Advanced Arithmetic Operations

More arithmetic rules: subtraction of a sum, subtraction of a difference, division by product, and division by quotient

In this article, we will dive into the world of essential arithmetic rules that are fundamental for tackling a wide variety of mathematical exercises. Mastering these rules will provide you with a solid foundation and allow you to solve problems with greater confidence and precision. From basic operations like addition and subtraction to more advanced concepts like the division of products and quotients, we will explore each of these rules in detail. Are you ready to deepen your mathematical skills?
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$$ 100-(5+55)= $$

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Subtraction of a sum

Sometimes we need to subtract a sum of elements from another element.
Rule:
$a−(b+c)=a−b−c$

This is also true in algebraic expressions.

We can operate according to the rule: apply the subtraction sign to each of the elements included in the parentheses.
Likewise, we can act according to the order of mathematical operations starting with the parentheses - calculate the sum and only then subtract it.

For example, in the exercise:
$21-(7+2)=$

Option 1 - according to the rule:

We will subtract each element in the parentheses separately and it will give us:
$21-7-2=12$

Option 2 - according to the order of operations:

$21-9=12$

Click here for a more detailed explanation about subtracting a sum.

Subtraction of a difference

It is valid when we need to subtract a difference of elements from another element.
Rule:
$a−(b-c)=a-b+c$

This is also valid in algebraic expressions.

We can operate according to the rule: apply the subtraction sign to each of the elements included in the parentheses and always remember that, minus times minus gives plus.
Likewise, we can act according to the order of mathematical operations starting with the parentheses - calculate the difference and only then subtract it.

For example, in the exercise:
$33-(9-3)=$

Option 1 - according to the rule:

We will separately subtract each element in the parentheses and it will give us:
$33-9+3=$

$24+3=27$

Option 2 - according to the order of operations:

$33-6=27$

Click here for a more detailed explanation about subtracting a difference.

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Test your knowledge

Question 1

$$ 10:(10:5)= $$

Question 2

$15:(2\times5)=$ ?

Question 3

$18:(6\times3)=$

Division by product

It is also true when we need to divide a certain element by the product of others.
Rule:
$a:(b\cdot c)=a:b:c$

This is also valid in algebraic expressions.

We can operate according to the rule: apply the division to each of the elements included in the parentheses.
Likewise, we can act according to the order of mathematical operations starting with the parentheses - calculate the multiplication and only then divide by the product.

For example, in the exercise:
$50:(2\cdot5)$

Option 1 - according to the rule:

We will divide separately for each element of the parentheses and it will give us:
$50:2:5=$
First, we will divide $50:2$ and rewrite the exercise:
$25:5=5$

Option 2 - according to the order of operations:

$50:10=5$

Click here for a more detailed explanation about division by product.

Division by quotient

It is valid when we need to divide a certain element by the quotient of others.
Rule:
$a:(b:c)=a:b\cdot c$

This is also valid in algebraic expressions.

We can operate according to the rule: apply the division to the first element inside the parentheses and then apply the multiplication to the second element of the parentheses.
Likewise, we can act according to the order of mathematical operations starting with the parentheses - calculate the quotient and only then divide by it.

For example, in the exercise:
$48:(6:2)=$

Option 1 - according to the rule:

We will apply division to the first element inside the parentheses and then multiply by the second element of the parentheses.

$48:6\cdot2=$
First, we will divide $48:6$ and rewrite the exercise:
$8\cdot 2=16$

Option 2 - according to the order of operations:

$48:(6:2)=$
$48:3=16$

Click here for a more detailed explanation about division by quotient.

Examples and exercises with solutions of arithmetic rules

Exercise #1

$15:(2\times5)=$ ?

Video Solution

Step-by-Step Solution

First we need to apply the following formula:

$a:(b\times c)=a:b:c$

Therefore, we get:

$15:2:5=$

Now, let's rewrite the exercise as a fraction:

$\frac{\frac{15}{2}}{5}=$

Then we'll convert it to a multiplication of two fractions:

$\frac{15}{2}\times\frac{1}{5}=$

Finally, we multiply numerator by numerator and denominator by denominator, leaving us with:

$\frac{15}{10}=1\frac{5}{10}=1\frac{1}{2}$

Answer

$1\frac{1}{2}$

Exercise #2

$10:(10:5)=$

Video Solution

Step-by-Step Solution

To solve the expression $10 : (10 : 5)$ , we will apply the order of operations systematically.

Step 1: Evaluate the inner division $10 : 5$ .
When we compute $10 : 5$ , we are finding how many times 5 fits into 10. This calculation can be expressed as:
$\frac{10}{5} = 2$ .

Step 2: Substitute the result from step 1 into the outer division.
Now, we substitute $10 : (10 : 5)$ with $10 : 2$ . Once again, we apply division:
$\frac{10}{2} = 5$ .

Therefore, the solution to the expression $10 : (10 : 5)$ is $5$ .

Answer

$5$

Exercise #3

$18:(6\times3)=$

Video Solution

Step-by-Step Solution

To solve the expression $18 \div (6 \times 3)$ , we need to follow the order of operations, which specifies that multiplication should be performed before division. Therefore, we proceed as follows:

Step 1: Calculate the operation inside the parentheses: $(6 \times 3)$ .
We multiply $6$ by $3$ to get $18$ .
Step 2: Replace the multiplication expression in the original division: $18 \div 18$ .
Step 3: Perform the division: $18 \div 18 = 1$ .

Thus, the result of the expression $18 \div (6 \times 3)$ is $\mathbf{1}$ .

Answer

Exercise #4

$2-(1+1)=$

Video Solution

Step-by-Step Solution

To solve the expression $2 - (1 + 1)$ , follow these steps:

First, evaluate the expression inside the parentheses: $1 + 1$ .
This gives $2$ .
Now replace the parentheses with this result, transforming the expression to $2 - 2$ .
The result of $2 - 2$ is $0$ .

Therefore, the solution to the expression is $0$ .

Answer

Exercise #5

$19-(5+11)=$

Video Solution

Step-by-Step Solution

To solve the problem $19 - (5 + 11)$ , we will follow these steps:

Step 1: Evaluate the expression inside the parentheses. This means we need to calculate $5 + 11$ .
Step 2: Once the sum inside the parentheses is found, subtract this sum from 19.

Let's work through each step:

Step 1: Calculate $5 + 11$ which equals 16.

Step 2: Substitute 16 in place of $5 + 11$ in the original expression. You have $19 - 16$ .

Now, solve $19 - 16$ , which equals 3.

Therefore, the solution to the problem is $3$ .

Answer

Do you know what the answer is?

Question 1

$$ 19-(5+11)= $$

Question 2

$$ 2-(1+1)= $$

Question 3

$$ 21:(30:10)= $$

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Advanced Arithmetic Operations

More arithmetic rules: subtraction of a sum, subtraction of a difference, division by product, and division by quotient

Test yourself on additional arithmetic rules!

Subtraction of a sum

Option 1 - according to the rule:

Option 2 - according to the order of operations:

Subtraction of a difference

Option 1 - according to the rule:

Option 2 - according to the order of operations:

Division by product

Option 1 - according to the rule:

Option 2 - according to the order of operations:

Division by quotient

Option 1 - according to the rule:

Option 2 - according to the order of operations:

Examples and exercises with solutions of arithmetic rules

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

More Questions

Additional Arithmetic Rules