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.