课程： 7.1分数和整数的乘法

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选择节分数和整数的乘法

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分数和整数的乘法
Julie is writing a report on rainforests. A rainforest can receive at least $\begin{aligned} \frac{1}{5} \end{aligned}$ of an inch of rain each day. She wants to calculate the amount of rainfall that occurs in one week. How much rainfall occurs in the forest over the course of one week?
::朱莉正在写一篇关于雨林的报告。雨林每天至少能得到15英寸的降雨量。她想计算一个星期的降雨量。一个星期的森林降雨量是多少?

In this concept, you will learn to multiply fractions and whole numbers .
::在这个概念中,你会学会乘以分数和整数。

Multiplying Fractions and Whole Numbers
::乘数分数和整数

Remember that multiplication is a shortcut for repeated addition. It is a way to add the same number of groups, several or "multiple" times.
::记住乘法是重复添加的快捷键。它是一种添加相同数目的组, 数次或“ 多次” 的方法。

Here is an example.
::举一个例子。

5 + 5 + 5 + 5 = 20

5 is added 4 times. Therefore, 5 × 4 = 20.
::因此,5×4=20。

The same can be applied towards fractions. Here is a repeated addition problem involving a fraction.
::同样的情况也适用于分数。这里是一个重复增加的问题,涉及一个分数。

$\begin{aligned} \frac{1}{9} + \frac{1}{9} + \frac{1}{9} + \frac{1}{9} = \underline{} \end{aligned}$

The fraction $\begin{aligned} \frac{1}{9} \end{aligned}$ is added 4 times.
::第19项增加4次。

$\begin{aligned} \frac{1}{9} \times 4 \end{aligned}$

To find the product of a fraction and a whole number, convert the whole number to a fraction before multiplying. Remember that a whole number can be written into a fraction as the number over a denominator of 1.
::要找到一个分数和整数的产物, 请在乘法前将整数转换成一个分数。记住, 整数可以写成一个分数, 作为分数乘以 1 的分母。

$\begin{aligned} n = \frac{n}{1} \end{aligned}$

First, change the whole number to a fraction.
::首先,将整个数字修改为一个分数。

$\begin{aligned} 4 = \frac{4}{1} \end{aligned}$

Rewrite the problem with the fraction.
::用分数重写问题。

$\begin{aligned} \frac{1}{9} \times \frac{4}{1} \end{aligned}$

Then, multiply the two fractions. Multiply the numerator by the numerator and the denominator by the denominator.
::然后乘以两个分数。乘以分子,乘以分母,乘以分子,乘以分母,乘以分母。

$\begin{aligned} \frac{1 \times 4}{9 \times 1} = \frac{4}{9} \end{aligned}$

The resulting fraction is in simplest form . The product is $\begin{aligned} \frac{4}{9} \end{aligned}$ .
::结果的分数以最简单的形式出现,产品为4 9。

The commutative property of multiplication states that the order in which you multiply numbers does not matter as the product will be the same.
::乘法的通量属性表示,乘法数的顺序与产品相同无关。

$\begin{aligned} a \times b = b \times a \end{aligned}$

A fraction times a whole number is the same as the whole number times a fraction.
::一个整数的分数乘以一个整数与整个数乘以一个分数相同。

$\begin{aligned} \frac{1}{9} \times 4 = 4 \times \frac{1}{9} \end{aligned}$

Therefore, if $\begin{aligned} \frac{1}{9} \end{aligned}$ times 4 is equal to $\begin{aligned} \frac{4}{9} \end{aligned}$ , then 4 times $\begin{aligned} \frac{1}{9} \end{aligned}$ is also equal to $\begin{aligned} \frac{4}{9} \end{aligned}$ .
::因此,如果1 9乘以4等于4 9,那么4乘以1 9等于4 9,那么4乘以1 9也等于4 9。

$\begin{aligned} 4 \times \frac{1}{9} = \frac{4}{9} \end{aligned}$

The word “of” in a problem can also indicate multiplication.
::问题中的“of”一词也可以表示乘法。

Here is an expression.
::这是一种表达方式。

$\begin{aligned} \frac{1}{2} of 4 \end{aligned}$

To write this as a numerical expression , change the word “of” to a multiplication sign.
::要将此写成数字表达式,请将“of”改为乘号。

$\begin{aligned} \frac{1}{2} \times 4 \end{aligned}$

Then, rewrite the whole number as a fraction.
::然后,重写整个数字作为一个分数。

$\begin{aligned} 4 = \frac{4}{1} \\ \frac{1}{2} \times \frac{4}{1} \end{aligned}$

