Multiplicative Inverse

Two numbers are multiplicative inverses when their product results in $1$ .

For example:

${\Large {1 \over 2}}$ and $2$ are multiplicative inverses because ${\Large 2 \cdot {1 \over 2}=1}$

Formulation of the Rule for Multiplicative Inverse Numbers:

Whenever a is different from $0$ , it follows that ${\Large a\cdot{1 \over a} = 1}$

Multiplication and Division of Multiplicative Inverses

Division is equivalent to multiplication by its multiplicative inverse,

That is: ${\Large {{2 \over {1 \over 3}} = 2 \cdot 3 = 6}}$

Because $3$ is the multiplicative inverse of ${\Large {1 \over 3}}$

Generally: $\frac{a}{\frac{1}{b}}=a⋅b$

Detailed explanation

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Test yourself on special cases (0 and 1, inverse, fraction line)!

$(3\times5-15\times1)+3-2=$

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Two numbers are multiplicative inverses when their multiplication results in $1$ .

For example:

${\Large {1 \over 2}}$ and $2$ are multiplicative inverses because ${\Large 2 \cdot {1 \over 2}=1}$

More examples:

The multiplicative inverse of $5$ is ${\Large {1 \over 5}}$
${\Large 5 \cdot {1 \over 5}=1}$

The multiplicative inverse of $3$ is ${\Large {1 \over 3}}$
${\Large 3 \cdot {1 \over 3}=1}$

The multiplicative inverse of ${\Large {5 \over 7}}$ is ${\Large {7 \over 5}}$
${\Large {7 \over 5} \cdot {5 \over 7}=1}$

The multiplicative inverse of ${\Large {9 \over 23}}$ is ${\Large {23 \over 9}}$
${\Large {23 \over 9} \cdot {9 \over 23}=1}$

The multiplicative inverse of $0.5$ is $2$

${\Large 2 \cdot 0.5=1}$

The multiplicative inverse of $0.25$ is $4$

${\Large 4 \cdot 0.25=1}$

Formulation of the Rule for Multiplicative Inverse Numbers:

Whenever a is different from $0$ , it happens that ${\Large a\cdot{1 \over a} = 1}$

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Test your knowledge

Question 1

$(5\times4-10\times2)\times(3-5)=$

Question 2

$(5+4-3)^2:(5\times2-10\times1)=$

Question 3

$8\times(5\times1)=$

Multiplication and Division of Multiplicative Inverses

Division is equivalent to multiplication by the multiplicative inverse,

that is: ${\Large {{2 \over {1 \over 3}} = 2 \cdot 3 = 6}}$

This is because $3$ is the multiplicative inverse of ${\Large {1 \over 3}}$

In general: ${\Large a /{1 \over b} = a \cdot b}$

Exercises on Multiplicative Inverses

Solve the following exercises

${\Large {5+{{4 \over 7} \over 2} =}}$
${\Large {{6 \over 0.75} - 2 \cdot 3 =}}$
${\Large {{3{1 \over 2}-{{1 \over 3} \over {1 \over 6}}}=}}$
${\Large {{{{10 \over 7} \over 2 } + {2 \over {7 \over 8} }}=}}$
${\Large {{{3 \over 5} \over {9 \over 10}} + {{7 \over 9} \over {1 \over 3}}=}}$

Examples and Exercises with Solutions for Multiplicative Inverse

Exercise #1

$8\times(5\times1)=$

Video Solution

Step-by-Step Solution

According to the order of operations, we first solve the expression in parentheses:

$5\times1=5$

Now we multiply:

$8\times5=40$

Answer

Exercise #2

$7\times1+\frac{1}{2}=$

Video Solution

Step-by-Step Solution

According to the order of operations rules, we first insert the multiplication exercise into parentheses:

$(7\times1)+\frac{1}{2}=$

Let's solve the exercise inside the parentheses:

$7\times1=7$

And now we get the exercise:

$7+\frac{1}{2}=7\frac{1}{2}$

Answer

$7\frac{1}{2}$

Exercise #3

$\frac{6}{3}\times1=$

Video Solution

Step-by-Step Solution

According to the order of operations rules, we will solve the exercise from left to right, since there are only multiplication and division operations:

$\frac{6}{3}=2$

$2\times1=2$

Answer

$2$

Exercise #4

$(3\times5-15\times1)+3-2=$

Video Solution

Step-by-Step Solution

This simple rule is the order of operations which states that exponentiation precedes multiplication and division, which precede addition and subtraction, and that operations enclosed in parentheses precede all others,

Following the simple rule, multiplication comes before division and subtraction, therefore we calculate the values of the multiplications and then proceed with the operations of division and subtraction

$3\cdot5-15\cdot1+3-2= \\ 15-15+3-2= \\ 1$ Therefore, the correct answer is answer B.

Answer

$1$

Exercise #5

$(5\times4-10\times2)\times(3-5)=$

Video Solution

Step-by-Step Solution

This simple rule is the order of operations which states that multiplication precedes addition and subtraction, and division precedes all of them,

In the given example, a multiplication occurs between two sets of parentheses, thus we simplify the expressions within each pair of parentheses separately,

We start with simplifying the expression within the parentheses on the left, this is done in accordance with the order of operations mentioned above, meaning that multiplication comes before subtraction, we perform the multiplications in this expression first and then proceed with the subtraction operations within it, in reverse we simplify the expression within the parentheses on the right and perform the subtraction operation within them:

What remains for us is to perform the last multiplication that was deferred, it is the multiplication that occurred between the expressions within the parentheses in the original expression, we perform it while remembering that multiplying any number by 0 will result in 0:

Therefore, the correct answer is answer d.

Answer

$0$

Do you know what the answer is?

Question 1

$7\times1+\frac{1}{2}=$

Question 2

$\frac{6}{3}\times1=$

Question 3

$12+3\times0=$

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