1.3整数乘法: 整数乘法

Han is going on vacation and is leaving his cat, Hippo, with his neighbor. Han will be gone for a total of 21 days and will need to leave enough food for his neighbor to give the cat. Hippo eats 4 small cans of cat food every day. How can Han calculate how many cans of cat food to leave with his neighbor?
::Han去度假了,正要离开他的猫,希波和邻居一起离开。汉将离开21天,需要留足够的食物给邻居喂猫。希波每天要吃4小罐猫食。汉如何计算有多少罐猫食可以和邻居一起离开?

In this concept, you will learn how to multiply whole numbers .
::在这个概念中,你会学会如何乘以整个数字。

Multiplying Whole Numbers
::乘数全数数

Did you know that addition and multiplication are related? Let's explore what this means through an example.
::你知道加法和乘法是相关的吗?

\begin{aligned} 5 \times 6 = \underline{} \end{aligned}

You can use your times tables to complete this problem using mental math, but let’s look at what it means to multiply 5 by 6. $\begin{aligned} 5 \times 6 \end{aligned}$ means that you need five groups of six.
::你可以用时间表来用心理数学来解决这个问题, 但让我们看看乘以5乘以6、5×6意味着你需要五组六。

You can also think of this as adding 5 six times.
::也可以认为这增加了5次6次。

\begin{aligned} 5 + 5 + 5 + 5 + 5 + 5 = \underline{} \end{aligned}

Looked at this way, you can see that multiplication is a shortcut for repeated addition.
::以这种方式来看,你可以看到乘法是重复添加的捷径。

\begin{aligned} 5 \times 6 = 30 \end{aligned}

In this problem, 5 and 6 are the factors . Factors are numbers that are multiplied by each other. The product is the answer to a multiplication problem, in this case, the product is 30. You can say that 5 and 6 are factors of 30 because the product of 5 and 6 is 30
::在这个问题中,5和6是因素。因素是相互乘以的数字。产品是乘数问题的答案。产品是乘数问题的答案,在这种情况下,产品是30。你可以说5和6是30的因数,因为5和6的产数是30。

When multiplying larger numbers, it may help you to think of multiplication as just a short cut for repeated addition.
::当乘以较大数字时,它可以帮助你把乘法看作是重复加法的捷径。

Things to remember when multiplying:
::当乘以时要记住的事情 :

It is generally easier to multiply vertically, with the larger number on top.
::纵向乘法一般比较容易,上面的乘数较多。

Multiply each digit of the top number by each digit of the bottom number, from right to left.
::从右到左,将最高数字的每一个数字乘以底数的每一个数字。

Multiply the first digit in the bottom number by every digit in the top number, from right to left, then move down to the next line and add in a zero for each digit that you have already multiplied before repeating the process with the next digit.
::从右向左乘以数字顶端的每个数字,乘以数字底部的第一个数字,然后向下移到下一行,再将您已经乘以的每个数字加为零,然后再用下一个数字重复进程。

Here is an example:
::以下是一个例子:

\begin{aligned} 567 \times 3 = \underline{} \end{aligned}

First, line up the numbers according to place value , with the larger number on top.
::首先,根据地点值排列数字,在上方数字较大。

\begin{aligned} 567 \\ \underline{\times 3} \end{aligned}

Multiply each digit of the larger number by each digit of the smaller number, from right to left. In this case, since there is only one digit in the bottom number, multiply 3 by each digit of the top number. Since 7 is the first number in the upper row, multiply that by 3 first.
::从右向左按较小数字的每个位数乘以较大数字的每个位数乘以较大数字的每个位数。在此情况下,由于底数只有一个位数,因此将最高数字的每个位数乘以3。由于7是上行的第一个数字,先乘以3。

\begin{aligned} 7 \times 3 = 21 \end{aligned}

Put the 1 in the ones place and carry the 2 (which is really 2 tens) to the next column over, where it can be added to the other tens after the next multiplication step.
::将 1 放在那些位置,并将 2 (其实是 2 至 10 ) 携带到下一列, 可以在下一个乘法步骤之后将其添加到其他 10 列中。

\begin{aligned} _{2} \\ 567 \\ \underline{\times 3} \\ 1 \end{aligned}

Next, multiply the 3 by 6 and add the 2 you carried.
::接下来,乘以3乘以6,加上你携带的2。

\begin{aligned} 6 \times 3 + 2 = 20 \end{aligned}

Leave the 0 in the tens place and carry the two over to the hundreds column.
::把0放在十号位置上将两架抬到几百列的柱子上

\begin{aligned} _{2}_{2} \\ 567 \\ \underline{\times 3} \\ 01 \end{aligned}

Then multiply the 3 by 5 and add the two you carried.
::然后乘以3乘以5 加上你携带的两个

\begin{aligned} 5 \times 3 + 2 = 17 \end{aligned}

Finally, leave the 7 in the hundreds column, and put the 1 in the thousands column (because 17 hundreds is really 1 thousand and 7 hundreds).
::最后,在数百列中留下7个,在千列中留下1个(因为17百个其实是1千个和7百个)。

\begin{aligned} _{2}_{2} \\ 567 \\ \underline{\times 3} \\ 1701 \end{aligned}

The product is 1,701.
::产品为1 701美元。

You can practice multiplying three digits by one digit here:
::您可以在此练习将3位数乘以1位数 :

