Whole Numbers: Learn It 6

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Whole Numbers: Learn It 6

Multiplying Whole Numbers

Suppose you were asked to count all these pennies shown below.

An image of 3 horizontal rows of pennies, each row containing 8 pennies.

Would you count the pennies individually? Or would you count the number of pennies in each row and add that number [latex]3[/latex] times?

[latex]8+8+8[/latex]

Multiplication is a way to represent repeated addition. So instead of adding [latex]8[/latex] three times, we could write a multiplication expression. [latex]3\times 8[/latex].

We call each number being multiplied a factor and the result the product. We read [latex]3\times 8[/latex] as three times eight, and the result as the product of three and eight. There are several symbols that represent multiplication. These include the symbol [latex]\times[/latex] as well as the dot, [latex]\cdot[/latex], and parentheses [latex]\left(\right)[/latex].

multiplication notation

To describe multiplication, we can use symbols and words.

Operation	Notation	Expression	Read as	Result
Multiplication
	[latex]\times[/latex]	[latex]3\times 8[/latex]	three times eight	the product of [latex]3[/latex] and [latex]8[/latex]
	[latex]\cdot[/latex]	[latex]3\cdot 8[/latex]	three times eight	the product of [latex]3[/latex] and [latex]8[/latex]
	[latex]\left(\right)[/latex]	[latex]3\left(8\right)[/latex]	three times eight	the product of [latex]3[/latex] and [latex]8[/latex]

You may see other words in problems besides “times” that indicate the need for multiplication. Some of the words that indicate multiplication are given in the table below.

Operation	Word Phrase	Example	Expression
Multiplication
	times	[latex]3[/latex] times [latex]8[/latex]	[latex]3\times 8, 3\cdot 8, (3)(8)[/latex], [latex](3)8, \text{ or } 3(8)[/latex]
	product	the product of [latex]3[/latex] and [latex]8[/latex]	[latex]3\times 8, 3\cdot 8, (3)(8)[/latex], [latex](3)8, \text{ or } 3(8)[/latex]
	twice	twice [latex]4[/latex]	[latex]2\cdot 4[/latex]

Translate each of the following from math notation to words:

[latex]7\times 6[/latex]
[latex]12\cdot 14[/latex]
[latex]6\left(13\right)[/latex]

In order to multiply without using models, you need to know all the one-digit multiplication facts. Make sure you know them fluently before proceeding in this section.

The table below shows the multiplication facts. Each box shows the product of the number down the left column and the number across the top row. It is important that you memorize any number facts you do not already know so you will be ready to multiply larger numbers. Start with memorizing the multiplication facts through [latex]9 \times 9[/latex]. Knowing the times tables up to [latex]12 \times 12[/latex] will allow your brain to focus on problem-solving in future math questions so you’re not stuck on the arithmetic.

