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Multiply Whole Numbers

Use multiplication notation

Use Multiplication Notation

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

Would you count the pennies individually? Or would you count the number of pennies in each row and add that number \(3\) times.

\[8+8+8\]

Multiplication is a way to represent repeated addition. So instead of adding \(8\) three times, we could write a multiplication expression.

\[3\ \times \ 8\]

We call each number being multiplied a factor and the result the product. We read \(3\ \times \ 8\) 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 \(\times\) as well as the dot, \(\cdot\), and parentheses \((\ \text{).}\)

Example

Try it.

Translate from math notation to words:

ⓐ \(7\ \times \ 6\)
ⓑ \(12\cdot 14\)
ⓒ \(6(13)\)

Solution

ⓐ We read this as seven times six and the result is the product of seven and six.
ⓑ We read this as twelve times fourteen and the result is the product of twelve and fourteen.
ⓒ We read this as six times thirteen and the result is the product of six and thirteen.

Model Multiplication of Whole Numbers

There are many ways to model multiplication. Unlike in the previous sections where we used \(\text{base-10}\) blocks, here we will use counters to help us understand the meaning of multiplication. A counter is any object that can be used for counting. We will use round blue counters.

Example

Try it.

Model: \(3\ \times \ 8.\)

Solution

To model the product \(3\ \times \ 8,\) we’ll start with a row of \(8\) counters.

The other factor is \(3,\) so we’ll make \(3\) rows of \(8\) counters.

Now we can count the result. There are \(24\) counters in all.

\(3\ \times \ 8=24\)

If you look at the counters sideways, you’ll see that we could have also made \(8\) rows of \(3\) counters. The product would have been the same. We’ll get back to this idea later.

Multiply Whole Numbers

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.

shows the multiplication facts. Each box shows the product of the number down the left column and the number across the top row. If you are unsure about a product, model it. It is important that you memorize any number facts you do not already know so you will be ready to multiply larger numbers.

×	1	2	3	4	5	6	7	8	9
0	0	0	0	0	0	0	0	0	0
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	10	12	14	16	18
3	3	6	9	12	15	18	21	24	27
4	4	8	12	16	20	24	28	32	36
5	5	10	15	20	25	30	35	40	45
6	6	12	18	24	30	36	42	48	54
7	7	14	21	28	35	42	49	56	63
8	8	16	24	32	40	48	56	64	72
9	9	18	27	36	45	54	63	72	81

What happens when you multiply a number by zero? You can see that the product of any number and zero is zero. This is called the Multiplication Property of Zero.

Example

Try it.

Multiply:

ⓐ \(0\cdot 11\)
ⓑ \((42)0\)

Solution

ⓐ	\(0\cdot 11\)
The product of any number and zero is zero.	\(0\)
ⓑ	\((42)0\)
Multiplying by zero results in zero.	\(0\)

What happens when you multiply a number by one? Multiplying a number by one does not change its value. We call this fact the Identity Property of Multiplication, and \(1\) is called the multiplicative identity.

Example

Try it.

Multiply:

ⓐ \((11)1\)
ⓑ \(1\cdot 42\)

Solution

ⓐ	\((11)1\)
The product of any number and one is the number.	\(11\)
ⓑ	\(1\cdot 42\)
Multiplying by one does not change the value.	\(42\)

Earlier in this chapter, we learned that the Commutative Property of Addition states that changing the order of addition does not change the sum. We saw that \(8+9=17\) is the same as \(9+8=17.\)

Is this also true for multiplication? Let’s look at a few pairs of factors.

\[4\cdot 7=28\ 7\cdot 4=28\]\[9\cdot 7=63\ 7\cdot 9=63\]\[8\cdot 9=72\ 9\cdot 8=72\]

When the order of the factors is reversed, the product does not change. This is called the Commutative Property of Multiplication.

Example

Try it.

Multiply:

ⓐ \(8\cdot 7\)
ⓑ \(7\cdot 8\)

Solution

ⓐ	\(8\cdot 7\)
Multiply.	\(56\)
ⓑ	\(7\cdot 8\)
Multiply.	\(56\)

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

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

Condensed — the full section is in OpenStax Prealgebra 2e.

Translate Word Phrases to Math Notation

Earlier in this section, we translated math notation into words. Now we’ll reverse the process and translate word phrases into math notation. Some of the words that indicate multiplication are given in .

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

Example

Try it.

