Addition and Subtraction of Directed Numbers: Solving the problem

Let's add 8 to the number line and move 12 steps to the right.

Note that our result is a positive number:

Now solve the following exercise:

$8+12=20$

Let's locate -8 on the number line and move 12 steps to the left.

Let's note that our result is a negative number:

Let's remember the rule:

Now let's write the exercise in the appropriate form and solve it:

$-8-12=-20$

Let's locate -10 on the number line and move 13 steps to the left.

Let's note that our result is a negative number:

Let's remember the rule:

$-(+x)=-x$

Now let's write the exercise in the appropriate form and solve it:

$-10-13=-23$

Let's remember the rule:

$-(-x)=+x$

Now let's write the exercise in the appropriate form:

$-8+13=$

We'll use the substitution law and solve:

$13-8=5$

First we need to locate the number 8 on the number line and move 4.5 steps to the left from it:

Remember the rule:

$+(-x)=-x$

Now let's rewrite the problem in the appropriate form and solve:

$8-4.5=3.5$

Let's remember the rule:

$-(-x)=+x$

Now let's write the exercise in the appropriate form:

$567+69=$

Let's solve the exercise vertically:

$567\\+69\\636$

First, remember the rule:

$+(-x)=-x$

Place 301 on the number line and move 51 steps to the left:

Rewrite the exercise in the appropriate form and solve it:

$301-51=250$

Let's remember the rule:

$-(+x)=-x$

Now let's write the exercise in the appropriate form:

$6-11=$

We'll locate the number 6 on the number line and from there we'll move 11 steps to the left:

The answer is minus 5.

First, let's remember the rule:

$+(+x)=+x$

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

$-8+12=$

We'll draw a number line and place minus 8 on it, then move 12 steps to the right:

Therefore:

$-8+12=4$

Let's position minus $\frac{1}{7}$ on the number line and move one step to the right, since $\frac{7}{7}=\frac{1}{1}=1$

We should note that our result is a positive number:

Let's remember the rule:

$-(-x)=+x$

Now let's write the exercise in the appropriate form and solve it:

$-\frac{1}{7}+1=\frac{6}{7}$

First, locate the number $\frac{1}{4}$ on the number line and move $2\frac{3}{4}$ steps to the right from it.

This means the resulting number will be positive:

Finally, solve the exercise:

$\frac{1}{4}+2\frac{3}{4}=3$

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

• Step 1: Convert $3\frac{1}{3}$ to an improper fraction.
• Step 2: Find a common denominator for $-\frac{1}{5}$ and the improper fraction.
• Step 3: Add the fractions.
• Step 4: Simplify the result, converting it back to a mixed number if necessary.

Now, let's work through each step:

Step 1: Convert $3\frac{1}{3}$ to an improper fraction.
$3\frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3}$

Step 2: Find a common denominator for $-\frac{1}{5}$ and $\frac{10}{3}$.
The least common denominator of 5 and 3 is 15.

Step 3: Express each fraction with the common denominator:
$-\frac{1}{5} = -\frac{3}{15}$ (multiply the numerator and denominator by 3)
$\frac{10}{3} = \frac{50}{15}$ (multiply the numerator and denominator by 5)

Step 4: Add the fractions:
$-\frac{3}{15} + \frac{50}{15} = \frac{-3 + 50}{15} = \frac{47}{15}$

Step 5: Simplify $\frac{47}{15}$ back to a mixed number if needed:
Performing the division, 47 divided by 15 is 3 with a remainder of 2.
Therefore, $\frac{47}{15} = 3\frac{2}{15}$.

Therefore, the solution to the problem is $3\frac{2}{15}$.

