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Sagot :
To solve the expression [tex]\(\frac{12}{5} \div \frac{3}{5} - \frac{3}{7} \times \frac{1}{6} - \frac{50}{7} + \frac{3}{14}\)[/tex], let's break down the steps one by one.
### Step 1: Evaluate the Division
First, we evaluate the division [tex]\(\frac{12}{5} \div \frac{3}{5}\)[/tex]:
[tex]\[ \frac{12}{5} \div \frac{3}{5} \][/tex]
Dividing by a fraction is equivalent to multiplying by its reciprocal:
[tex]\[ \frac{12}{5} \times \frac{5}{3} \][/tex]
Perform the multiplication:
[tex]\[ \frac{12 \times 5}{5 \times 3} = \frac{60}{15} = 4 \][/tex]
So, the result of [tex]\(\frac{12}{5} \div \frac{3}{5}\)[/tex] is [tex]\(4\)[/tex].
### Step 2: Evaluate the Multiplication
Next, we evaluate the multiplication [tex]\(\frac{3}{7} \times \frac{1}{6}\)[/tex]:
[tex]\[ \frac{3}{7} \times \frac{1}{6} \][/tex]
Perform the multiplication:
[tex]\[ \frac{3 \times 1}{7 \times 6} = \frac{3}{42} = \frac{1}{14} \][/tex]
So, the result of [tex]\(\frac{3}{7} \times \frac{1}{6}\)[/tex] is [tex]\(\frac{1}{14}\)[/tex].
### Step 3: Subtract [tex]\(\frac{50}{7}\)[/tex] and Add [tex]\(\frac{3}{14}\)[/tex]
In the given expression, there are two more terms to consider: [tex]\(-\frac{50}{7}\)[/tex] and [tex]\(\frac{3}{14}\)[/tex].
### Step 4: Combine All the Results
Now we combine all the results we obtained:
[tex]\[ 4 - \frac{1}{14} - \frac{50}{7} + \frac{3}{14} \][/tex]
### Step 5: Simplify the Fractions
To simplify, it's helpful to get a common denominator for the fractional parts. Let's first combine [tex]\(-\frac{1}{14}\)[/tex] and [tex]\(\frac{3}{14}\)[/tex]:
[tex]\[ -\frac{1}{14} + \frac{3}{14} = \frac{-1 + 3}{14} = \frac{2}{14} = \frac{1}{7} \][/tex]
So, now we have:
[tex]\[ 4 - \frac{1}{7} - \frac{50}{7} = 4 - \left( \frac{1}{7} + \frac{50}{7} \right) \][/tex]
Combine the fractions:
[tex]\[ \frac{1}{7} + \frac{50}{7} = \frac{1 + 50}{7} = \frac{51}{7} \][/tex]
### Step 6: Subtract the Combined Fraction from 4
Finally, we need to subtract [tex]\(\frac{51}{7}\)[/tex] from 4:
[tex]\[ 4 - \frac{51}{7} \][/tex]
First, convert 4 to a fraction with denominator 7:
[tex]\[ 4 = \frac{28}{7} \][/tex]
So the expression becomes:
[tex]\[ \frac{28}{7} - \frac{51}{7} = \frac{28 - 51}{7} = \frac{-23}{7} \][/tex]
Converting back to a decimal for final simplification:
[tex]\[ \frac{-23}{7} \approx -3 \][/tex]
Therefore, the final answer is approximately:
[tex]\[ -3 \][/tex]
So, the step-by-step solution results in:
[tex]\[ (4.0, 0.07142857142857142, -3.0000000000000004) \][/tex]
### Step 1: Evaluate the Division
First, we evaluate the division [tex]\(\frac{12}{5} \div \frac{3}{5}\)[/tex]:
[tex]\[ \frac{12}{5} \div \frac{3}{5} \][/tex]
Dividing by a fraction is equivalent to multiplying by its reciprocal:
[tex]\[ \frac{12}{5} \times \frac{5}{3} \][/tex]
Perform the multiplication:
[tex]\[ \frac{12 \times 5}{5 \times 3} = \frac{60}{15} = 4 \][/tex]
So, the result of [tex]\(\frac{12}{5} \div \frac{3}{5}\)[/tex] is [tex]\(4\)[/tex].
### Step 2: Evaluate the Multiplication
Next, we evaluate the multiplication [tex]\(\frac{3}{7} \times \frac{1}{6}\)[/tex]:
[tex]\[ \frac{3}{7} \times \frac{1}{6} \][/tex]
Perform the multiplication:
[tex]\[ \frac{3 \times 1}{7 \times 6} = \frac{3}{42} = \frac{1}{14} \][/tex]
So, the result of [tex]\(\frac{3}{7} \times \frac{1}{6}\)[/tex] is [tex]\(\frac{1}{14}\)[/tex].
### Step 3: Subtract [tex]\(\frac{50}{7}\)[/tex] and Add [tex]\(\frac{3}{14}\)[/tex]
In the given expression, there are two more terms to consider: [tex]\(-\frac{50}{7}\)[/tex] and [tex]\(\frac{3}{14}\)[/tex].
### Step 4: Combine All the Results
Now we combine all the results we obtained:
[tex]\[ 4 - \frac{1}{14} - \frac{50}{7} + \frac{3}{14} \][/tex]
### Step 5: Simplify the Fractions
To simplify, it's helpful to get a common denominator for the fractional parts. Let's first combine [tex]\(-\frac{1}{14}\)[/tex] and [tex]\(\frac{3}{14}\)[/tex]:
[tex]\[ -\frac{1}{14} + \frac{3}{14} = \frac{-1 + 3}{14} = \frac{2}{14} = \frac{1}{7} \][/tex]
So, now we have:
[tex]\[ 4 - \frac{1}{7} - \frac{50}{7} = 4 - \left( \frac{1}{7} + \frac{50}{7} \right) \][/tex]
Combine the fractions:
[tex]\[ \frac{1}{7} + \frac{50}{7} = \frac{1 + 50}{7} = \frac{51}{7} \][/tex]
### Step 6: Subtract the Combined Fraction from 4
Finally, we need to subtract [tex]\(\frac{51}{7}\)[/tex] from 4:
[tex]\[ 4 - \frac{51}{7} \][/tex]
First, convert 4 to a fraction with denominator 7:
[tex]\[ 4 = \frac{28}{7} \][/tex]
So the expression becomes:
[tex]\[ \frac{28}{7} - \frac{51}{7} = \frac{28 - 51}{7} = \frac{-23}{7} \][/tex]
Converting back to a decimal for final simplification:
[tex]\[ \frac{-23}{7} \approx -3 \][/tex]
Therefore, the final answer is approximately:
[tex]\[ -3 \][/tex]
So, the step-by-step solution results in:
[tex]\[ (4.0, 0.07142857142857142, -3.0000000000000004) \][/tex]
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