Here is another example. This one has two digits in the second number:
::下面是另一个例子。这个第二个数字有两个位数:

\begin{aligned} 234 \times 12 = \underline{} \end{aligned}

First, line up the digits according to place value.
::首先,根据地点值排列数字。

\begin{aligned} 234 \\ \underline{\times 12} \end{aligned}

Next, multiply each digit of the top number by each digit of the bottom number, 12. Start by multiplying the 2 of the bottom number by each digit of the top number:
::接下来,将最高数字的每个数字乘以下限数字的每个数字,12。开始, 将下限数字的2乘以最高数字的每个数字:

\begin{aligned} 234 \\ \underline{\times 12} \\ 468 \end{aligned}

Next, multiply the 1, which is in the tens place, by each digit in the top number. Because you are multiplying by a number in the "tens" place, you start the second row of numbers with a zero so that the answer to the multiplication is kept in the correct place value for the addition we will do next. Here is what this looks like:
::接下来,乘以十位数中的每个位数,乘以十位数,乘以十位数。由于你在十位数中乘以一个数字,您以零开始第二行数字,这样乘法的答案将保留在正确的位置值中,用于我们下一步将增加的数值。这里是这样的:

\begin{aligned} 234 \\ \underline{\times 12} \\ 468 \\ 2340 \end{aligned}

Finally, add the products of the two rounds of multiplication:
::最后,加上两轮乘法的产品:

\begin{aligned} 234 \\ \underline{\times 12} \\ 468 \\ \underline{+ 2340} \\ 2808 \end{aligned}

The final product is 2,808.
::最终产品为2 808。

Practice multiplying three digits by two digits until you are comfortable with the process:
::练习将3位数乘以2位数,直到您适应这个过程:

Examples
::实例实例实例实例

Example 1
::例1

Earlier, you were given a problem about Han and his hungry cat, Hippo.
::之前,你被问起关于汉和他饥饿的猫,希波的问题。

Hippo eats 4 small cans of cat food every day, and will need enough food for 21 days.
::希波每天吃4小罐猫食, 21天需要足够的食物。

To figure out the product of 4 cans per day for 21 days, Han has to multiply.
::要想在21天的时间里找到每天4罐的产物 Han必须乘以乘法

4 x 21
::4 x 21 4 x 21

First, write the problem vertically, lining up place values with the larger number on top.
::首先,垂直写出问题, 将位置值排成一列, 上面的数值更大。

\begin{aligned} 21 \\ \underline{\times 4} \end{aligned}

Next, multiply the multiplier , 4, by each digit of the top number.
::下一个乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘以乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘乘

Multiply 4 by 1:
::乘以 4 乘以 1:

\begin{aligned} 21 \\ \underline{\times 4} \\ 4 \end{aligned}

Then, multiply 4 by 2:
::然后乘以4乘以2乘以2

\begin{aligned} 21 \\ \underline{\times 4} \\ 84 \end{aligned}

The product is 84.
::产品为84。

Han will need to leave 84 cans of food for hungry Hippo.
::Han需要留84罐食物给饥饿的河马

Example 2
::例2

Multiply the following values.
::乘以以下值。

\begin{aligned} 214 \times 362 \end{aligned}

First, you will write the problem vertically, lining up the place values, and multiply the first digit of the multiplier by each of the three digits of the top number.
::首先,您将垂直写出问题, 将位置值排成一行, 并将乘数乘以乘数的第一个位数乘以最高数的三个位数。

\begin{aligned} 214 \\ \underline{\times 362} \\ 428 \end{aligned}

Next, you will multiply the second digit of the multiplier by all three digits of the top number. Don’t forget that placeholder zero!
::接下来,您将把乘数的第二位乘数乘以最高数字的所有三个位数。不要忘记占位符零!

\begin{aligned} 214 \\ \underline{\times 362} \\ 428 \\ 12840 \end{aligned}

Then, you will multiply the third digit of the multiplier by all three digits of the top number. Use two zeros since you are now multiplying by a number in the "hundreds" place.
::然后,您将乘以乘数的三位数乘以最高数的三位数。使用两个零,因为您现在在“ 百” 位置乘以一个数字。

\begin{aligned} 214 \\ \underline{\times 362} \\ 428 \\ 12840 \\ 64200 \end{aligned}

Finally, sum the three products to get the answer.
::最后,总结这三个产品,以获得答案。

\begin{aligned} 214 \\ \underline{\times 362} \\ 428 & H e r e i s t h e r e s u l t o f m u l t i p l y i n g b y 2. \\ 12840 & H e r e i s t h e r e s u l t o f m u l t i p l y i n g b y 6. Y o u h a d t o c a r r y a n d a d d i n a z e r o . \\ \underline{+ 64200} & H e r e i s t h e r e s u l t o f m u l t i p l y i n g b y 3. Y o u h a d t o a d d i n t w o z e r o s . \\ 77468 & T h i s i s t h e a n s w e r . \end{aligned}

