Data Analysis - Elementary School Math

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


Answer:

30% Blue Shirts

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


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

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



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

Equation of a Circle - College Level

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Now we have solved the equation and are results are:

(4 , -3) , 6



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

Interpreting Line Graphs - High School Math

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

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

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

Solution: 1970

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

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

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

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

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

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

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

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

Similar Objects - Middle School Math

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

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

Answer: 6 units ( c )

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

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

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

Adding and Subtracting Fractions - Elementary Math

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

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

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

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

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

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

The above problems are solved as follows:

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

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

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

Solving equations for indicated variables - College Level

Solve each of the following formula for the indicated variable:

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

Solution:

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

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

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

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

Measures and rates - High School Math

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

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

Answer : c

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

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

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