To solve the problem of subtracting $+7\frac{1}{4}$ from $+13\frac{1}{3}$, we follow these steps:

• Convert $13\frac{1}{3}$ to an improper fraction:

$13\frac{1}{3} = \frac{39}{3} + \frac{1}{3} = \frac{39 + 1}{3} = \frac{40}{3}$

• Convert $7\frac{1}{4}$ to an improper fraction:

$7\frac{1}{4} = \frac{28}{4} + \frac{1}{4} = \frac{28 + 1}{4} = \frac{29}{4}$

• Find a common denominator for $\frac{40}{3}$ and $\frac{29}{4}$. The least common multiple of 3 and 4 is 12.
• Convert $\frac{40}{3}$ to a fraction with a denominator of 12:

$\frac{40}{3} \times \frac{4}{4} = \frac{160}{12}$

• Convert $\frac{29}{4}$ to a fraction with a denominator of 12:

$\frac{29}{4} \times \frac{3}{3} = \frac{87}{12}$

• Subtract the fractions:

$\frac{160}{12} - \frac{87}{12} = \frac{160 - 87}{12} = \frac{73}{12}$

• Convert $\frac{73}{12}$ back to a mixed fraction:

$73 \div 12 = 6$ remainder $1$, so it equals $6\frac{1}{12}$.

Therefore, the solution to the problem is $6\frac{1}{12}$.

Let's mark minus $\frac{1}{5}$ on the number line and move $3\frac{4}{5}$ steps to the left, meaning our result will be a negative number:

Let's remember the rule:

$+(-x)=-x$

Now let's write the exercise in the appropriate form and solve it:

$-\frac{1}{5}-3\frac{4}{5}=-4$

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

• Convert the mixed number to an improper fraction.

• Find a common denominator for the fractions.

• Perform the subtraction.

Now, let's work through each step:

Step 1: Convert the Mixed Number
Convert $3\frac{3}{4}$ to an improper fraction:

$3\frac{3}{4} = \frac{3 \times 4 + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4}$.

Step 2: Find a Common Denominator
Identify a common denominator for $\frac{1}{5}$ and $\frac{15}{4}$. The least common denominator (LCD) of 5 and 4 is 20.

Convert $\frac{1}{5}$ and $\frac{15}{4}$ to have denominator 20:

$\frac{1}{5} = \frac{1 \times 4}{5 \times 4} = \frac{4}{20}$ and $\frac{15}{4} = \frac{15 \times 5}{4 \times 5} = \frac{75}{20}$.

Step 3: Subtract the Fractions
Subtract the fractions:

$\frac{4}{20} - \frac{75}{20} = \frac{4 - 75}{20} = \frac{-71}{20}$.

The resulting fraction can be converted back to a mixed number:

$\frac{-71}{20} = -3\frac{11}{20}$.

Therefore, the solution to the problem is $-3\frac{11}{20}$.

Note the following rule:

$-(-x)=+x$

Now let's write the exercise in the appropriate form:

$-302+7.6=$

We'll locate -302 on the number line and go right 7.6 steps:

Note that our result will be negative.

Let's solve the exercise carefully by adding a decimal point to the number 302 to avoid confusion during the solution:

$302.0\\-7.6\\294.4$

Note that the final answer is negative, meaning:

$-294.4$

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

• Step 1: Convert mixed numbers to improper fractions.
• Step 2: Find the least common denominator.
• Step 3: Perform the addition and convert back to a mixed number.

Let's work through each step:

Step 1: Convert mixed numbers to improper fractions.

$-13\frac{1}{4}$ = $-\left( 13 + \frac{1}{4} \right) = -\left( \frac{52}{4} + \frac{1}{4} \right) = -\frac{53}{4}$

$-9\frac{2}{7}$ = $-\left( 9 + \frac{2}{7} \right) = -\left( \frac{63}{7} + \frac{2}{7} \right) = -\frac{65}{7}$

Step 2: Determine a common denominator. The least common multiple of 4 and 7 is 28.

Convert the fractions:

$-\frac{53}{4} = -\frac{53 \times 7}{4 \times 7} = -\frac{371}{28}$

$-\frac{65}{7} = -\frac{65 \times 4}{7 \times 4} = -\frac{260}{28}$

Step 3: Add the fractions:

$-\frac{371}{28} + \left(-\frac{260}{28}\right) = -\frac{371 + 260}{28} = -\frac{631}{28}$

Converting $-\frac{631}{28}$ back to a mixed number:

Perform the division: $631 \div 28 \approx 22$ remainder 15.