The answer is 77,468.
::答案是77 468

Example 3
::例3

Find the product.
::找到产品。

$\begin{aligned} 456 \times 9 = \underline{} \end{aligned}$

First, rewrite the problem vertically, lining up the place values.
::首先,垂直重写问题, 排列地方价值。

Then, multiply each digit of 456 by 9, from right to left, carrying as needed.
::然后,将456乘以9的每位数从右向左乘以456乘以9,视需要携带。

\begin{aligned} 456 \\ \underline{\times 9} \\ 4104 \end{aligned}

The answer is 4,104.
::答案是4,104。

Example 4
::例4

Find the product.
::找到产品。

$\begin{aligned} 321 \times 18 = \underline{} \end{aligned}$

First rewrite the problem vertically, lining up place values.
::首先垂直重写问题, 排列位置值。

Next, multiply each digit of 321 by 8, carrying as needed.
::其次,将321乘以每位数乘以8,视需要携带。

Then, begin a new row of numbers with a 0 and multiply 321 by 1. Finally, sum the results to get the answer.
::然后用 0 乘以 321 乘以 1 开始一排新的数字, 最后, 将结果相加, 以获得答案。

\begin{aligned} 321 \\ \underline{\times 18} \\ 2568 \\ \underline{+ 3210} \\ 5778 \end{aligned}

The answer is 5,778.
::答案是5 778

Example 5
::例5

Find the product.
::找到产品。

$\begin{aligned} 562 \times 248 = \underline{} \end{aligned}$

First, write the problem vertically, lining up the place values.
::首先,垂直写出问题, 排成一列地值。

Next, multiply 562 by 8, carrying as needed.
::接下来,乘以562乘以8,视需要携带。

Next, multiply 562 by 4, writing the answers in the next row and starting with a single 0.
::下一个乘以562乘以4, 在下一行写答案, 从一个零开始。

Then, multiply 562 by 2 and put the answers in another new row, this time starting with two 0's.
::然后乘以562乘以2 然后把答案放进新的一排这次从两张零开始

Finally, sum the three products to get the answer.
::最后,总结这三个产品,以获得答案。

\begin{aligned} 562 \\ \underline{\times 248} \\ 4496 \\ 22480 \\ \underline{+ 112400} \\ 139376 \end{aligned}

The answer is 139,376
::答案是139 376

Review
::审查审查审查审查

Find the product.
::找到产品。

$\begin{aligned} 34 \times 8 = \underline{} \end{aligned}$
$\begin{aligned} 67 \times 12 = \underline{} \end{aligned}$
$\begin{aligned} 34 \times 87 = \underline{} \end{aligned}$
$\begin{aligned} 124 \times 9 = \underline{} \end{aligned}$
$\begin{aligned} 345 \times 11 = \underline{} \end{aligned}$
$\begin{aligned} 6, 721 \times 9 = \underline{} \end{aligned}$
$\begin{aligned} 8, 723 \times 31 = \underline{} \end{aligned}$
$\begin{aligned} 9, 802 \times 22 = \underline{} \end{aligned}$
$\begin{aligned} 345 \times 123 = \underline{} \end{aligned}$
$\begin{aligned} 617 \times 234 = \underline{} \end{aligned}$
$\begin{aligned} 534 \times 78 = \underline{} \end{aligned}$
$\begin{aligned} 834 \times 228 = \underline{} \end{aligned}$
$\begin{aligned} 1, 134 \times 68 = \underline{} \end{aligned}$
$\begin{aligned} 2, 434 \times 218 = \underline{} \end{aligned}$
$\begin{aligned} 6, 734 \times 208 = \underline{} \end{aligned}$
$\begin{aligned} 8, 934 \times 1, 238 = \underline{} \end{aligned}$
$\begin{aligned} 2, 334 \times 3, 408 = \underline{} \end{aligned}$

Review (Answers)
::审查(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Resources
::资源资源资源资源资源资源资源资源资源资源

最后修改: 2025年03月1日星期六 13:37

Multiplying Whole Numbers ::乘数全数数

You can practice multiplying three digits by one digit here: ::您可以在此练习将3位数乘以1位数 :

Practice multiplying three digits by two digits until you are comfortable with the process: ::练习将3位数乘以2位数,直到您适应这个过程:

Examples ::实例实例实例实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::审查审查审查审查

Review (Answers) ::审查(答复)

Resources ::资源资源资源 资源资源资源 资源资源 资源资源

Multiplying Whole Numbers
::乘数全数数

You can practice multiplying three digits by one digit here:
::您可以在此练习将3位数乘以1位数 :

Practice multiplying three digits by two digits until you are comfortable with the process:
::练习将3位数乘以2位数,直到您适应这个过程:

Examples
::实例实例实例实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::审查审查审查审查

Review (Answers)
::审查(答复)

Resources
::资源资源资源资源资源资源资源资源资源资源