## Practice Test on Multiplying Integers

In the previous post, we have learned the rules in multiplying integers.  Below is a 10-item practice test on multiplying integers. Note that in multiplication, we can use the x and () symbols in multiplying two numbers. This means that 3 x 4 is the same as 3 (4) and (3)(4). We will use both symbols in the practice test below to familiarize yourselves with them.  Any of the two symbols can be used in the Civil Service exams.

##### Practice Test on Multiplying Integers

1. -4 x -8

2. 3 x -7

3. -12 x -7

4.  -8 x -1 x -4 » Read more

## How to Multiply Signed Numbers

In the two previous post in Mathematics, we have discussed how to add and subtract signed numbers. In this post, we are going to learn how to multiply signed numbers particularly integers. Signed means positive and negative.

Positive Integer x Positive Integer

Clearly, the product is positive. We had been multiplying positive integers since Grade school and we all know that the product is positive.

Positive Integer x Negative Integer

When you multiply, notice that you are actually adding repeated. When we say 2 x 3, we are actually saying twice three or 2 groups of 3 or 3 + 3. When we say, thrice 11, we are saying 11 + 11 + 11. With this in mind, 3 x – 5 = -5 + -5 + -5. Since we are adding integers which are negative, the sum is also negative or -15. This means that 3 x -5 = -15. If we generalize this, we can say that the product of a positive integer and a negative integer is negative. » Read more

## Solutions to Subtracting Integers Practice Test

Blow are the solutions to the Civil Service Exam subtracting integers practice test.  As I have mentioned before, you do not have to memorize the rules of subtraction. All you have to do is convert it to addition. We can do this by converting a subtraction of integers problem  to addition of integers by changing the signs: minus “-: to plus negative “+ (-“  and minus negative “– –” or “– (-” to plus “+“. Observe how the subtraction problems are converted into addition problems in the solutions below. If you have any questions, please use the comment box below.

1. 12 23 = 12 + (-23) = -11

2. -21 – (-4) = -21 + 4 = -17

3. 17 – (-36) = 17 + 36 = 53

4. -18 13 = -18 + (-13) =-31

5. 22 35 = 22 + (-35) = -13

6. -34 – (-21) = -34 + 21 = – 13

7. 29 – (-6) = 29 + 6 = 35

8. 98 – 14 = 84 (no need to change signs, this is ordinary subtraction since the minuend is larger than the subtrahend).

9. -87 53 = -87 + (-53) = -140

10. 63 92 = 63 + (-92) = -29

In the next post, we are going to discuss about multiplication and division of integers.

## Civil Service Practice Test on Subtracting Integers

Three days ago, we have learned how to subtract positive and negative integers or signed numbers. In this post, I am going to give you a practice test on subtracting integers. For convenience, I have colored the negative signs blue instead of raising them to an exponent.

1. 12 – 23

2. 21 -(4)

3. 17 – (36)

4. 18 – 13

5. 22 – 35

6. 34 -( 21)

7. 29 -( 6)

8. 98 – 14

9.  87 – 53

10. 63 – 92

## How to Subtract Positive and Negative Integers

This is the continuation of the series of Civil Service review in mathematics particularly on operations of integers. In this post, we are going to discuss the most complicated operation on integers. I have taught people of all ages about this topic and it seems that for many, this is the most difficult among the four operations. In this post, we are going to learn how to subtract positive and negative integers or signed numbers. Note that in subtracting integers, there are only  four forms. If a and b are positive, the subtraction are of the following forms.

Case 1: positive minus positive (ab)
Case 2: negative minus positive (ab)
Case 3: positive minus negative (ab)
Case 4: negative minus negative (ab)

##### How to Subtract Positive and Negative Integers

What most people don’t know that ab is the same as ab, or subtracting a number is the same as adding its negative. That means that you only have to memorize the steps in addition of integers. Given a subtraction sentence, you then transform it  into addition. Here are a few examples.

Case 1 Exampe 1: 5 – 8

Subtracting is the same as adding its negative, so 5 – 8 = 5 + 8. Note that 5 + 8 is already addition and 5 + 8 = 3.

Case 2 Example: 10 – 4

The expression 10 – 4 is the same as 10 + 4 = 14.

Remember also that if you see two consecutive – signs or a minus and a negative sign, you can transform it to +. That is, -(a) = + a and -(a) + a. In most exam, the negative signs are not usually superscript, so you will likely -(-a).

Case 3 Example: 5 – 6

The above expression might be written in 5- -6 or 5-(-6). In any case, two negative signs, a minus and a negative sign can be transformed into a plus sign so, 5 (6) = 5 + 6 = 11. Notice that the last equation is also an addition sentence.