Translate and simplify: the product of \(12\) and \(27.\)

Solution

The word product tells us to multiply. The words of \(12\) and \(27\) tell us the two factors.

	the product of 12 and 27
Translate.	\(12⋅27\)
Multiply.	\(324\)

Example

Try it.

Translate and simplify: twice two hundred eleven.

Solution

The word twice tells us to multiply by \(2.\)

	twice two hundred eleven
Translate.	2(211)
Multiply.	422

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.

Example

Try it.

Humberto bought \(4\) sheets of stamps. Each sheet had \(20\) stamps. How many stamps did Humberto buy?

Solution

We are asked to find the total number of stamps.

Write a phrase for the total.	the product of 4 and 20
Translate to math notation.	\(4⋅20\)
Multiply.
Write a sentence to answer the question.	Humberto bought 80 stamps.

Example

Try it.

When Rena cooks rice, she uses twice as much water as rice. How much water does she need to cook \(4\) cups of rice?

Solution

We are asked to find how much water Rena needs.

Write as a phrase.	twice as much as 4 cups
Translate to math notation.	\(2⋅4\)
Multiply to simplify.	8
Write a sentence to answer the question.	Rena needs 8 cups of water for 4 cups of rice.

Example

Try it.

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

Solution

We are asked to find the total number of tiles.

Write a phrase.	the product of 8 and 14
Translate to math notation.	\(8⋅14\)
Multiply to simplify.	\(\begin{array}{l}\ \\ \overset{3}{1}4 \\ \underset{\text{___}}{\times 8} \\ 112\end{array}\)
Write a sentence to answer the question.	Van needs 112 tiles for his patio.

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

Operation	Notation	Expression	Read as	Result
\(\text{Multiplication}\)	\(\times\) \(\cdot\) \((\ )\)	\(3\ \times \ 8\) \(3\cdot 8\) \(3(8)\)	\(\text{three times eight}\)	\(\text{the product of 3 and 8}\)

Multiplication Property of Zero
- The product of any number and 0 is 0.
  \(a⋅0=0\)
  \(0⋅a=0\)
Identity Property of Multiplication
- The product of any number and 1 is the number.
  \(1⋅a=a\)
  \(a⋅1=a\)
Commutative Property of Multiplication
- Changing the order of the factors does not change their product.
  \(a⋅b=b⋅a\)
Multiply two whole numbers to find the product.
1. Write the numbers so each place value lines up vertically.
2. Multiply the digits in each place value.
3. Work from right to left, starting with the ones place in the bottom number.
4. Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on.
5. If a product in a place value is more than 9, carry to the next place value.
6. 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.
7. Insert a zero as a placeholder with each additional partial product.
8. Add the partial products.

Multiply Whole Numbers

Use Multiplication Notation

In the following exercises, translate from math notation to words.

Try it.

\(4\ \times \ 7\)

Solution

four times seven; the product of four and seven

Try it.

\(8\ \times \ 6\)

Try it.

\(5\cdot 12\)

Solution

five times twelve; the product of five and twelve

Try it.

\(3\cdot 9\)

Try it.

\((10)(25)\)

Solution

ten times twenty-five; the product of ten and twenty-five

Try it.

\((20)(15)\)

Try it.

\(42(33)\)

Solution

forty-two times thirty-three; the product of forty-two and thirty-three

Try it.

\(39(64)\)

Model Multiplication of Whole Numbers

In the following exercises, model the multiplication.

Try it.

\(3\ \times \ 6\)

Solution

Try it.

\(4\ \times \ 5\)

Try it.

\(5\ \times \ 9\)

Solution

Try it.

\(3\ \times \ 9\)

Multiply Whole Numbers

In the following exercises, fill in the missing values in each chart.

Try it.

Solution

Try it.

Solution

Try it.

Solution

Try it.

Solution

Try it.

In the following exercises, multiply.

Try it.

\(0\cdot 15\)

Solution

Try it.

\(0\cdot 41\)

Try it.

\((99)0\)

Solution

Try it.

\((77)0\)

Try it.

\(1\cdot 43\)

Solution

Try it.

\(1\cdot 34\)

Try it.

\((28)1\)

Solution

Try it.

\((65)1\)

Try it.

\(1(240,055)\)

Solution

240,055

Try it.

\(1(189,206)\)

Try it.

ⓐ \(7\cdot 6\)
ⓑ \(6\cdot 7\)

Solution

ⓐ 42
ⓑ 42

Try it.

ⓐ \(8\ \times \ 9\)
ⓑ \(9\ \times \ 8\)

Try it.

