Additional Arithmetic Rules: Using multiple rules

To solve the given expression $130 : \frac{10}{2x} - (23x - 204)$, follow these steps:

• Firstly, simplify the fraction $\frac{10}{2x}$ to get $\frac{5}{x}$.
• Next, perform the division $130 \div \frac{5}{x}$. This is equivalent to multiplying $130$ by the reciprocal, resulting in $130 \times \frac{x}{5}$. Simplify this to get $26x$.
• The expression now becomes $26x - (23x - 204)$.
• Distribute the negative sign inside the parentheses to get $26x - 23x + 204$.
• Simplify by combining like terms, which yields $3x + 204$.

Therefore, the solution to the problem is $3x + 204$. This matches option 1 from our choices, confirming it as the correct answer.

To solve the problem $300:(20\cdot8)-(13+200)$, follow these steps:

• Step 1: Evaluate the multiplication $20 \cdot 8$.
  $20 \cdot 8 = 160$
• Step 2: Perform the division $300:160$.
  $300 \div 160 = 1.875$
• Step 3: Calculate the addition inside the parentheses $13 + 200$.
  $13 + 200 = 213$
• Step 4: Subtract the result of the addition from the division.
  $1.875 - 213 = -211.125$
• Step 5: Convert the decimal to a fraction, if necessary.
  Since $-211.125$ is equivalent to $-211\frac{1}{8}$.

Therefore, the solution to the problem is $-211\frac{1}{8}$, which corresponds to choice 3.

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

• Step 1: Calculate the expression inside the parentheses by performing the division $210:15$.
• Step 2: Add the numbers inside the other parentheses, $12 + 42$.
• Step 3: Use the result from Step 1 to perform the division $28:\text{(result of Step 1)}$.
• Step 4: Subtract the result of Step 2 from the result of Step 3.

Now, let's work through each step:
Step 1: Compute $210 : 15 = 14$.
Step 2: Compute $12 + 42 = 54$.
Step 3: Compute the division $28 : 14 = 2$.
Step 4: Finally, compute the subtraction $2 - 54 = -52$.

Therefore, the solution to the problem is $-52$.

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

• Step 1: Simplify each division separately.
• Step 2: Combine the results.

Now, let's work through each step:

Step 1: Simplify $39 : (x \cdot 3)$.

This is equivalent to $\frac{39}{x \cdot 3} = \frac{39}{3x}$.

Step 2: Simplify $\frac{y}{x} : \frac{y}{4}$.

This is equivalent to $\frac{y}{x} \times \frac{4}{y} = \frac{4y}{xy} = \frac{4}{x}$.

Step 3: Add the results from Steps 1 and 2.

We have:

$\frac{39}{3x} + \frac{4}{x}$

Simplifying further, find a common denominator for the fractions, which is $3x$:

$\frac{39}{3x} + \frac{4 \cdot 3}{3x} = \frac{39 + 12}{3x} = \frac{51}{3x} = \frac{17}{x}$.

Therefore, the solution to the problem is $\frac{17}{x}$.

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

• Step 1: Evaluate the division $400 : \frac{20}{13}$.
• Step 2: Subtract 12 from the result obtained in Step 1.
• Step 3: Subtract the entire result from 35.

Now, let's work through each step:
Step 1: Evaluate the division $400 : \frac{20}{13}$. Division by a fraction is the same as multiplying by its reciprocal, so we have:
$400 \times \frac{13}{20} = \frac{400 \times 13}{20} = \frac{5200}{20} = 260$

Step 2: Now subtract 12 from 260:
$260 - 12 = 248$

Step 3: Finally, subtract the result from 35:
$35 - 248 = -213$

Thus, the solution to the problem is $-213$.

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

• Step 1: Solve the division inside the parentheses: $7 : \frac{2}{26}$ is equivalent to $7 \times \frac{26}{2}$.
• Step 2: Calculate $\frac{26}{2} = 13$.
• Step 3: Thus, $7 \times 13 = 91$.
• Step 4: Determine the full product expression: $4 \cdot 13 \cdot 91$.
• Step 5: $4 \cdot 13 = 52$.
• Step 6: Therefore, $52 \cdot 91 = 4732$.
• Step 7: Compute the second product: $13 \cdot 13 \cdot 7 \cdot 4$.
• Step 8: Begin with $13 \cdot 13 = 169$.
• Step 9: Continue with $169 \cdot 7 = 1183$.
• Step 10: Finally, $1183 \cdot 4 = 4732$.
• Step 11: Subtract the two calculated products: $4732 - 4732 = 0$.
• Step 12: Subtract this result from 1, yielding $1 - 0 = 1$.