[latex]×[/latex]	[latex]0[/latex]	[latex]1[/latex]	[latex]2[/latex]	[latex]3[/latex]	[latex]4[/latex]	[latex]5[/latex]	[latex]6[/latex]	[latex]7[/latex]	[latex]8[/latex]	[latex]9[/latex]	[latex]10[/latex]	[latex]11[/latex]	[latex]12[/latex]
[latex]0[/latex]	[latex]0[/latex]	[latex]0[/latex]	[latex]0[/latex]	[latex]0[/latex]	[latex]0[/latex]	[latex]0[/latex]	[latex]0[/latex]	[latex]0[/latex]	[latex]0[/latex]	[latex]0[/latex]	[latex]0[/latex]	[latex]0[/latex]	[latex]0[/latex]
[latex]1[/latex]	[latex]0[/latex]	[latex]1[/latex]	[latex]2[/latex]	[latex]3[/latex]	[latex]4[/latex]	[latex]5[/latex]	[latex]6[/latex]	[latex]7[/latex]	[latex]8[/latex]	[latex]9[/latex]	[latex]10[/latex]	[latex]11[/latex]	[latex]12[/latex]
[latex]2[/latex]	[latex]0[/latex]	[latex]2[/latex]	[latex]4[/latex]	[latex]6[/latex]	[latex]8[/latex]	[latex]10[/latex]	[latex]12[/latex]	[latex]14[/latex]	[latex]16[/latex]	[latex]18[/latex]	[latex]20[/latex]	[latex]22[/latex]	[latex]24[/latex]
[latex]3[/latex]	[latex]0[/latex]	[latex]3[/latex]	[latex]6[/latex]	[latex]9[/latex]	[latex]12[/latex]	[latex]15[/latex]	[latex]18[/latex]	[latex]21[/latex]	[latex]24[/latex]	[latex]27[/latex]	[latex]30[/latex]	[latex]33[/latex]	[latex]36[/latex]
[latex]4[/latex]	[latex]0[/latex]	[latex]4[/latex]	[latex]8[/latex]	[latex]12[/latex]	[latex]16[/latex]	[latex]20[/latex]	[latex]24[/latex]	[latex]28[/latex]	[latex]32[/latex]	[latex]36[/latex]	[latex]40[/latex]	[latex]44[/latex]	[latex]48[/latex]
[latex]5[/latex]	[latex]0[/latex]	[latex]5[/latex]	[latex]10[/latex]	[latex]15[/latex]	[latex]20[/latex]	[latex]25[/latex]	[latex]30[/latex]	[latex]35[/latex]	[latex]40[/latex]	[latex]45[/latex]	[latex]50[/latex]	[latex]55[/latex]	[latex]60[/latex]
[latex]6[/latex]	[latex]0[/latex]	[latex]6[/latex]	[latex]12[/latex]	[latex]18[/latex]	[latex]24[/latex]	[latex]30[/latex]	[latex]36[/latex]	[latex]42[/latex]	[latex]48[/latex]	[latex]54[/latex]	[latex]60[/latex]	[latex]66[/latex]	[latex]72[/latex]
[latex]7[/latex]	[latex]0[/latex]	[latex]7[/latex]	[latex]14[/latex]	[latex]21[/latex]	[latex]28[/latex]	[latex]35[/latex]	[latex]42[/latex]	[latex]49[/latex]	[latex]56[/latex]	[latex]63[/latex]	[latex]70[/latex]	[latex]77[/latex]	[latex]84[/latex]
[latex]8[/latex]	[latex]0[/latex]	[latex]8[/latex]	[latex]16[/latex]	[latex]24[/latex]	[latex]32[/latex]	[latex]40[/latex]	[latex]48[/latex]	[latex]56[/latex]	[latex]64[/latex]	[latex]72[/latex]	[latex]80[/latex]	[latex]88[/latex]	[latex]96[/latex]
[latex]9[/latex]	[latex]0[/latex]	[latex]9[/latex]	[latex]18[/latex]	[latex]27[/latex]	[latex]36[/latex]	[latex]45[/latex]	[latex]54[/latex]	[latex]63[/latex]	[latex]72[/latex]	[latex]81[/latex]	[latex]90[/latex]	[latex]99[/latex]	[latex]108[/latex]
[latex]10[/latex]	[latex]0[/latex]	[latex]10[/latex]	[latex]20[/latex]	[latex]30[/latex]	[latex]40[/latex]	[latex]50[/latex]	[latex]60[/latex]	[latex]70[/latex]	[latex]80[/latex]	[latex]90[/latex]	[latex]100[/latex]	[latex]110[/latex]	[latex]120[/latex]
[latex]11[/latex]	[latex]0[/latex]	[latex]11[/latex]	[latex]22[/latex]	[latex]33[/latex]	[latex]44[/latex]	[latex]55[/latex]	[latex]66[/latex]	[latex]77[/latex]	[latex]88[/latex]	[latex]99[/latex]	[latex]110[/latex]	[latex]121[/latex]	[latex]132[/latex]
[latex]12[/latex]	[latex]0[/latex]	[latex]12[/latex]	[latex]24[/latex]	[latex]36[/latex]	[latex]48[/latex]	[latex]60[/latex]	[latex]72[/latex]	[latex]84[/latex]	[latex]96[/latex]	[latex]108[/latex]	[latex]120[/latex]	[latex]132[/latex]	[latex]144[/latex]

When multiplying whole numbers we have to keep a few properties in mind.

properties of multiplication

Multiplication Property of Zero

The product of any number and [latex]0[/latex] is [latex]0[/latex].

[latex]\begin{array}{}\\ a\cdot 0=0\hfill \\ 0\cdot a=0\end{array}[/latex]

Identity Property of Multiplication

The product of any number and [latex]1[/latex] is the number.

[latex]\begin{array}{c}1\cdot a=a\\ a\cdot 1=a\end{array}[/latex]

Commutative Property of Multiplication

Changing the order of the factors does not change their product.

