Chapter 4.6, Problem 1P

Solve the multiplication problem

   $\underset{\_}{}$

in three different ways: by using the common method for writing the standard algorithm, by using the partial-products method for writing the standard algorithm, and by writing the numbers in expanded forms and using properties of arithmetic. For each of the three methods, discuss how the steps in that method are related to the steps in the other methods.

To determine

To calculate: The multiplication problem $\underset{\_}{\begin{array}{l}\text{ 9 6}\\ \times \text{ 8}\end{array}}$ by the use of distributive property and expanded form and then relate the equations with partial product method.

Answer to Problem 1P

The solution of the problem $96\times 8$ is $768$ .

Explanation of Solution

Given information:

The expression is $\underset{\_}{\begin{array}{l}\text{ 9 6}\\ \times \text{ 8}\end{array}}$ .

Concept used:

Distributive property is used to simplify the algebraic expression of this kind $a\left(b+c\right)$ . Here, $a,b\text{ and }c$ are real numbers.

The algebraic expression $a\left(b+c\right)$ can be calculated with the help of distributive property as,

  $a\left(b+c\right)=a\cdot b+a\cdot c$

Calculation:

The multiplication of two numbers $\underset{\_}{\begin{array}{l}\text{ 9 6}\\ \times \text{ 8}\end{array}}$ .can be done by three methods,

Method I: Common method

Solve the problem by common method, first multiply the number 8 by 6, write the number 8 and carry the number 4now multiply 8 by 9 and then add the carried number 4 and write it as $76$ in the first line.

  $\begin{array}{l}\underset{\_}{\begin{array}{l}\text{ 9 6}\\ \times \text{ 8}\end{array}}\\ 768\end{array}$

The solution of the multiplication problem $\underset{\_}{\begin{array}{l}\text{ 9 6}\\ \times \text{ 8}\end{array}}$ is $768$ .

Method II: Partial product method

The solution of the problem $\underset{\_}{\begin{array}{l}\text{ 9 6}\\ \times \text{ 8}\end{array}}$ by the use of partial product algorithm can be explained by the following steps.

First multiply the number 8 with 6 and it can be calculated as,

  $8\cdot 6=48$

Second multiply the number 8 with 9 tens and it can be calculated as,

  $8\cdot 90=720$

Now add all the calculations that is done in above steps as,

  $48+72=768$

The solution of the multiplication problem $\underset{\_}{\begin{array}{l}\text{ 9 6}\\ \times \text{ 8}\end{array}}$ is $768$ .

Method III: Arithmetic properties

The number $96$ can be written as $96=90+6$ .

Therefore, the expression $96\cdot 8$ can be written in the expanded form as $\left(90+6\right)\cdot 8$ .

Use the distributive property of multiplication over addition in the expression $\left(90+6\right)\cdot 8$ as,

  $\begin{array}{c}\left(90+6\right)\cdot 8=90\cdot 8+6\cdot 8\\ =720+48\\ =768\end{array}$

The solution of the multiplication problem $\underset{\_}{\begin{array}{l}\text{ 9 6}\\ \times \text{ 8}\end{array}}$ is $768$ .

It is observed from the calculation that is done by the distributive property $90\cdot 8+6\cdot 8$ corresponds to the steps of the partial although it is the different order product algorithm.

Conclusion:

Thus, the solution of the problem $96\times 8$ is $768$ .