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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.