[latex]a\cdot b=b\cdot a[/latex]

It is important to note that the identity property of addition and the identity property of multiplication are different. Although their names are similar, their properties are different.

To multiply numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition and subtraction.

[latex]\begin{array}{c}\hfill 27\\ \hfill \underset{\text{___}}{\times 3}\end{array}[/latex]

We start by multiplying [latex]3[/latex] by [latex]7[/latex].

[latex]3\times 7=21[/latex]

We write the [latex]1[/latex] in the ones place of the product. We carry the [latex]2[/latex] tens by writing [latex]2[/latex] above the tens place.

Then we multiply the [latex]3[/latex] by the [latex]2[/latex], and add the [latex]2[/latex] above the tens place to the product. So [latex]3\times 2=6[/latex], and [latex]6+2=8[/latex]. Write the [latex]8[/latex] in the tens place of the product.

The product is [latex]81[/latex].

When we multiply two numbers with a different number of digits, it’s usually easier to write the smaller number on the bottom. You could write it the other way, too, but this way is easier to work with.

Multiply the following:

[latex]15\cdot 4[/latex]
[latex]286\cdot 5[/latex]

Write the numbers so the digits [latex]5[/latex] and [latex]4[/latex] line up vertically.	[latex]\begin{array}{c}\hfill 15\\ \hfill \underset{\text{_____}}{\times 4}\end{array}[/latex]
Multiply [latex]4[/latex] by the digit in the ones place of [latex]15[/latex]. [latex]4\cdot 5=20[/latex].
Write [latex]0[/latex] in the ones place of the product and carry the [latex]2[/latex] tens.	[latex]\begin{array}{c}\hfill \stackrel{2}{1}5\\ \hfill \underset{\text{_____}}{\times 4}\\ \hfill 0\end{array}[/latex]
Multiply [latex]4[/latex] by the digit in the tens place of [latex]15[/latex]. [latex]4\cdot 1=4[/latex] .Add the [latex]2[/latex] tens we carried. [latex]4+2=6[/latex] .
Write the [latex]6[/latex] in the tens place of the product.	[latex]\begin{array}{c}\hfill \stackrel{2}{1}5\\ \hfill \underset{\text{_____}}{\times 4}\\ \hfill 60\end{array}[/latex]

Write the numbers so the digits [latex]5[/latex] and [latex]6[/latex] line up vertically.	[latex]\begin{array}{c}\hfill 286\\ \hfill \underset{\text{_____}}{\times 5}\end{array}[/latex]
Multiply [latex]5[/latex] by the digit in the ones place of [latex]286[/latex].[latex]5\cdot 6=30[/latex]
Write the [latex]0[/latex] in the ones place of the product and carry the [latex]3[/latex] to the tens place.Multiply [latex]5[/latex] by the digit in the tens place of [latex]286[/latex].[latex]5\cdot 8=40[/latex]	[latex]\begin{array}{}\\ \hfill 2\stackrel{3}{8}6\\ \hfill \underset{\text{_____}}{\times 5}\\ \hfill 0\end{array}[/latex]
Add the [latex]3[/latex] tens we carried to get [latex]40+3=43[/latex] .Write the [latex]3[/latex] in the tens place of the product and carry the [latex]4[/latex] to the hundreds place.	[latex]\begin{array}{c}\hfill \stackrel{4}{2}\stackrel{3}{8}6\\ \hfill \underset{\text{_____}}{\times 5}\\ \hfill 30\end{array}[/latex]
Multiply [latex]5[/latex] by the digit in the hundreds place of [latex]286[/latex]. [latex]5\cdot 2=10[/latex].Add the [latex]4[/latex] hundreds we carried to get [latex]10+4=14[/latex]. Write the [latex]4[/latex] in the hundreds place of the product and the [latex]1[/latex] to the thousands place.	[latex]\begin{array}{c}\hfill \stackrel{4}{2}\stackrel{3}{8}6\\ \hfill \underset{\text{_____}}{\times 5}\\ \hfill 1,430\end{array}[/latex]

When we multiply by a number with two or more digits, we multiply by each of the digits separately, working from right to left. Each separate product of the digits is called a partial product. When we write partial products, we must make sure to line up the place values.

While multiplying two whole numbers to find a product can initially appear daunting, it becomes easier once you employ the correct problem-solving tactics! Whenever you’re tasked with multiplying two numbers, consider these key steps.