\((79)(5)\)

Solution

395

Try it.

\((58)(4)\)

Try it.

\(275\cdot 6\)

Solution

1,650

Try it.

\(638\cdot 5\)

Try it.

\(3,421\ \times \ 7\)

Solution

23,947

Try it.

\(9,143\ \times \ 3\)

Try it.

\(52(38)\)

Solution

1,976

Try it.

\(37(45)\)

Try it.

\(96\cdot 73\)

Solution

7,008

Try it.

\(89\cdot 56\)

Try it.

\(27\ \times \ 85\)

Solution

2,295

Try it.

\(53\ \times \ 98\)

Try it.

\(23\cdot 10\)

Solution

230

Try it.

\(19\cdot 10\)

Try it.

\((100)(36)\)

Solution

3,600

Try it.

\((100)(25)\)

Try it.

\(1,000(88)\)

Solution

88,000

Try it.

\(1,000(46)\)

Try it.

\(50\ \times \ 1,000,000\)

Solution

50,000,000

Try it.

\(30\ \times \ 1,000,000\)

Try it.

\(247\ \times \ 139\)

Solution

34,333

Try it.

\(156\ \times \ 328\)

Try it.

\(586(721)\)

Solution

422,506

Try it.

\(472(855)\)

Try it.

\(915\cdot 879\)

Solution

804,285

Try it.

\(968\cdot 926\)

Try it.

\((104)(256)\)

Solution

26,624

Try it.

\((103)(497)\)

Try it.

\(348(705)\)

Solution

245,340

Try it.

\(485(602)\)

Try it.

\(2,719\ \times \ 543\)

Solution

1,476,417

Try it.

\(3,581\ \times \ 724\)

Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

Try it.

the product of \(18\) and \(33\)

Solution

18 · 33; 594

Try it.

the product of \(15\) and \(22\)

Try it.

fifty-one times sixty-seven

Solution

51(67); 3,417

Try it.

forty-eight times seventy-one

Try it.

twice \(249\)

Solution

2(249); 498

Try it.

twice \(589\)

Try it.

ten times three hundred seventy-five

Solution

10(375); 3,750

Try it.

ten times two hundred fifty-five

Mixed Practice

In the following exercises, simplify.

Try it.

\(38\ \times \ 37\)

Solution

1,406

Try it.

\(86\ \times \ 29\)

Try it.

\(415-267\)

Solution

148

Try it.

\(341-285\)

Try it.

\(6,251+4,749\)

Solution

11,000

Try it.

\(3,816+8,184\)

Try it.

\((56)(204)\)

Solution

11,424

Try it.

\((77)(801)\)

Try it.

\(947\cdot 0\)

Solution

Try it.

\(947+0\)

Try it.

\(15,382+1\)

Solution

15,383

Try it.

\(15,382\cdot 1\)

In the following exercises, translate and simplify.

Try it.

the difference of 50 and 18

Solution

50 − 18; 32

Try it.

the difference of 90 and 66

Try it.

twice 35

Solution

2(35); 70

Try it.

twice 140

Try it.

20 more than 980

Solution

20 + 980; 1,000

Try it.

65 more than 325

Try it.

the product of 12 and 875

Solution

12(875); 10,500

Try it.

the product of 15 and 905

Try it.

subtract 74 from 89

Solution

89 − 74; 15

Try it.

subtract 45 from 99

Try it.

the sum of 3,075 and 95

Solution

3,075 + 95; 3,170

Try it.

the sum of 6,308 and 724

Try it.

366 less than 814

Solution

814 − 366; 448

Try it.

388 less than 925

Multiply Whole Numbers in Applications

In the following exercises, solve.

Try it.

Party supplies Tim brought 9 six-packs of soda to a club party. How many cans of soda did Tim bring?

Solution

Tim brought 54 cans of soda to the party.

Try it.

Sewing Kanisha is making a quilt. She bought 6 cards of buttons. Each card had four buttons on it. How many buttons did Kanisha buy?

Try it.

Field trip Seven school busses let off their students in front of a museum in Washington, DC. Each school bus had 44 students. How many students were there?

Solution

There were 308 students.

Try it.

Gardening Kathryn bought 8 flats of impatiens for her flower bed. Each flat has 24 flowers. How many flowers did Kathryn buy?

Try it.

Charity Rey donated 15 twelve-packs of t-shirts to a homeless shelter. How many t-shirts did he donate?

Solution

Rey donated 180 t-shirts.

Try it.