Next, multiply the fractions. Multiply the numerator by the numerator and the denominator by the denominator.
::下一步,乘以分数。乘以分子,乘以分数。乘以分子,乘以分数,乘以分数,乘以分数。

$\begin{aligned} \frac{1 \times 4}{2 \times 1} = \frac{4}{2} \end{aligned}$

Finally, simplify the fraction. Convert the improper fraction to a mixed number by dividing the numerator by the denominator. 2 divides evenly into 4 so the product will be a whole number.
::最后,简化分数。将不适当的分数转换成混合数, 将分子除以分母。 2 均分分为 4 , 这样产品将是一个整数。

$\begin{aligned} \frac{4}{2} = 4 \div 2 = 2 \end{aligned}$

The product is 2.
::产品为2。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Julie's rainforest report.
::早些时候,你得到一个问题关于朱莉的雨林报告。

Julie wants to calculate how much it rains in a week if the rainforest receives an average of $\begin{aligned} \frac{1}{5} \end{aligned}$ of an inch of rain per day. Multiply the average rainfall per day times 7 to find the average rainfall in a week.
::如果雨林每天平均降雨量为15英寸,朱莉想计算一周雨量。将平均每天降雨量乘以7乘以一周的平均降雨量。

$\begin{aligned} \frac{1}{5} \times 7 \end{aligned}$

First, rewrite the whole number as a fraction.
::首先,将整个数字重写为分数。

$\begin{aligned} 7 = \frac{7}{1} \\ \frac{1}{5} \times \frac{7}{1} \end{aligned}$

Then, multiply the fractions.
::然后,乘以分数。

$\begin{aligned} \frac{1 \times 7}{5 \times 1} = \frac{7}{5} \end{aligned}$

Finally, simplify the fraction. Convert the improper fraction to a mixed number.
::最后,简化分数,将不适当的分数转换成混合数。

$\begin{aligned} \frac{7}{5} = 1 \frac{2}{5} \end{aligned}$

In one week, it rains about $\begin{aligned} 1 \frac{2}{5} \end{aligned}$ of an inch in the rainforest.
::一周后雨雨雨雨雨中雨雨下约125英寸

Example 2
::例2

Answer the problem in simplest form.
::以最简单的方式回答问题。

Jessie handed out $\begin{aligned} \frac{2}{9} \end{aligned}$ of the cake to each of her 3 friends. How much cake was given out altogether?
::杰西把29块蛋糕交给了3个朋友总共给了多少蛋糕?

Normally the word “altogether” tells you to add. However, since the same portion of the cake is given out to each friend, you can multiply instead of add.
::通常,“一起”一词指您要添加。但是,既然蛋糕的同一部分是给每个朋友的,您可以乘数而不是加数。

First, write an expression for the total amount of cake given.
::首先,给定的蛋糕总数量写一个表达式。

$\begin{aligned} \frac{2}{9} \times 3 \end{aligned}$

Then, rewrite the whole number as a fraction.
::然后,重写整个数字作为一个分数。

$\begin{aligned} 3 & = \frac{3}{1} \\ \frac{2}{9} & \times \frac{3}{1} \end{aligned}$

Next, multiply the fractions. Multiply the numerator by the numerator and the denominator by the denominator.
::下一步,乘以分数。乘以分子,乘以分数。乘以分子,乘以分数,乘以分数,乘以分数。

$\begin{aligned} \frac{2 \times 3}{9 \times 1} = \frac{6}{9} \end{aligned}$

Our final step is to simplify. We can simplify $\begin{aligned} \frac{6}{9} \end{aligned}$ by dividing the numerator and denominator by the greatest common factor of 3.
::我们的最后一步是简化。我们可以通过将分子和分母除以3这一最大共同系数来简化 6 9 。

$\begin{aligned} \frac{6 \div 3}{9 \div 3} = \frac{2}{3} \end{aligned}$

Jessie handed out $\begin{aligned} \frac{2}{3} \end{aligned}$ of the cake to her friends.
::杰西把23块蛋糕交给了朋友

Example 3
::例3

Find the product: $\begin{aligned} \frac{1}{4} \times 5 = \underline{} \end{aligned}$ . Answer in simplest form.
::查找产品:1 4× 5 = _. 回答最简单。

First, rewrite the whole number as a fraction.
::首先,将整个数字重写为分数。

$\begin{aligned} 5 = \frac{5}{1} \\ \frac{1}{4} \times \frac{5}{1} \end{aligned}$

Then, multiply the fractions.
::然后,乘以分数。

$\begin{aligned} \frac{1 \times 5}{4 \times 1} = \frac{5}{4} \end{aligned}$

Next, simplify the product. Convert the improper fraction to a mixed number.
::接下来,简化产品。将不当的分数转换成混合数。

$\begin{aligned} \frac{5}{4} = 1 \frac{1}{4} \end{aligned}$

The product is $\begin{aligned} 1 \frac{1}{4} \end{aligned}$ .

