is a subtraction problem. We take the number of T-shirts Amanda bought and subtract the 3 less that Marcy bought to get our answer.

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

Square Roots - College Level

Solve for x:

Answer :

Explanation:
The steps for solving this equation are as follows:
1. Take square root of both sides.
2. Isolate variable on one side of equation.
3. Combine like terms.
4. Solve for x.

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

Mean - High School Math

Jane’s scores on six math exams were as follows:
84.5, 88.0, 92.5, 89.5, 94.0, 90.0
What is the mean of her scores?

a. 88.5
b. 89.75
c. 90.25
d. 90.75
Answer : b

Explanation:The mean value of any set of numbers indicates the average of those numbers. The average of the numbers is calculated by adding all the numbers and dividing them by how many times they occurred. For this example it will be as follows:
84.5 + 88.0 + 92.5 + 89.5 + 94.0 + 90.0 = 538.5
538.5 / 6 = 89.75
(see post 08-31)
For additional information on this subject go to : How to Find the Mean

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

Solving Equations - Middle grade

What value of A makes this equation true?

A – 7 x 4 = 12

a. 20
b. 32
c. 40
d. 48

Answer : c

Explanation:
Lets solve this problem by combining like terms and then isolating the variable on one side of the equal sign as follows:

A – 28 = 12 First step was to use the order of Operations and multiply
the 7 and 4.

A – 28 + 28 = 12 + 28 Second step involves adding 28 to both sides to isolate the
Variable.

A = 40

We can check the answer by substituting 40 in the original equation and solving as follows:

40 – 7 x 4 = 12

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

Addition and Subtraction - Elementary Math

One of the problems on the homework is as follows: 12 + 6. When Ann solved the problem she got an answer or 18. Which of the following methods can she use to check her answer?

a. 18 + 6
b. 18 x 6
c. 18 – 6
d. 18 + 12

Answer : c

Explanation:
The original problem is an addition problem. In this problem we combined or added the numbers 12 and 6 to come up with a total value of 18.
An addition operation is inverse to a subtraction operation because both operations are considered as equations since they have an equal sign (=) separating the left side from the right side. An equal sign means that what is on the left side should always equal what is on the right side.
Lets go back to our problem:
12 + 6 = 18 To check this we use the inverse which is subtraction and it could be either
18 – 6 = 12 or
18 – 12 = 6
From the list of answers above we choose C because that is the only correct inverse operation for our addition.

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

Rational Expressions - College Level

Multiply and Divide the following rational expression:

Answer :

Explanation:
The steps to solve this problem are as follows:
1. Multiply the numerators
2. Multiply the denominators
3. Add or subtract the exponents for the variables. In this problem we subtract the
exponents since we are dividing the variables.
4. Simplify the expression by dividing the product of the numerators by the product
of the denominator.
Lets apply the above steps in the above problem:

For additional information go to: Multiplying and Dividing Rational Expressions.

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

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.