School There are 28 classrooms at Anna C. Scott elementary school. Each classroom has 26 student desks. What is the total number of student desks?

Try it.

Recipe Stephanie is making punch for a party. The recipe calls for twice as much fruit juice as club soda. If she uses 10 cups of club soda, how much fruit juice should she use?

Solution

Stephanie should use 20 cups of fruit juice.

Try it.

Gardening Hiroko is putting in a vegetable garden. He wants to have twice as many lettuce plants as tomato plants. If he buys 12 tomato plants, how many lettuce plants should he get?

Try it.

Government The United States Senate has twice as many senators as there are states in the United States. There are 50 states. How many senators are there in the United States Senate?

Solution

There are 100 senators in the U.S. senate.

Try it.

Recipe Andrea is making potato salad for a buffet luncheon. The recipe says the number of servings of potato salad will be twice the number of pounds of potatoes. If she buys 30 pounds of potatoes, how many servings of potato salad will there be?

Try it.

Painting Jane is painting one wall of her living room. The wall is rectangular, 13 feet wide by 9 feet high. What is the area of the wall?

Solution

The area of the wall is 117 square feet.

Try it.

Home décor Shawnte bought a rug for the hall of her apartment. The rug is 3 feet wide by 18 feet long. What is the area of the rug?

Try it.

Room size The meeting room in a senior center is rectangular, with length 42 feet and width 34 feet. What is the area of the meeting room?

Solution

The area of the room is 1,428 square feet.

Try it.

Gardening June has a vegetable garden in her yard. The garden is rectangular, with length 23 feet and width 28 feet. What is the area of the garden?

Try it.

NCAA basketball According to NCAA regulations, the dimensions of a rectangular basketball court must be 94 feet by 50 feet. What is the area of the basketball court?

Solution

The area of the court is 4,700 square feet.

Try it.

NCAA football According to NCAA regulations, the dimensions of a rectangular football field must be 360 feet by 160 feet. What is the area of the football field?

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Add: \(1,683+479.\)
If you missed this problem, review .

Odhalte odpověď

\(2,162\)
Subtract: \(605-321.\)
If you missed this problem, review .

Odhalte odpověď

\(284\)
Translate from math notation to words:
1. ⓐ \(7\ \times \ 6\)
2. ⓑ \(12\cdot 14\)
3. ⓒ \(6(13)\)
Odhalte odpověď
- ⓐ We read this as seven times six and the result is the product of seven and six.
- ⓑ We read this as twelve times fourteen and the result is the product of twelve and fourteen.
- ⓒ We read this as six times thirteen and the result is the product of six and thirteen.
Translate from math notation to words:
1. ⓐ \(8\ \times \ 7\)
2. ⓑ \(18\cdot 11\)
Odhalte odpověď
1. ⓐ eight times seven ; the product of eight and seven
2. ⓑ eighteen times eleven ; the product of eighteen and eleven
Translate from math notation to words:
1. ⓐ \((13)(7)\)
2. ⓑ \(5(16)\)
Odhalte odpověď
1. ⓐ thirteen times seven ; the product of thirteen and seven
2. ⓑ five times sixteen; the product of five and sixteen
Model: \(3\ \times \ 8.\)

Odhalte odpověď

To model the product \(3\ \times \ 8,\) we’ll start with a row of \(8\) counters.

The other factor is \(3,\) so we’ll make \(3\) rows of \(8\) counters.

Now we can count the result. There are \(24\) counters in all.
\(3\ \times \ 8=24\)
If you look at the counters sideways, you’ll see that we could have also made \(8\) rows of \(3\) counters. The product would have been the same. We’ll get back to this idea later.
Model each multiplication: \(4\ \times \ 6.\)

Odhalte odpověď
Model each multiplication: \(5\ \times \ 7.\)

Odhalte odpověď
Multiply:
1. ⓐ \(0\cdot 11\)
2. ⓑ \((42)0\)
Odhalte odpověď

ⓐ \(0\cdot 11\)
The product of any number and zero is zero. \(0\)
ⓑ \((42)0\)
Multiplying by zero results in zero. \(0\)
Find each product:
1. ⓐ \(0\cdot 19\)
2. ⓑ \((39)0\)
Odhalte odpověď
1. ⓐ \(0\)
2. ⓑ \(0\)
Find each product:
1. ⓐ \(0\cdot 24\)
2. ⓑ \((57)0\)
Odhalte odpověď
1. ⓐ \(0\)
2. ⓑ \(0\)
Multiply:
1. ⓐ \((11)1\)
2. ⓑ \(1\cdot 42\)
Odhalte odpověď