Case 4 Example: 8 – 6

The expression 8 – 6 = 8 + 6 = 2.

Observe that the four forms are already completed in the examles. From the strategy above, we only remember two strategies: (1) transform any subtraction sentence to addition sentence and (2) replace two consecutive negatives or a minus and a negative with + sign.

## Solutions to Practice Exercises on Addition of Integers

We have learned how to add integers and in the previous post, I have given you practice exercises that you can use to evaluate your understanding of the topic. Below is the complete solutions to the practice exercises on adding integers. Share to me how many did you get right.

Solutions to Practice Exercises on Addition of Integers

1. 28 + 12 = 40.

2. 14 + 11 = 25

3. 24 + 15 = 9

Solution: We pair 15 and 15  to get 0. We have 9  left. So, the correct answer is 9.

4. –16 + 31 = 15

We pair 16  and 16 to get 0. We are left with positive 15.
5. 23 + 46 + 15 = 54

We add the two positive numbers first: 23 + 46 = 69. Next, we add 69 and 15. We get 15 from 69 and pair it with 15 resulting to 0, so we have 54 left.

6. 45 + 12 + 16 = 17

We add the negative numbers first: 12 + 16 = 28. We add the result to 45. We get 28 from 45 and pair it with  28 to get 0 leaving 17.

7. 12 +15 + 62 =  89

Explanation: They are negative, so we just added them. Of course negative added to negative is always negative.

8. 22 +  36 + 36 = 22

Explanation: 36 + 36 = 0, so we are left with 22.

9. 12 + 18 + 12 + 18 = 0

Explanation: 12 and 12 and 18 and 18 = 0.

10. 31 + 55 + 41 +  32 + 10 = 3

Explanation: Adding the positive integers, we have 31 + 55  = 86. Adding the negative integers, we have  41 +  32 + 10 =83. Now, 86 + 83 = 3

## Practice Exercises on Addition of Integers

In the previous post, we have learned about adding positive and negative integers particularly on how to add integers with different signs. In this post, I am going to give you 10 exercises on adding integers. I will give the answers below for you to be able to check if your answers are correct. Also, I am going to omit the + sign before the positive integers because this is not usually shown in the exam. This means that 3 will automatically mean +3 unless a – sign precedes it.

1. 28 + 12
2. 14 + 11
3. 24 + 15
4. 16 + 31
5. 23 + 46 + 15
6. 45 + 12 + 16
7.  12 + 15 + 62
8. 22 +  36 + 36
9. 12 + 18 + 12 + 18
10. 31 + 55 + 41 + 32 + 10

## How to Add Positive and Negative Integers

One of the topics in basic mathematics  that will likely be included in the the Philippine Civil Service Exam both professional and subprofessional are operations on integers. Although a few Civil Service test items may be given from this topic,  it is important that you master it because a lot of calculation in other topics will need knowledge of integers and its operations (addition, subtraction, multiplication, division). For example, solving some word problems in mathematics and solving equations will need knowledge on operations of integers.

Integers are whole numbers that are either positive or negative. Examples of integers are -5, 6, 0, and 10. If we place this on the number line, negative integers are the integers that are below 0 (left of 0), while the positive integers are the integers above 0 (right of 0).

##### Adding Integers that Are Both Positive

When you add integers that are both positive, it is just like adding whole numbers. Below are the examples.

Example 1: +2 + +4 = +6

Example 2: +9 + +41 + +6 + = +56

Example 3: +120 + +13 + +12 + = +145

Although we have created a small + before the number to indicate that it is positive, in reality, only negative numbers have signs. This means that +2 + +4 = +6 is just written as 2 + 4 = 6.

##### Adding Integers that Are Both Negative

Adding number that are both negative is just the same as adding numbers that are both positive. The only difference is that if you add two negative numbers, the result is negative.

Example 1: 5 + 8 = 13

Example 2: 10 + 18 + 32 + = 60

Example 3: 220 + 11 + 16 + = 247

##### How to Add Positive and Negative Integers

Before adding, you should always remember that +1 and 1 cancel out each other, or +1 + 1 is 0. So the strategy is to pair the positive and negative numbers and take out what’s left.

Example 1:  What is +13 + 8?

Solution:

We pair 8 positives and 8 negatives to cancel out. Then what’s left is of +13 is +5. In equation form, we have

+138 = ++ +8 + 8+5 + (+88) = +5 + (0) = +5

Example 2: What is +17 + 20?

Solution:

We pair 17 negatives and 17 positives. What’s left of 20 is 3. In equation form, we have

+1720 = +17 + (173) =  (+17 + 17) + 3 = 0 + 33

Example 3: What is +16 + +37 + 20 + 3 +9 ?