Course: 3.10带分数积 | The Way To Learn

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带分数积
Cynthia just got the recipe for her grandmother’s famous Italian bread. The recipe makes 3 loaves of bread, which is more than Cynthia wants right now. Cynthia only wants to make 2 loaves of bread so she is going to need $\begin{aligned} \frac{2}{3} \end{aligned}$ of all of the ingredients listed in the recipe. The recipe calls for $\begin{aligned} 4 \frac{1}{2} \end{aligned}$ tablespoons of olive oil along with 7 other ingredients. How can Cynthia figure out how much olive oil she should use to make only 2 loaves of bread?
::辛西娅刚刚得到了她祖母著名的意大利面包的食谱。食谱制造了三块面包,这比辛西娅现在想要的要多得多。辛西娅只想要两块面包,所以她需要食谱中列出的所有成分中的23个。食谱需要412个橄榄油和7个其他成分。辛西娅如何知道她应该用多少橄榄油来做两块面包呢?

In this concept, you will learn how to multiply fractions and mixed numbers.
::在这个概念中,你会学会如何乘以分数和混合数字。

Multiplying Fractions and Mixed Numbers
::乘数和混合数字

Recall that in a fraction, the number written above the bar is the numerator and the number written below the bar is the denominator . When multiplying fractions, you will multiply the numerators and multiply the denominators.
::回想到,在一个分数中,在横条上方写的数字是分子,在横条下写的数字是分母。在乘数分数时,将乘数乘数,再乘数分母。

To multiply fractions:
::乘以分数 :

Multiply the numerators to find the numerator of your answer. Multiply the denominators to find the denominator of your answer.
::乘乘数以找到您答案的分子。乘数分母以找到您答案的分母。

Simplify your answer. Rewrite as a mixed number if necessary.
::简化您的答复。如有必要, 重写为混合数。

Here is an example.
::举一个例子。

Find the product of $\begin{aligned} \frac{1}{2} \cdot \frac{4}{5} \end{aligned}$ .
::查找1 2 4 5 的产物。

First, multiply the numerators and multiply the denominators. Note that you do not need to find a common denominator first!
::首先, 乘乘数数并乘数分母。请注意, 您不需要先找到共同的分母 !

$\begin{aligned} \frac{1}{2} \cdot \frac{4}{5} = \frac{1 \cdot 4}{2 \cdot 5} = \frac{4}{10} \end{aligned}$

Next, simplify your answer. The of 4 and 10 is 2, so divide by 4 and 10 by 2 to simplify the fraction.
::接下来,简化你的回答。4和10是2, 所以除以4和10乘以2来简化分数。

$\begin{aligned} \frac{4}{10} = \frac{4 \div 2}{10 \div 2} = \frac{2}{5} \end{aligned}$

The answer is $\begin{aligned} \frac{1}{2} \cdot \frac{4}{5} = \frac{2}{5} \end{aligned}$ .
::答案是 1 2 4 5 = 2 5 。

You can also multiply fractions and whole numbers . You will first need to turn the whole number into a fraction by writing it over 1.
::您也可以乘以分数和整数。您首先需要将整数以 1 的写法将整数转换成一个分数。

Here is an example.
::举一个例子。

Find the product of $\begin{aligned} 5 \cdot \frac{1}{2} \end{aligned}$ .
::查找 5 1 2 的产物。

First, rewrite 5 as $\begin{aligned} \frac{5}{1} \end{aligned}$ .
::首先,将5改写为5 1。

$\begin{aligned} \frac{5}{1} \cdot \frac{1}{2} \end{aligned}$

Next, multiply the numerators and multiply the denominators.
::接下去,乘乘乘乘数数,乘乘乘分母。

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

Now, simplify your answer. You can rewrite $\begin{aligned} \frac{5}{2} \end{aligned}$ as a mixed number.
::现在,简化你的回答。你可以重写52,作为一个混合数字。

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

The answer is $\begin{aligned} 5 \cdot \frac{1}{2} = 2 \frac{1}{2} \end{aligned}$ .
::答案是 5 1 2 = 2 2 。

