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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$$ 100-(30-21)= $$

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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

$$ 100-(5+55)= $$

Question 2

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

Question 3

$12:(2\times2)=$

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

$100-(30-21)=$

Video Solution

Step-by-Step Solution

According to the order of operations, we first solve the exercise within parentheses:

$30-21=9$

Now we obtain:

$100-9=91$

Answer

$91$

Exercise #2

$12:(2\times2)=$

Video Solution

Step-by-Step Solution

According to the order of operations, we first solve the exercise within parentheses:

$2\times2=4$

Now we divide:

$12:4=3$

Answer

$3$

Exercise #3

$13-(7+4)=$

Video Solution

Step-by-Step Solution

According to the order of operations, we first solve the exercise within parentheses:

$7+4=11$

Now we subtract:

$13-11=2$

Answer

$2$

Exercise #4

$15:(2\times5)=$ ?

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 #5

$21:(30:10)=$

Video Solution

Step-by-Step Solution

We will use the formula:

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

Therefore, we will get:

$21:30\times10=$

Let's write the division exercise as a fraction:

$\frac{21}{30}=\frac{7}{10}$

Now let's multiply by 10:

$\frac{7}{10}\times\frac{10}{1}=$

We'll reduce the 10 and get:

$\frac{7}{1}=7$

Answer

$7$

Do you know what the answer is?

Question 1

$$ 13-(7+4)= $$

Question 2

$15:(2\times5)=$ ?

Question 3

$18:(6\times3)=$

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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