Examples with solutions for Addition and Subtraction of Directed Numbers: Number of terms

Exercise #1

Solve the following equation:

30(2)+(8)+5= -30-(-2)+(-8)+5=

Video Solution

Step-by-Step Solution

Let's begin by applying the following rule in order to rewrite the equation:

(x)=+ -(-x)=+

30+2+(8)+5= -30+2+(-8)+5=

28+(8)+5= -28+(-8)+5=

Next we will locate negative 28 on the number line and proceed 8 steps to the left (since negative 8 is less than zero):

-27-27-27-28-28-28-29-29-29-30-30-30-31-31-31-32-32-32-33-33-33-34-34-34-35-35-35-36-36-36-37-37-37-38-38-38-39-39-39-40-40-40

We reach negative 36.

Resulting in the following exercise:

36+5= -36+5=

\

Next we locate negative 36 on the number line and proceed 5 steps to the right (since 5 is greater than zero):

-27-27-27-28-28-28-29-29-29-30-30-30-31-31-31-32-32-32-33-33-33-34-34-34-35-35-35-36-36-36-37-37-37-38-38-38-39-39-39-40-40-40

We reach negative 31

Answer

31 -31

Exercise #2

Solve the following equation:

30+4+(8)= -30+4+(-8)=

Video Solution

Step-by-Step Solution

Let's begin by locating negative 30 on the number line and moving 4 steps to the right (since 4 is greater than zero):

-27-27-27-28-28-28-29-29-29-30-30-30-31-31-31-32-32-32-33-33-33-34-34-34-35-35-35-36-36-36-37-37-37-38-38-38-39-39-39-40-40-40-26-26-26

We reach negative 26.

Resulting in the following exercise:

26+(8)= -26+(-8)=

Now let's locate negative 26 on the number line and move 8 steps to the left (since negative 8 is less than zero):

-27-27-27-28-28-28-29-29-29-30-30-30-31-31-31-32-32-32-33-33-33-34-34-34-35-35-35-36-36-36-37-37-37-38-38-38-39-39-39-40-40-40-26-26-26We reach negative 34

Answer

34 -34

Exercise #3

4+3+(6)+(9)= 4+3+(-6)+(-9)=

Video Solution

Step-by-Step Solution

First, let's look at the first exercise:

4+3 4+3

We will locate the number 4 on the axis, and move right three steps, where each step represents a whole number in the following way:

-6-6-6-5-5-5-4-4-4-3-3-3-2-2-2-1-1-1000111222333444555666777

We can see that we reached the number 7.

Now we get the exercise:

7+(6)+(9)= 7+(-6)+(-9)=

Let's look at the exercise:

7+(6)= 7+(-6)=

We will locate the number 7 on the axis, and move left six steps, where each step represents a whole number in the following way:

-6-6-6-5-5-5-4-4-4-3-3-3-2-2-2-1-1-1000111222333444555666777

We can see that we reached the number 1.

Now we got the exercise:

1+(9)= 1+(-9)=

We will locate the number 1 on the axis, and move left nine steps, where each step represents a whole number in the following way:

-6-6-6-5-5-5-4-4-4-3-3-3-2-2-2-1-1-1000111222333-7-7-7-8-8-8-9-9-9

We can see that we reached the number minus 8.

Answer

8 -8

Exercise #4

(4)+2+(3)+(5)= (-4)+2+(-3)+(-5)=

Video Solution

Step-by-Step Solution

Let's start with the leftmost exercise:

(4)+2 (-4)+2

We'll locate negative 4 on the axis and move two steps to the right, where each step represents one whole number:

-6-6-6-5-5-5-4-4-4-3-3-3-2-2-2-1-1-1000111222333

We can see that we've reached negative 2.

Now we'll get the exercise:

(2)+(3)+(5)= (-2)+(-3)+(-5)=

Let's focus on the exercise:

(2)+(3) (-2)+(-3)

We'll locate negative 2 on the axis and move three steps to the left, where each step represents one whole number:

-6-6-6-5-5-5-4-4-4-3-3-3-2-2-2-1-1-1000111222333

We can see that we've reached negative 5.