This gives us a mixed number: $-22\frac{15}{28}$

Therefore, the solution to the problem is $-22\frac{15}{28}$.

First we need to remember the rule:

$-(+x)=-x$

Now let's rewrite the exercise in the appropriate form:

$0.76-13.04=$

Next we will solve the exercise vertically, keeping in mind that the final result will be negative since we are subtracting a smaller number from a larger number:

$13.04\\-0.76\\12.28$

Remember that the answer must be a negative number, so we will add a minus sign to get:

$-\text{12}.28$

To solve the problem $(-17\frac{1}{4}) - (+11\frac{1}{8})$, we need to perform the following steps:

• Step 1: Convert each mixed number to an improper fraction.
  $-17\frac{1}{4}$ converts to an improper fraction as follows:
  $-17\frac{1}{4} = -\left(\frac{17 \times 4 + 1}{4}\right) = -\left(\frac{68 + 1}{4}\right) = -\left(\frac{69}{4}\right).$
  $+11\frac{1}{8}$ converts to an improper fraction as follows:
  $+11\frac{1}{8} = \left(\frac{11 \times 8 + 1}{8}\right) = \left(\frac{88 + 1}{8}\right) = \left(\frac{89}{8}\right).$
• Step 2: Ensure both fractions have a common denominator. Here, the least common denominator (LCD) between 4 and 8 is 8.
  $-\left(\frac{69}{4}\right)$ must be converted to have a denominator of 8:
  $-\left(\frac{69 \times 2}{4 \times 2}\right) = -\left(\frac{138}{8}\right).$
• Step 3: Perform the subtraction:
  Subtraction: $-\left(\frac{138}{8}\right) - \left(\frac{89}{8}\right) = -\left(\frac{138 + 89}{8}\right) = -\left(\frac{227}{8}\right).$
• Step 4: Convert the result back to a mixed number, if necessary.
  Divide 227 by 8: $227 \div 8 = 28$ with a remainder of 3. Hence, the result is:
  $-\left(28 + \frac{3}{8}\right) = -28\frac{3}{8}.$

Therefore, the solution to the problem is $-28\frac{3}{8}$.

Let's solve the problem step-by-step:

• Step 1: Convert mixed numbers to improper fractions.

For $-12\frac{1}{4}$, convert to an improper fraction:

$-12\frac{1}{4} = -\left(12 + \frac{1}{4}\right) = -\left(\frac{48}{4} + \frac{1}{4}\right) = -\frac{49}{4}$

For $-8\frac{2}{7}$, convert to an improper fraction:

$-8\frac{2}{7} = -\left(8 + \frac{2}{7}\right) = -\left(\frac{56}{7} + \frac{2}{7}\right) = -\frac{58}{7}$

• Step 2: Subtract the second improper fraction from the first.

The operation becomes $-\frac{49}{4} - (-\frac{58}{7})$, which simplifies to $-\frac{49}{4} + \frac{58}{7}$.

Find a common denominator for $\frac{49}{4}$ and $\frac{58}{7}$. The least common denominator is 28.

Convert $-\frac{49}{4}$ and $\frac{58}{7}$ to equivalents with a denominator of 28:

$-\frac{49}{4} = -\frac{49 \times 7}{4 \times 7} = -\frac{343}{28}$

$\frac{58}{7} = \frac{58 \times 4}{7 \times 4} = \frac{232}{28}$

So, $-\frac{49}{4} + \frac{58}{7}$ becomes:

$-\frac{343}{28} + \frac{232}{28} = \frac{-343 + 232}{28} = \frac{-111}{28}$

• Step 3: Simplify the improper fraction $\frac{-111}{28}$ to a mixed number form.

Divide $-111$ by 28, which gives $-3$ and a remainder of 27. Thus:

$-3\frac{27}{28}$

Therefore, the solution to the problem is $-3\frac{27}{28}$.