::产品为1,14。

Example 4
::例4

Find the product: $\begin{aligned} \frac{2}{3} \times 2 = \underline{} \end{aligned}$ . Answer in simplest form.
::查找产品: 2 3 × 2 = _. 回答最简单。

First, rewrite the whole number as a fraction.
::首先,将整个数字重写为分数。

$\begin{aligned} 2 = \frac{2}{1} \\ \frac{2}{3} \times \frac{2}{1} \end{aligned}$

Then, multiply the fractions.
::然后,乘以分数。

$\begin{aligned} \frac{2 \times 2}{3 \times 1} = \frac{4}{3} \end{aligned}$

Next, simplify the product. Convert the improper fraction to a mixed number.
::接下来,简化产品。将不当的分数转换成混合数。

$\begin{aligned} \frac{4}{3} = 1 \frac{1}{3} \end{aligned}$

The product is $\begin{aligned} 1 \frac{1}{3} \end{aligned}$ .
::产品为1,13。

Example 5
::例5

Find the product: $\begin{aligned} \frac{1}{7} \times 8 = \underline{} \end{aligned}$ . Answer in simplest form.
::查找产品: 1 7 × 8 = _. 回答最简单。

First, rewrite the whole number as a fraction.
::首先,将整个数字重写为分数。

$\begin{aligned} 8 = \frac{8}{1} \\ \frac{1}{7} \times \frac{8}{1} \end{aligned}$

Then, multiply the fractions.
::然后,乘以分数。

$\begin{aligned} \frac{1 \times 8}{7 \times 1} = \frac{8}{7} \end{aligned}$

Next, simplify the product. Convert the improper fraction to a mixed number.
::接下来,简化产品。将不当的分数转换成混合数。

$\begin{aligned} \frac{8}{7} = 1 \frac{1}{7} \end{aligned}$

The product is $\begin{aligned} 1 \frac{1}{7} \end{aligned}$ .

::产品是1,17。

Review
::回顾

Find the product. Answer in simplest form.
::找到产品,回答最简单

$\begin{aligned} 6 \times \frac{1}{2} = \underline{} \end{aligned}$

$\begin{aligned} 19 \times \frac{1}{2} = \underline{} \end{aligned}$

$\begin{aligned} 26 \times \frac{1}{4} = \underline{} \end{aligned}$

$\begin{aligned} 8 \times \frac{2}{4} = \underline{} \end{aligned}$

$\begin{aligned} 10 \times \frac{2}{3} = \underline{} \end{aligned}$

$\begin{aligned} 6 \times \frac{2}{3} = \underline{} \end{aligned}$

$\begin{aligned} 5 \times \frac{2}{5} = \underline{} \end{aligned}$

$\begin{aligned} \frac{1}{2} \times 9 = \underline{} \end{aligned}$

$\begin{aligned} \frac{2}{7} \times 9 = \underline{} \end{aligned}$

$\begin{aligned} \frac{1}{3} \times 7 = \underline{} \end{aligned}$

$\begin{aligned} \frac{3}{4} \times 10 = \underline{} \end{aligned}$

$\begin{aligned} \frac{3}{4} \times 12 = \underline{} \end{aligned}$

$\begin{aligned} \frac{3}{5} \times 12 = \underline{} \end{aligned}$

$\begin{aligned} \frac{1}{9} \times 63 = \underline{} \end{aligned}$

$\begin{aligned} \frac{1}{2} o f 11 = \underline{} \end{aligned}$
::1 2o f 11 = _

$\begin{aligned} \frac{1}{2} o f 24 = \underline{} \end{aligned}$
::1 2 o f 24 = _

$\begin{aligned} \frac{1}{4} o f 45 = \underline{} \end{aligned}$
::1 4 o f 45 = _

$\begin{aligned} \frac{2}{5} o f 35 = \underline{} \end{aligned}$
::2 5 o f 35 = _

Review (Answers)
::回顾(答复)

章节大纲

常规

分数和整数的乘法

Multiplying Fractions and Whole Numbers ::乘数分数和整数

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Multiplying Fractions and Whole Numbers
::乘数分数和整数

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)