Now we'll get the exercise:

(5)+(5)= (-5)+(-5)=

We'll locate negative 5 on the axis and move five steps to the left, where each step represents one whole number:

-6-6-6-5-5-5-4-4-4-3-3-3-2-2-2-1-1-1000111-7-7-7-8-8-8-9-9-9-10-10-10

We can see that we've reached negative 10.

Answer

10 -10

Exercise #5

(30)+(2)+30+(5)= (-30)+(-2)+30+(-5)=

Video Solution

Step-by-Step Solution

First, let's organize the exercise in a way that will make it easier and more convenient to solve.

Notice that the number 30 appears twice in the exercise, so let's start with it:

(30)+30+(2)+(5)= (-30)+30+(-2)+(-5)=

Let's look at the exercise:

30+30 -30+30

Since we move left from zero to minus 30, and then return right 30 steps, we will arrive at the same number we started from: 0

-30-30-30000+30

Now let's continue the exercise in the following way:

0+(2)+(5)= 0+(-2)+(-5)=

We'll locate the number minus 2 on the number line, and move left five steps where each step represents one whole number:

-5-5-5000-1-1-1-2-2-2-3-3-3-4-4-4222111-8-8-8-7-7-7-6-6-6

We can see that we arrived at minus 7.

Answer

7 -7

Exercise #6

Solve the following problem:

(10)+(3)+4+(12)= (-10)+(-3)+4+(-12)=

Video Solution

Step-by-Step Solution

Begin by observing the first exercise:

(10)+(3) (-10)+(-3)

We will locate the number minus 10 on the number line, and move three steps to the left, where each step represents one whole number:

-16-16-16-11-11-11-12-12-12-13-13-13-14-14-14-15-15-15-9-9-9-10-10-10

We reached the number minus 13.

Obtaining the following exercise:

(13)+4+(12)= (-13)+4+(-12)=

The next exercise is:

(13)+4 (-13)+4

Locate the number minus 13 on the number line, and move four steps to the right where each step represents one whole number:

-16-16-16-11-11-11-12-12-12-13-13-13-14-14-14-15-15-15-9-9-9-10-10-10

We reached the number minus 9.

Obtaining the following exercise:

(9)+(12)= (-9)+(-12)=

Locate the number minus 9 on the number line, and move twelve steps to the left where each step represents one whole number:

-16-16-16-11-11-11-12-12-12-13-13-13-14-14-14-15-15-15-9-9-9-10-10-10-21-21-21-20-20-20-19-19-19-18-18-18-17-17-17

We reached the number minus 21.

Answer

21 -21

Exercise #7

3+(12)+(38)+58= -3+(-\frac{1}{2})+(\frac{3}{8})+\frac{5}{8}=

Video Solution

Step-by-Step Solution

To solve the given problem of adding 3+(12)+38+58 -3 + (-\frac{1}{2}) + \frac{3}{8} + \frac{5}{8} , we will use the following steps:

  • Step 1: Calculate 38+58\frac{3}{8} + \frac{5}{8}
  • Step 2: Subtract 12-\frac{1}{2} from the result of step 1
  • Step 3: Add the final result to 3-3

Now, let us work through each step:

Step 1: Calculate 38+58\frac{3}{8} + \frac{5}{8}. Since these fractions have the same denominator, we simply add their numerators: 3+58=88=1\frac{3 + 5}{8} = \frac{8}{8} = 1.

Step 2: Now we subtract 12-\frac{1}{2} from 1. We can rewrite 11 as 88\frac{8}{8} and 12-\frac{1}{2} as 48-\frac{4}{8} (since their least common denominator is 8). So: 1(12)=88(48)=8+48=128=32.1 - \left(-\frac{1}{2}\right) = \frac{8}{8} - \left(-\frac{4}{8}\right) = \frac{8 + 4}{8} = \frac{12}{8} = \frac{3}{2}.

Step 3: Finally, we add this result to 3-3. 3-3 can be expressed as 62-\frac{6}{2} and 32\frac{3}{2} remains the same: 3+32=62+32=6+32=32=52.-3 + \frac{3}{2} = -\frac{6}{2} + \frac{3}{2} = \frac{-6 + 3}{2} = \frac{-3}{2} = -\frac{5}{2}.

