Prealgebra Practice Problems

Get at least 9/10 (or 10/11 if you count the two step problem as two questions) correct to understand the concept!

Starred (*) problems are challenge problems, which could have multiple steps.

1. What is the median of the following numbers?
   -1, 3, 2.25, 0, 5, -1.2, and 4


2. Is 187,233,642 a multiple of 9?


3. What is the LCM (least common multiple) of 33, 6, and 18?


4. *What is the product of the GCF (greatest common factor) and median of the number set 24, 92, 16?


5. Solve for b in the following equation:
   3b + 7 = 28


6. Solve for z in the following equation:
   4z - 21 = 9


7. a). If Sarah walks at 3 miles per hour for 20 minutes, and runs at 5 miles per hour for 12 minutes, how many miles will she have completed in total?
   b). If John jogs at 3.5 miles per hour for 40 minutes, who covers more distance in total?


8. What is the mean of the following numbers?
   1, 3, 4, 7, -2, 3


9. What is the mode of the following numbers?
   0, 3, 0, 4, 5, 1, 6, 1, 3, 0, 2, 8, 5, 2, 0


10. Joana's siblings' heights are shown below, in inches:
    49, 52, 58, 60
    What is the mean height of Joana and her siblings, assuming she is 61 inches? What is the median height of Joana and her siblings? Is there a unique mode height?

Prealgebra Practice Solutions

1. To find the median of the number, you must order the number set from least to greatest. When we do this, we get -1.2, -1, 0, 2.25, 3, 4, 5.
   For our next step, we must find the numbers in the middle of the data set - in this case, the number in the middle is 2.25. Therefore, our median is 2.25.


2. To find if a number is a multiple of nine, you simply add the digits up. If the sum of the digits is a multiple of 9, the entire number is a multiple of 9. Otherwise, it isn't. Our number is 187,233,642. Adding up the digits, we get 36, which is a multiple of 9, (4 • 6 = 36) so our answer is yes.


3. We are asked to find the least common multiple of the numbers 33, 6, and 18. We immediately see that each number is a multiple of 3. We can, therefore divide each number by three. Here is a visual:
   3 | 11, 2, 6
   Now we multiply all the numbers in the visual together to get 396.


4. In this challenge problem, we are asked to take the product of the GCF (greatest common factor) and median of the numbers 24, 16, and 92. We start by taking the median.
   
   Our median is 24. This is because, when we arrange the numbers from least to greatest, 24 is in the middle.
   Now we must find the greatest common factor. The largest factor is common between 16, 24, and 92 is 4. We cannot go farther because we are left with the numbers 4, 6, and 23 - 23 is prime, so we cannot divide any more.
   Our final step is now to multiple the median (24) by the greatest common factor (4). Multiplying, we get 24 • 4 = 96.


5. To solve linear equations, we must isolate the variable (b in this case) to find its value.
   3b + 7 = 28
   Subtract 7 from both sides.
   3b = 21
   Divide both sides by 3 to isolate b.
   b = 7

   ---

   Now, we have found that b is 7.


6. To solve linear equations, we must isolate the variable (z in this case) to find its value.
   4z - 21 = 9
   Add 21 to both sides.
   4z = 30
   Divide both sides by 4 to isolate z.
   z = 30/4
   Simplifying, we now get:
   z = 7 1/2, or z = 7.5

   ---

   Now, we have found that z is 7.5.


7. a). Sarah walks at 3 miles per hour for 20 minutes. Since 20 minutes is 20/60, or 1/3 of an hour, and Sarah goes at 3 miles per hour, Sarah will have walked 3 miles/60 minutes • 20 minutes = 1/3 • 3 miles = 1 mile.
   Now let's find out how much she ran. She ran at 5 miles per hour for 12 minutes. Since 12 minutes is 12/60, or 1/5 of an hour, and Sarah goes at 5 miles per hour, Sarah will have run 5 miles/ 60 minutes • 12 minutes = 1/5 • 5 miles = 1 mile.
   Adding these two numbers, we get 1 + 1 = 2, and so Sarah has travelled 2 miles.
   
   b.) Now, we need to find how far John has moved. John has jogged at 3.5 miles per hour for 40 minutes. This is 3.5 miles/60 minutes • 40 minutes, which is simplified to 2/3 • 3.5 miles, which is 7/3 miles. 7/3 miles can be simplified to 2 1/3 miles, which is more than 2 miles. Therefore, John has covered more distance than Sarah.


8. Since the mean is just the average, we just need to find the average of the numbers 1, 3, 4, 7, -2, 3. To find the average/mean, we need to add up all the numbers and then divide by the number of numbers in the number set. First, when we add up the numbers, we get 16. Next, we need to see how many numbers there are. There are 6 numbers. All we now have to do is divide 16 by 6, to get 2 2/3.


9. The mode is simply the most common number. To make the problem a bit easier, we can start by ordering the numbers in ascending order. This will put all the same values right next to each other. When we do this, we get:
   0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 8
   Now that we have arranged the numbers like this, it is easy to see that 0 appears the most times (4 times), so our mode is 0.


10. The problem asks us to find the mean of Joana and her siblings' heights. Therefore, we can include Joana's height in the data set:
    49, 52, 58, 60, 61.
    (We know that her height was not included, since the list did not include 61 inches as a value, and her height is 61 inches). Now we just need to find the mean/average of the data set. We can do that by finding the sum of the numbers and dividing that by the number of numbers. After adding, we get 280, and since there are 5 numbers, the mean is 280/5 = 56. Now we need to find the median. To find the median, we normally need to order the numbers from least to greatest (or greatest to least) and find the middle number. Here, however, it is already ordered for us. We now just need to find the middle number, and that is 58, so the median height of Joana's siblings is 58 inches. Lastly, we now must see if there is a unique mode. Since there is only 1 of each height in the data set, there is no unique mode in the data set.