Write the numbers so each place value lines up vertically.
Multiply the digits in each place value.
- Work from right to left, starting with the ones place in the bottom number.
  - Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on.
  - If a product in a place value is more than [latex]9[/latex], carry to the next place value.
  - Write the partial products, lining up the digits in the place values with the numbers above.
- Repeat for the tens place in the bottom number, the hundreds place, and so on.
- Insert a zero as a placeholder with each additional partial product.
Add the partial products.

Multiply:

[latex]47\cdot 10[/latex]
[latex]47\cdot 100[/latex]

Multiply the following:

[latex]62\left(87\right)[/latex]
[latex]\left(354\right)\left(438\right)[/latex]

When there are three or more factors, we multiply the first two and then multiply their product by the next factor.

For example:

to multiply	[latex]8\cdot 3\cdot 2[/latex]
first multiply [latex]8\cdot 3[/latex]	[latex]24\cdot 2[/latex]
then multiply [latex]24\cdot 2[/latex]	[latex]48[/latex]

Multiply Whole Numbers in Applications

We will use the same strategy we used previously to solve applications of multiplication. First, we need to determine what we are looking for. Then we write a phrase that gives the information to find it. We then translate the phrase into math notation and simplify to get the answer. Finally, we write a sentence to answer the question.

Van is planning to build a patio. He will have [latex]8[/latex] rows of tiles, with [latex]14[/latex] tiles in each row. How many tiles does he need for the patio?

When Rena cooks rice, she uses twice as much water as rice. How much water does she need to cook [latex]4[/latex] cups of rice?

Write the numbers so each place lines up vertically.
Start by multiplying 7 by 62. Multiply 7 by the digit in the ones place of 62.[latex]7\cdot 2=14[/latex]. Write the [latex]4[/latex] in the ones place of the product and carry the [latex]1[/latex]to the tens place.
Multiply [latex]7[/latex] by the digit in the tens place of [latex]62[/latex]. [latex]7\cdot 6=42[/latex]. Add the 1 ten we carried.[latex]42+1=43[/latex]latex]. Write the [latex]3[/latex] in the tens place of the product and the [latex]4[/latex] in the hundreds place.
The first partial product is [latex]434[/latex].
Now, write a [latex]0[/latex] under the [latex]4[/latex] in the ones place of the next partial product as a placeholder since we now multiply the digit in the tens place of [latex]87[/latex] by [latex]62[/latex].Multiply [latex]8[/latex] by the digit in the ones place of [latex]62[/latex] [latex]8\cdot 2=16[/latex]. Write the [latex]6[/latex] in the next place of the product, which is the tens place. Carry the [latex]1[/latex] to the tens place.
Multiply [latex]8[/latex] by [latex]6[/latex], the digit in the tens place of [latex]62[/latex], then add the [latex]1[/latex] ten we carried to get [latex]49[/latex].Write the [latex]9[/latex] in the hundreds place of the product and the [latex]4[/latex] in the thousands place.
The second partial product is [latex]4960[/latex]. Add the partial products.
The product is [latex]5,394[/latex].

Write a phrase.	the product of [latex]8[/latex] and [latex]14[/latex]
Translate to math notation.	[latex]8\cdot 14[/latex]
Multiply to simplify.	[latex]\begin{array}{c}\\ \stackrel{3}{1}4\hfill \\ \underset{\text{___}}{\times 8}\hfill \\ 112\hfill \end{array}[/latex]
Write a sentence to answer the question.	Van needs [latex]112[/latex] tiles for his patio.

Write as a phrase.	twice as much as 4 cups
Translate to math notation.	[latex]2\cdot 4[/latex]
Multiply to simplify.	[latex]8[/latex]
Write a sentence to answer the question.	Rena needs [latex]8[/latex] cups of water for cups of rice.

[latex]47\cdot 10[/latex]	[latex]\begin{array}{c}\hfill 47\\ \hfill \underset{\text{___}}{\times 10}\\ \hfill 00\\ \hfill \underset{\text{___}}{470}\\ \hfill 470\end{array}[/latex]
[latex]47\cdot 100[/latex]	[latex]\begin{array}{c}\hfill 47\\ \hfill \underset{\text{_____}}{\times 100}\\ \hfill 00\\ \hfill \underset{\text{_____}}{\begin{array}{c}\hfill 000\\ \hfill 4700\end{array}}\\ \hfill 4,700\end{array}[/latex]