Therefore, the solution to the problem is $1$.

To solve this problem, we begin by simplifying the expression inside the parentheses: $y \cdot z + 3z : \frac{z}{x}$.

First, evaluate the division: $\frac{3z}{\frac{z}{x}}$. This can be simplified by multiplying by the reciprocal, yielding $3z \cdot \frac{x}{z} = 3x$.

The expression inside the parentheses becomes $y \cdot z + 3x$.

Now substitute this back into the entire expression: $3x - (y \cdot z + 3x)$.

Apply the distributive property to the negative sign, resulting in: $3x - y \cdot z - 3x$.

Combine like terms: $3x - 3x - y \cdot z$ simplifies to $- y \cdot z$.

Thus, the solution to the given expression is $-yz$.

Let's solve the expression $48:\frac{10}{x}-(x+5)$.

Step 1: Change the division to multiplication by the reciprocal.

We have $48:\frac{10}{x} = 48 \times \frac{x}{10}$.

Simplifying this, we obtain:

$48 \times \frac{x}{10} = \frac{48x}{10} = 4.8x$.

Step 2: Incorporate the subtraction operation.

We now have $4.8x - (x + 5)$.

Distribute the negative sign:

This gives us $4.8x - x - 5$.

Step 3: Combine like terms.

Simplifying further, we have:

$4.8x - x = 3.8x$.

Thus, we end with:

$3.8x - 5$.

Therefore, the solution to the problem is $3.8x - 5$.

Upon reviewing the given choices, choice 1 corresponds to our calculated result: $3.8x - 5$.

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

• Step 1: Evaluate the expression inside the parentheses, focusing first on multiplication.
• Step 2: Perform the division inside the parentheses.
• Step 3: Simplify the expression by completing the subtraction inside the parentheses.
• Step 4: Subtract the result from 35.

Now, let's work through each step:

Step 1: Evaluate the multiplication inside the parentheses.
The expression is $3 \cdot 2$, which equals $6$.

Step 2: Perform the division inside the parentheses.
Substitute the result into $39 : 6$, which is $39 \div 6 = 6.5$.

Step 3: Simplify the expression inside the parentheses by performing the subtraction.
The expression becomes $82 - 6.5 = 75.5$.

Step 4: Subtract the result from 35.
Hence, $35 - 75.5 = -40.5$.

Therefore, the solution to the problem is $-40.5$, which corresponds to the first choice given in the possible answers as $-40\frac{1}{2}$.

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

• Step 1: Evaluate the expression inside the first parentheses, $(38-92)$.
• Step 2: Evaluate the expression inside the second parentheses, $(56+33)$.
• Step 3: Substitute these results back into the main expression and simplify.

Now, let's work through each step:

Step 1: Calculate $38 - 92$.
This results in $-54$.

Step 2: Calculate $56 + 33$.
This equals $89$.

Step 3: Substitute these values into the original expression:
$124 - (-54) - 89$.

• First, simplify $124 - (-54)$.
  Subtracting a negative is equivalent to adding its positive, so this becomes:
  $124 + 54 = 178$.
• Next, subtract 89:
  $178 - 89 = 89$.

Therefore, the solution to the problem is $89$.

To solve the problem $78-(39-47)-(95+3:\frac{3}{5})$, follow these steps:

• Step 1: First, evaluate the expression within the first parentheses: $39-47 = -8$. Thus, the expression becomes $78 - (-8) - (95 + 3:\frac{3}{5})$.
• Step 2: Simplifying $78 - (-8)$ gives $78 + 8 = 86$. The expression is now $86 - (95 + 3:\frac{3}{5})$.
• Step 3: Calculate the division within the second parentheses $3:\frac{3}{5}$ by multiplying 3 by the reciprocal of $\frac{3}{5}$, resulting in $3 \times \frac{5}{3} = 5$. Thus, $95 + 5 = 100$.
• Step 4: Substitute back into the problem, yielding $86 - 100$, which equals $-14$.

Thus, the solution to the problem is $-14$.

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

• Step 1: Simplify the nested expressions and handle the negative signs.

• Step 2: Perform the division operation involving the fraction.

• Step 3: Combine the results from the two operations.

Now, let's work through each step:

Step 1: Address the nested subtraction.