ⓐ \((11)1\)
The product of any number and one is the number. \(11\)
ⓑ \(1\cdot 42\)
Multiplying by one does not change the value. \(42\)
Find each product:
1. ⓐ \((19)1\)
2. ⓑ \(1\cdot 39\)
Odhalte odpověď
1. ⓐ \(19\)
2. ⓑ \(39\)
Find each product:
1. ⓐ \((24)(1)\)
2. ⓑ \(1\ \times \ 57\)
Odhalte odpověď
1. ⓐ \(24\)
2. ⓑ \(57\)
Multiply:
1. ⓐ \(8\cdot 7\)
2. ⓑ \(7\cdot 8\)
Odhalte odpověď

ⓐ \(8\cdot 7\)
Multiply. \(56\)
ⓑ \(7\cdot 8\)
Multiply. \(56\)
Changing the order of the factors does not change the product.
Multiply:
1. ⓐ \(9\cdot 6\)
2. ⓑ \(6\cdot 9\)
Odhalte odpověď

54 and 54; both are the same.
Multiply:
1. ⓐ \(8\cdot 6\)
2. ⓑ \(6\cdot 8\)
Odhalte odpověď

48 and 48; both are the same.

Multiply: \(15\cdot 4.\)

Odhalte odpověď

Write the numbers so the digits \(5\) and \(4\) line up vertically.	\(\begin{array}{l}15\ \\ \underset{\text{_____}}{\times \ 4}\end{array}\)
Multiply \(4\) by the digit in the ones place of \(15.\) \(4⋅5=20.\)
Write \(0\) in the ones place of the product and carry the \(2\) tens.	\(\begin{array}{l}\overset{2}{1}5\ \\ \underset{\text{_____}}{\times \ 4} \\ 0\ \end{array}\)
Multiply \(4\) by the digit in the tens place of \(15.\) \(4⋅1=4\). Add the \(2\) tens we carried. \(4+2=6\).
Write the \(6\) in the tens place of the product.	\(\begin{array}{l}\overset{2}{1}5\ \\ \underset{\text{_____}}{\times \ 4} \\ 60\ \end{array}\)

Multiply: \(64\cdot 8.\)

Odhalte odpověď

\(512\)
Multiply: \(57\cdot 6.\)

Odhalte odpověď

\(342\)

Multiply: \(286\cdot 5.\)

Odhalte odpověď

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

Multiply: \(347\cdot 5.\)

Odhalte odpověď

\(1,735\)
Multiply: \(462\cdot 7.\)

Odhalte odpověď

\(3,234\)

Multiply: \(62(87).\)

Odhalte odpověď

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. \(7⋅2=14.\) Write the 4 in the ones place of the product and carry the 1 to the tens place.
Multiply 7 by the digit in the tens place of 62. \(7⋅6=42.\) Add the 1 ten we carried. \(42+1=43\). Write the 3 in the tens place of the product and the 4 in the hundreds place.
The first partial product is 434.
Now, write a 0 under the 4 in the ones place of the next partial product as a placeholder since we now multiply the digit in the tens place of 87 by 62. Multiply 8 by the digit in the ones place of 62. \(8⋅2=16.\) Write the 6 in the next place of the product, which is the tens place. Carry the 1 to the tens place.
Multiply 8 by 6, the digit in the tens place of 62, then add the 1 ten we carried to get 49. Write the 9 in the hundreds place of the product and the 4 in the thousands place.
The second partial product is 4960. Add the partial products.

The product is \(5,394.\)

Multiply: \(43(78).\)

Odhalte odpověď

3,354
Multiply: \(64(59).\)

Odhalte odpověď

3,776

Multiply:

ⓐ \(47\cdot 10\)
ⓑ \(47\cdot 100.\)

Odhalte odpověď

ⓐ \(47\cdot 10\).	\(\begin{array}{l}47\ \\ \underset{\text{___}}{\times 10} \\ 00 \\ \underset{\text{___}}{470} \\ 470\end{array}\)
ⓑ \(47\cdot 100\)	\(\begin{array}{l}47\ \\ \underset{\text{_____}}{\times 100}\ \\ 00\ \\ \underset{\text{_____}}{\begin{array}{l}000 \\ 4700\end{array}} \\ 4,700\ \end{array}\)

When we multiplied \(47\) times \(10,\) the product was \(470.\) Notice that \(10\) has one zero, and we put one zero after \(47\) to get the product. When we multiplied \(47\) times \(100,\) the product was \(4,700.\) Notice that \(100\) has two zeros and we put two zeros after \(47\) to get the product.