Hence, the solution to the problem is 52-\frac{5}{2}.

Answer

52 -\frac{5}{2}

Exercise #8

12+34+15+(45)= -\frac{1}{2}+\frac{3}{4}+-\frac{1}{5}+(-\frac{4}{5})=

Video Solution

Step-by-Step Solution

To solve this problem, we must simplify the expression 12+34+(15)+(45) -\frac{1}{2} + \frac{3}{4} + (-\frac{1}{5}) + (-\frac{4}{5}) .

First, we need to find the least common denominator (LCD) for the fractions 2, 4, and 5. The LCD is 20.

Next, we convert each fraction to an equivalent fraction with the common denominator of 20:

  • 12-\frac{1}{2} becomes 1020-\frac{10}{20}
  • 34\frac{3}{4} becomes 1520\frac{15}{20}
  • 15-\frac{1}{5} becomes 420-\frac{4}{20}
  • 45-\frac{4}{5} becomes 1620-\frac{16}{20}

Now we perform the addition and subtraction:

1020+15204201620-\frac{10}{20} + \frac{15}{20} - \frac{4}{20} - \frac{16}{20}

Combine the numerators:

10+15416=15-10 + 15 - 4 - 16 = -15

Thus, the resulting fraction is:

1520-\frac{15}{20}

We simplify 1520-\frac{15}{20} by dividing both the numerator and the denominator by their greatest common divisor, which is 5:

1520=34-\frac{15}{20} = -\frac{3}{4}

Therefore, the solution to the problem is 34 -\frac{3}{4} , which corresponds to choice 2.

Answer

34 -\frac{3}{4}

Exercise #9

49+5+(2)+59= -\frac{4}{9}+5+(-2)+\frac{5}{9}=

Video Solution

Step-by-Step Solution

To solve this problem, we'll perform operations involving both fractions and whole numbers:

  • Step 1: Combine the fractional parts 49-\frac{4}{9} and 59\frac{5}{9}.

  • Step 2: Add the remaining whole numbers 55 and 2-2.

  • Step 3: Sum the results from Step 1 and Step 2.

Let's start:
Step 1: Work with the fractions together. - We have 49-\frac{4}{9} and 59\frac{5}{9}, both have the same denominator, thus can be directly added: 49+59=4+59=19 -\frac{4}{9} + \frac{5}{9} = \frac{-4+5}{9} = \frac{1}{9}

Step 2: Add the integer components 55 and 2-2: 5+(2)=52=3. 5 + (-2) = 5 - 2 = 3.

Step 3: Combine results from Step 1 and Step 2: 3+19=279+19=289 3+\frac{1}{9}=\frac{27}{9}+\frac{1}{9}=\frac{28}{9} .

Therefore, the final result of the expression is 289\frac{28}{9}.

Answer

289 \frac{28}{9}

Exercise #10

5+12+10+(34)= -5+-\frac{1}{2}+10+(-\frac{3}{4})=

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Simplify the integers using addition.
  • Step 2: Simplify the fractions, ensuring a common denominator before adding.
  • Step 3: Combine results from integers and fractions for the final answer.

Let's work through each step:

Step 1: Combine the integers 5 -5 and 10 10 .

5+10=5 -5 + 10 = 5

Step 2: Simplify and add the fractions 12-\frac{1}{2} and 34-\frac{3}{4}.

To add the fractions, we need a common denominator. The denominators are 2 and 4. The least common denominator is 4.

Convert 12-\frac{1}{2} to an equivalent fraction with a denominator of 4:

12=24-\frac{1}{2} = -\frac{2}{4}

Now add 24-\frac{2}{4} and 34-\frac{3}{4}:

24+34=54-\frac{2}{4} + -\frac{3}{4} = -\frac{5}{4}

Step 3: Combine the result of the integer addition and the fraction addition.

The integer result is 5 and the fraction result is 54-\frac{5}{4}. Convert 5 to a fraction with the same denominator:

5=2045 = \frac{20}{4}

Combine the fractions:

204+54=2054=154\frac{20}{4} + -\frac{5}{4} = \frac{20 - 5}{4} = \frac{15}{4}

Therefore, the solution to the problem is 154 \frac{15}{4} .

