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Advanced Properties Of Multiplication

Multiplicative Inverse Property

  • Multiplicative inverse of a number is the reciprocal of that number.

For Example: Multiplicative inverse of \(5\) is \(\dfrac{1}{5}\).

  • Multiplicative inverse property states that when we multiply the given whole number with its multiplicative inverse, it gives \(1\) as the answer.
  • Mathematically, we can say that

\(a×\dfrac{1}{a}=1\)

where \(\dfrac{1}{a}\) is multiplicative inverse of '\(a\)' and vice versa.

Example:

1. \(3×\dfrac{1}{3}=1\)

2. \(6×\dfrac{1}{6}=1\)

3. \(7×\dfrac{1}{7}=1\)

Illustration Questions

Which equation shows the multiplicative inverse property for whole numbers?

A \(23×0=0\)

B \(23×\dfrac{1}{23}=1\)

C \(23×1=23\)

D \(23+0=23\)

×

Multiplicative inverse property says that any whole number multiplied by its multiplicative inverse [reciprocal of number] gives \(1\) as the answer.

Mathematically we can write multiplicative inverse property as

\(a×\dfrac{1}{a}=1\)

Among all the options only option (B) represents the product of a number(23) with its multiplicative inverse \(\dfrac{1}{23}\) and their result as 1.

Hence, option (B) is correct.

Which equation shows the multiplicative inverse property for whole numbers?

A

\(23×0=0\)

.

B

\(23×\dfrac{1}{23}=1\)

C

\(23×1=23\)

D

\(23+0=23\)

Option B is Correct

Multiplication Property of One

  • Multiplication property of one states that a number does not change when multiplying it with one.

\(a×1=a\\b×1=b\)

  • We always get that number as the answer.

Example:

\(25×1=25\)

\(100×1=100\)

\(1×1=1\)

Illustration Questions

Which one of the following illustrates multiplication property of one?

A \(29×1=29\)

B \(29×28=28×29\)

C \(28+1=29\)

D \(29+28=28+29\)

×

According to multiplicative property of one, a number does not change when multiplying it with one.

\(a×1=a \,\,\,\,\,;\;\;b×1=b\)

Among the choices, only equation i.e. \(29×1=29\), involves multiplication with one.

Also, it illustrates multiplication property of one.

Hence, option (A) is correct.

Which one of the following illustrates multiplication property of one?

A

\(29×1=29\)

.

B

\(29×28=28×29\)

C

\(28+1=29\)

D

\(29+28=28+29\)

Option A is Correct

The Distributive Property of Multiplication Over Addition

  • The distributive property of multiplication over addition states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the product together.

\(a(b+c)=a\,b+a\,c\)

  • For example: Suppose we want to multiply \(5\) to the sum of \(6\) and \(7\) i.e.

\(5(6+7)\)

In this method first we add the numbers \(6\) and \(7\), then multiply \(5\) to the sum \((13)\) i.e.

\(5(13)=65\)

  • In the second method, we first multiply each number by \(5\) (this is called distributing the \(5\)).

 \(5(6+7)\)

 \(=5(6)+5(7)\)

Now, we add the products

\(5(6)+5(7)\)

\(=30+35\)

\(=65\)

In both the methods, answer is same.

Illustration Questions

Solve: \(39×63+39×37\)

A \(3800\)

B \(6300\)

C \(3900\)

D \(1000\)

×

Given: \(39×63+39×37\) ... (1)

According to distributive property, multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the product together.

\(a(b+c)=a\,b+a\,c\) ... (2)

Rewriting equation (1) in form of equation (2) gives,

\(39×63+39×37=39(63+37)\)

\(=39(100)\)

Now, we multiply sum by \(39\),

\(39(100)\)

\(=3900\)

Hence, option (C) is correct.

Solve: \(39×63+39×37\)

A

\(3800\)

.

B

\(6300\)

C

\(3900\)

D

\(1000\)

Option C is Correct

The Distributive Property of Multiplication Over Subtraction

  • The distributive property of multiplication over subtraction is like the distributive property of multiplication over addition.
  • In this case, either we find out the difference first and then multiply or we first multiply with each number and then subtract.

\(a(b-c) = a (b) - a (c)\)

  •  But in the case, if we change the order in subtraction, then answer also changes.

\(a (b-c) \neq a (c) - a (b)\)

  • For example: Suppose we want to multiply \(10\) by the difference of \(12\) and \(6\).

\(10 (12-6)\)

  • According to this property, we can first subtract the numbers and then multiply the difference with \(10\).

\(10 (6) = 60\)

  • Also, we can first multiply each number by \(10\).

\(10 (12-6)\)

\(=10 (12) - 10 (6)\)

Now, we subtract the products

\(10 (12) - 10 (6)\)

\(=120 - 60 = 60\)

In both the methods, the answer is same. 

Illustration Questions

Rewrite the expression \(5 (6-3)\) using distributive property of multiplication over subtraction.

A \(5 (3) - 5 (6)\)

B \(5 (3)\)

C \(5 (3-6)\)

D \(5 (6) - 5 (3)\)

×

According to distributive property, either we find out the difference first and then multiply or first we multiply with each number and then subtract. But in the case, if we change the order in subtraction, then answer also changes.

\(a (b-c) = a (b) - a (c)\)

In the option (D), \(5\) is evenly distributed and also the order of subtraction is not changed.

So that \(5\) is used to multiply \(6\) and \(3\) separately i.e. \(5 (6) - 5 (3)\)

Hence, option (D) is correct.

Rewrite the expression \(5 (6-3)\) using distributive property of multiplication over subtraction.

A

\(5 (3) - 5 (6)\)

.

B

\(5 (3)\)

C

\(5 (3-6)\)

D

\(5 (6) - 5 (3)\)

Option D is Correct

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