Do you see the pattern? If we multiplied \(47\) times \(10,000,\) which has four zeros, we would put four zeros after \(47\) to get the product \(470,000.\)

Multiply:
1. ⓐ \(54\cdot 10\)
2. ⓑ \(54\cdot 100.\)
Odhalte odpověď
1. ⓐ 540
2. ⓑ 5,400
Multiply:
1. ⓐ \(75\cdot 10\)
2. ⓑ \(75\cdot 100.\)
Odhalte odpověď
1. ⓐ 750
2. ⓑ 7,500
Multiply: \((354)(438).\)

Odhalte odpověď

There are three digits in the factors so there will be \(3\) partial products. We do not have to write the \(0\) as a placeholder as long as we write each partial product in the correct place.
Multiply: \((265)(483).\)

Odhalte odpověď

127,995
Multiply: \((823)(794).\)

Odhalte odpověď

653,462
Multiply: \((896)201.\)

Odhalte odpověď

There should be \(3\) partial products. The second partial product will be the result of multiplying \(896\) by \(0.\)

Notice that the second partial product of all zeros doesn’t really affect the result. We can place a zero as a placeholder in the tens place and then proceed directly to multiplying by the \(2\) in the hundreds place, as shown.
Multiply by \(10,\) but insert only one zero as a placeholder in the tens place. Multiply by \(200,\) putting the \(2\) from the \(12.\) \(2\cdot 6=12\) in the hundreds place.
\[\begin{array}{l} \\ \\ 896\ \\ \underset{\text{_____}}{\times 201}\ \\ 896\ \\ \underset{\text{__________}}{17920}\ \\ 180,096\ \end{array}\]
Multiply: \((718)509.\)

Odhalte odpověď

365,462
Multiply: \((627)804.\)

Odhalte odpověď

504,108
Translate and simplify: the product of \(12\) and \(27.\)

Odhalte odpověď

The word product tells us to multiply. The words of \(12\) and \(27\) tell us the two factors.
the product of 12 and 27
Translate. \(12⋅27\)
Multiply. \(324\)
Translate and simplify: the product of \(13\) and \(28.\)

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13 · 28; 364
Translate and simplify: the product of \(47\) and \(14.\)

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47 · 14; 658
Translate and simplify: twice two hundred eleven.

Odhalte odpověď

The word twice tells us to multiply by \(2.\)
twice two hundred eleven
Translate. 2(211)
Multiply. 422
Translate and simplify: twice one hundred sixty-seven.

Odhalte odpověď

2(167); 334

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Multiply Whole Numbers

Use multiplication notation
Model multiplication of whole numbers
Multiply whole numbers
Translate word phrases to math notation
Multiply whole numbers in applications
Write the numbers so each place value lines up vertically.
Multiply the digits in each place value.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Zkuste si vlastní.

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Více v Arithmetic

Order of operations Fractions Percentages Powers and roots

×	1	2	3	4	5	6	7	8	9
0	0	0	0	0	0	0	0	0	0
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	10	12	14	16	18
3	3	6	9	12	15	18	21	24	27
4	4	8	12	16	20	24	28	32	36
5	5	10	15	20	25	30	35	40	45
6	6	12	18	24	30	36	42	48	54
7	7	14	21	28	35	42	49	56	63
8	8	16	24	32	40	48	56	64	72
9	9	18	27	36	45	54	63	72	81

×	1	2	3	4	5	6	7	8	9
0	0	0	0	0	0	0	0	0	0
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	10	12	14	16	18
3	3	6	9	12	15	18	21	24	27
4	4	8	12	16	20	24	28	32	36
5	5	10	15	20	25	30	35	40	45
6	6	12	18	24	30	36	42	48	54
7	7	14	21	28	35	42	49	56	63
8	8	16	24	32	40	48	56	64	72
9	9	18	27	36	45	54	63	72	81

×	1	2	3	4	5	6	7	8	9
0	0	0	0	0	0	0	0	0	0
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	10	12	14	16	18
3	3	6	9	12	15	18	21	24	27
4	4	8	12	16	20	24	28	32	36
5	5	10	15	20	25	30	35	40	45
6	6	12	18	24	30	36	42	48	54
7	7	14	21	28	35	42	49	56	63
8	8	16	24	32	40	48	56	64	72
9	9	18	27	36	45	54	63	72	81