Answer

154 \frac{15}{4}

Exercise #11

Solve the following problem:

25(5)+(6)4= 25-(-5)+(-6)-4=

Video Solution

Step-by-Step Solution

Let's remember the rule:

(x)=+x -(-x)=+x

Let's write the exercise in the appropriate form:

25+5+(6)4= 25+5+(-6)-4=

Let's solve the exercise from left to right:

25+5=30 25+5=30

We obtain the following exercise:

30+(6)4= 30+(-6)-4=

Let's remember the rule:

+(x)=x +(-x)=-x

Let's write the exercise in the appropriate form:

3064= 30-6-4=

Let's solve the exercise from left to right:

306=24 30-6=24

244=20 24-4=20

Answer

20 20

Exercise #12

Solve the following problem using the order of operations:

15+5(4)+(9)+84= 15+5-(-4)+(-9)+8-4=

Video Solution

Step-by-Step Solution

Let's remember the rule:

(x)=+x -(-x)=+x

We'll write the exercise in the appropriate form:

15+5+4+(9)+84= 15+5+4+(-9)+8-4=

Let's solve the exercise from left to right:

15+5=20 15+5=20

Now we'll obtain the exercise:

20+4+(9)+84= 20+4+(-9)+8-4=

Let's solve the exercise from left to right:

20+4=24 20+4=24

Now we'll obtain the exercise:

24+(9)+84= 24+(-9)+8-4=

Let's remember the rule:

+(x)=x +(-x)=-x

We'll write the exercise in the appropriate form:

249+84= 24-9+8-4=

Let's solve the exercise from left to right:

249=15 24-9=15

15+8=23 15+8=23

234=19 23-4=19

Answer

19 19

Exercise #13

27(7)+(6)+211= -27-(-7)+(-6)+2-11=

Video Solution

Step-by-Step Solution

First, we solve the multiplication exercise, that is where there is a plus or minus sign before another sign.

27+76+211= -27+7-6+2-11=

Now we solve as a common exercise from left to right:

27+7=20 -27+7=-20

206=26 -20-6=-26

26+2=24 -26+2=-24

2411=35 -24-11=-35

Answer

35 -35

Exercise #14

38(58)(12)= -\frac{3}{8}-(-\frac{5}{8})-(-\frac{1}{2})=

Video Solution

Step-by-Step Solution

To solve the problem 38(58)(12)-\frac{3}{8}-(-\frac{5}{8})-(-\frac{1}{2}), we will follow these steps:

  • Step 1: Address the negative signs. Note that subtracting a negative is the same as adding its positive counterpart:
    • 38-\frac{3}{8} remains the same.
    • (58)-(-\frac{5}{8}) becomes +58+\frac{5}{8}.
    • (12)-(-\frac{1}{2}) becomes +12+\frac{1}{2}.
  • Step 2: Write the expression with the adjusted signs: 38+58+12-\frac{3}{8} + \frac{5}{8} + \frac{1}{2}.
  • Step 3: Find a common denominator for the fractions. The denominators are 8 and 2. The least common denominator is 8.
  • Step 4: Convert all fractions to have this common denominator:
    • 38-\frac{3}{8} is already with a denominator of 8.
    • 58\frac{5}{8} is already with a denominator of 8.
    • 12=48\frac{1}{2} = \frac{4}{8}.
  • Step 5: Perform the arithmetic operations on the numerators while retaining the common denominator:
  • 38+58+48=3+5+48-\frac{3}{8} + \frac{5}{8} + \frac{4}{8} = \frac{-3+5+4}{8}.
  • Step 6: Compute the result: Calculate 3+5+4=6 -3 + 5 + 4 = 6 , therefore the fraction becomes 68\frac{6}{8}.
  • Step 7: Simplify the fraction 68\frac{6}{8} by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
  • 68=34\frac{6}{8} = \frac{3}{4}.

Therefore, the solution to the problem is 34 \frac{3}{4} .

Answer

34 \frac{3}{4}

Exercise #15

147+(3)12(14)= -\frac{14}{7}+(-3)-\frac{1}{2}-(-\frac{1}{4})=

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Simplify 147-\frac{14}{7}.
  • Step 2: Perform arithmetic operations in sequence, converting all parts as necessary.
  • Step 3: Combine the fractions into a single fraction and simplify.