First, simplify the innermost expression: $-94 - (-32)$.

Recall that subtracting a negative is equivalent to addition: $-94 - (-32) = -94 + 32 = -62$.

Now substitute this result back into the main expression: $-55 - (-62)$.

Again, subtracting a negative is addition: $-55 - (-62) = -55 + 62 = 7$.

Step 2: Resolve the division by a fraction.

Calculate $12 \div \frac{3}{4}$:

Dividing by a fraction is equivalent to multiplying by its reciprocal: $12 \times \frac{4}{3} = \frac{48}{3} = 16$.

Step 3: Combine results.

Now we add the results from steps 1 and 2: $7 + 16 = 23$.

Therefore, the solution to the problem is $23$.

To solve the given expression, we'll simplify it step-by-step:

• Step 1: Simplify the fraction
  $\frac{50}{-3}$ simplifies to $-\frac{50}{3}$.
• Step 2: Perform the division
  The expression $-450:\frac{50}{-3}$ can be rewritten as $-450 \times -\frac{3}{50}$ since division by a fraction is equivalent to multiplication by its reciprocal.
• Calculate the multiplication
  $-450 \times -\frac{3}{50} = 450 \times \frac{3}{50} = 450 \times 0.06 = 27$.
• Step 3: Simplify the expression within parentheses
  The expression $(-3x) + (-14)$ simplifies to $-3x - 14$.
• Step 4: Combine all parts
  The complete expression $(-450:\frac{50}{-3}) - ((-3x) + (-14))$ simplifies to $27 - (-3x - 14)$.
• Simplify further
  This becomes $27 + 3x + 14 = 41 + 3x$.

Therefore, the solution to the problem is $41 + 3x$.

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

• Step 1: Simplify $\frac{a^2}{2}:\frac{a}{3}$.
• Step 2: Simplify $3a + 12:(4a \cdot 3)$.
• Step 3: Perform the subtraction from Step 1 and Step 2 results.

Now, let's work through each step:

Step 1:
The expression $\frac{a^2}{2}:\frac{a}{3}$ translates to $\frac{a^2}{2} \div \frac{a}{3}$.
Dividing by a fraction is equivalent to multiplying by its reciprocal, thus:
$\frac{a^2}{2} \times \frac{3}{a} = \frac{a^2 \cdot 3}{2 \cdot a} = \frac{3a}{2}$.

Step 2:
Handle $3a + 12:(4a \cdot 3)$.
First, calculate $12:(4a \cdot 3)$, which translates to $\frac{12}{12a} = \frac{1}{a}$.
Thus, $3a + \frac{1}{a}$ remains unchanged as $3a + \frac{1}{a}$.

Step 3:
Subtract the result from Step 2 from the result in Step 1:
$\frac{3a}{2} - (3a + \frac{1}{a})$.
This simplifies to:
$\frac{3a}{2} - 3a - \frac{1}{a}$.
To combine $\frac{3a}{2}$ and $-3a$, find a common denominator:
Convert $-3a$ to $-\frac{6a}{2}$, then:
$\frac{3a}{2} - \frac{6a}{2} = -\frac{3a}{2}$.
Therefore, the expression simplifies to:
$-\frac{3a}{2} - \frac{1}{a}$.

Therefore, the solution to the problem is $-1.5a - \frac{1}{a}$.

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

• Step 1: Simplify $\frac{1}{4} \div \frac{8}{9} = \frac{1}{4} \times \frac{9}{8} = \frac{9}{32}$.
• Step 2: Simplify $\frac{3}{4} \div (4 \cdot 3) = \frac{3}{4} \div 12 = \frac{3}{4} \times \frac{1}{12} = \frac{3}{48} = \frac{1}{16}$.
• Step 3: Add the results from Steps 1 and 2: $\frac{9}{32} + \frac{1}{16}$. Convert $\frac{1}{16}$ to $\frac{2}{32}$ to have common denominators:
  $\frac{9}{32} + \frac{2}{32} = \frac{11}{32}$.
• Step 4: Subtract $\frac{11}{32}$ from $\frac{1}{2}$. Convert $\frac{1}{2}$ to $\frac{16}{32}$:
  $\frac{16}{32} - \frac{11}{32} = \frac{5}{32}$.

Therefore, the solution to the problem is $\frac{5}{32}$.

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

• Step 1: Simplify each term under the parentheses separately.
• Step 2: Calculate the overall subtraction outside the parentheses.