Now, let's work through each step:
Step 1: Simplify 147-\frac{14}{7}. Since 14÷7=214 \div 7 = 2, 147=2-\frac{14}{7} = -2.

Step 2: We rewrite the expression properly:
2+(3)12(14)-2 + (-3) - \frac{1}{2} - (-\frac{1}{4}).

Simplify and operate on each part:
Convert 3-3 to a fraction with a denominator of 1: 31-\frac{3}{1}.
Evaluate the subtraction of a negative: (14)=14-(-\frac{1}{4}) = \frac{1}{4}.

Rewrite the expression using fractions:
21+(31)12+14-\frac{2}{1} + (-\frac{3}{1}) - \frac{1}{2} + \frac{1}{4}.

Step 3: Add and subtract the fractions using a common denominator. The least common denominator for 1, 2, and 4 is 4.
21=84-\frac{2}{1} = -\frac{8}{4},
31=124-\frac{3}{1} = -\frac{12}{4},
12=24-\frac{1}{2} = -\frac{2}{4}.

Combining these, we get:
8412424+14=8122+14-\frac{8}{4} - \frac{12}{4} - \frac{2}{4} + \frac{1}{4} = \frac{-8 - 12 - 2 + 1}{4}.

Simplify the numerator: 8122+1=21 -8 - 12 - 2 + 1 = -21.
Thus, we have:
214\frac{-21}{4}.

Therefore, the solution to the problem is 214 -\frac{21}{4} , which corresponds to choice 3.

Answer

214 -\frac{21}{4}

Exercise #16

Solve:

416(38)+28+(14)= -\frac{4}{16}-(-\frac{3}{8})+\frac{2}{8}+(-\frac{1}{4})=

Video Solution

Step-by-Step Solution

To solve the problem, we will follow these steps:

  • Simplify each fraction where possible.
  • Find a common denominator for all fractions.
  • Convert each fraction to have this common denominator.
  • Perform the required operations: subtraction and addition.

Let's begin solving the problem:

Step 1: Simplify each fraction.
- 416-\frac{4}{16} simplifies to 14-\frac{1}{4} since both numerator and denominator can be divided by 4.
- The fractions (38)-(-\frac{3}{8}), 28\frac{2}{8}, and 14-\frac{1}{4} are already in their simplest forms.

Step 2: Find the common denominator.
The denominators are 4 and 8. The least common denominator (LCD) is 8.

Step 3: Convert each fraction to an equivalent fraction with this common denominator:
- 14-\frac{1}{4} becomes 28-\frac{2}{8} because 14×22=28\frac{1}{4} \times \frac{2}{2} = \frac{2}{8}.
- (38)-(-\frac{3}{8}) simplifies to 38\frac{3}{8} (due to subtracting a negative, which makes it positive).
- 28\frac{2}{8} remains unchanged, as it already has the common denominator.
- 14-\frac{1}{4} becomes 28-\frac{2}{8} for the same reason as above.

Step 4: Perform the operations:
28+38+2828-\frac{2}{8} + \frac{3}{8} + \frac{2}{8} - \frac{2}{8}.

Adding and subtracting these fractions with a common denominator:
- Combine them as (2+3+22)/8=1/8(-2 + 3 + 2 - 2)/8 = 1/8.

Therefore, the solution to the problem is 18 \frac{1}{8} .

Answer

18 \frac{1}{8}

Exercise #17

7.5+(9.5)+5+(13.5)=? 7.5+(-9.5)+5+(-13.5)=\text{?}

Video Solution

Step-by-Step Solution

Let's begin by expanding the parentheses whilst paying attention to the minus and plus signs, which change accordingly:

+(9.5)=9.5 +(-9.5)=-9.5

+(13)=13 +(-13)=-13

We should obtain the following:

7.59.5+513.5= 7.5-9.5+5-13.5=

Now let's solve the exercise from left to right:

2+513.5= -2+5-13.5=

313.5=10.5 3-13.5=-10.5

Answer

10.5 -10.5