Now, let's work through each step:

Step 1: Simplify each term inside the parentheses

First, consider the term $ab: \frac{2}{c}$. This can be rewritten using the division of a fraction as $ab \times \frac{c}{2} = \frac{abc}{2}$.

Next, consider $ab^2c^2 : (b \cdot c)$. This simplifies to:

$ab^2c^2 \div (bc) = ab^2c^2 \times \frac{1}{bc}$ which yields $ab^2c^2 \times \frac{1}{bc} = ab^2c$.

Step 2: Calculate the entire expression

We now substitute back these simplified terms into the expression:

The next step is to combine the terms inside the parentheses:

Substituting back into the main expression, we have:

We can now rewrite the subtraction:

Let's recombine these terms over a common denominator:

Simplify the terms:

It turns out the simplification $abc - abc \times 2bc$ simplifies directly:

This is consistent with the provided correct answer.

Therefore, the solution to the problem is $-\frac{1}{2}abc$.

The problem asks us to evaluate the expression $27-(140:\frac{35}{3}+360:(8\cdot9))$.

Following the order of operations (PEMDAS/BODMAS), we first focus on operations within the parentheses.

Calculate each division inside the parentheses:

• $140 : \frac{35}{3}$ means $140 \div \frac{35}{3}$, which is equivalent to $140 \times \frac{3}{35}$.

Simplifying $140 \times \frac{3}{35}$:

• Calculate $140 \div 35 = 4$.
• Then, multiply $4 \times 3 = 12$.

Now, calculate the second division:

• $360 : (8 \cdot 9)$ involves calculating $8 \cdot 9 = 72$.
• Then, divide $360 \div 72 = 5$.

So, our expression within the parentheses simplifies to:

• $12 + 5 = 17$.

Now go back to the main expression: $27 - 17$.

Calculate the final subtraction:

• $27 - 17 = 10$.

Thus, the solution to the problem is $10$.

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

• Step 1: Calculate the division inside the innermost parentheses: $560 : \frac{70}{3}$.
• Step 2: Use the reciprocal to perform division: Rewrite as $560 \times \frac{3}{70}$.
• Step 3: Continue simplifying the expression from inside to outside.

Now, let's work through each step:
Step 1: Evaluate $560 : \frac{70}{3}$. This is equivalent to $560 \times \frac{3}{70}$.
Step 2: Perform the multiplication: $(560 \times 3) / 70 = 1680 / 70 = 24$.
Step 3: Simplify $24 - 14$. This results in $10$.
Step 4: Calculate $10 - 9$. This equals $1$.
Step 5: Subtract this from 18: $18 - 1 = 17$.

Therefore, the answer to the problem is $17$.

To solve this problem, we'll follow these steps according to the order of operations:

• Step 1: Simplify the expression $35:(7 \cdot 5)-(2+9:\frac{3}{2})$.
• Step 2: Calculate $7 \cdot 5$ and $9:\frac{3}{2}$.
• Step 3: Simplify the division $35:35$ as well as $9:\frac{3}{2}$.
• Step 4: Evaluate the expression step-by-step.

Now, let's solve the expression:

Step 1: Calculate the multiplication: $7 \cdot 5 = 35$.

Step 2: Calculate the division: $9:\frac{3}{2} = 9 \times \frac{2}{3} = 6$. We use the reciprocal of $\frac{3}{2}$.

Step 3: Substitute back into the original expression: $35:35 - (2 + 6) = 35:35 - 8$.

Step 4: Simplify the division: $35:35 = 1$.

Step 5: Calculate the subtraction: $1 - 8 = -7$.

Therefore, the solution to the problem is $-7$.

To solve this problem, let's simplify the expression $79 - (35 - (-9)) - (10 + 43)$:

First, evaluate the expression inside the first parentheses:

• $35 - (-9)$ is equivalent to $35 + 9$ (since subtracting a negative number is equivalent to adding its positive counterpart).
  Thus, we have $35 + 9 = 44$.

Second, evaluate the expression inside the second parentheses:

• $10 + 43 = 53$.

Now substitute these values back into the original expression:

• $79 - 44 - 53$.

Finally, perform the subtraction sequentially:

• Calculate $79 - 44 = 35$.
• Then, calculate $35 - 53 = -18$.

Therefore, the solution to the expression is $-18$.

Checking against the given choices, the correct choice is:

• Choice 2: $18-$ corresponds to $-18$.