You can also multiply mixed numbers. Because mixed numbers involve both wholes and parts, multiplying mixed numbers requires an extra step. Before multiplying mixed numbers, you will convert them to improper fractions .
::您也可以乘以混合数字。由于混合数字涉及整数和部件,混合数字的乘法需要额外的一步。在乘以混合数字之前,您将把混合数字转换成不适当的分数。

To multiply mixed numbers:
::乘以混合数字:

Convert the mixed numbers to improper fractions.
::将混合数字转换成不适当的分数。

Multiply the numerators to find the numerator of your answer. Multiply the denominators to find the denominator of your answer.
::乘乘数以找到您答案的分子。乘数分母以找到您答案的分母。

Simplify your answer. Rewrite as a mixed number if necessary.
::简化您的答复。如有必要, 重写为混合数。

Here is an example.
::举一个例子。

Find the product of $\begin{aligned} 3 \frac{1}{2} \cdot 2 \frac{1}{3} \end{aligned}$ .
::查找3,1,2,2,2,1,3的产物。

First, rewrite both mixed numbers as improper fractions.
::首先,改写两个混合数字,作为不适当的分数。

$\begin{aligned} 3 \frac{1}{2} = \frac{(2 \cdot 3) + 1}{2} = \frac{7}{2} \\ 2 \frac{1}{3} = \frac{(3 \cdot 2) + 1}{3} = \frac{7}{3} \end{aligned}$

Next, multiply the numerators and multiply the denominators.
::接下去,乘乘乘乘数数,乘乘乘分母。

$\begin{aligned} \frac{7}{2} \cdot \frac{7}{3} = \frac{49}{6} \end{aligned}$

Now, simplify your answer. You can rewrite $\begin{aligned} \frac{49}{6} \end{aligned}$ as a mixed number.
::现在,简化您的答案。您可以重写 49 6 作为混合数字。

$\begin{aligned} \frac{49}{6} = 8 \frac{1}{6} \end{aligned}$

The answer is $\begin{aligned} 3 \frac{1}{2} \cdot 2 \frac{1}{3} = 8 \frac{1}{6} \end{aligned}$ .
::答案是3 1 2 2 1 3 = 8 1 6 。

Sometimes when you multiply fractions or mixed numbers you will end up with very large numbers. To avoid having to deal with such big numbers, you can simplify before multiplying. To do this, look for common factors and divide them out.
::有时,当您乘以分数或混合数字时,你最终会以非常大的数字结束。为了避免不得不处理如此大的数字,你可以在乘法之前简化。要做到这一点,可以寻找共同的因素并将其分开。

Here is an example.
::举一个例子。

Find the product of $\begin{aligned} \frac{2}{9} \cdot \frac{18}{30} \end{aligned}$ .
::查找2 9 18 30的产物。

First, write the numerator and the denominator as a product.
::首先,将分子和分母写成产品。

$\begin{aligned} \frac{2}{9} \cdot \frac{18}{30} = \frac{2 \cdot 18}{9 \cdot 30} \end{aligned}$

Next, look for common factors along the diagonals in the numerator and the denominator. First look at 2 and 30. Both 2 and 30 have a factor of 2. Divide both 2 and 30 by 2.
::接下来,在分子和分母的对角线上寻找共同因素。首先看2和30,2和30都有2的系数,将2和30除以2除以2。

$\begin{aligned} \frac{2 \cdot 18}{9 \cdot 30} = \frac{1 \cdot 18}{9 \cdot 15} \end{aligned}$

Now, look at 9 and 18. Both 9 and 18 have a factor of 9. Divide both 9 and 18 by 9.
::现在,看看9和18,9和18都有9.9的因数,9和18除以9除以9和18除以9。

$\begin{aligned} \frac{1 \cdot 18}{9 \cdot 15} = \frac{1 \cdot 2}{1 \cdot 15} \end{aligned}$

Now, multiply the numbers in the numerator and the numbers in the denominator.
::现在,乘以分子中的数字和分母中的数字。

$\begin{aligned} \frac{1 \cdot 2}{1 \cdot 15} = \frac{2}{15} \end{aligned}$

The answer is $\begin{aligned} \frac{2}{9} \cdot \frac{18}{30} = \frac{2}{15} \end{aligned}$ .
::答案是 2 9 18 30 = 2 15 。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Cynthia and her Italian bread.
::早些时候,你得到一个问题关于辛西娅和她的意大利面包。

Her recipe is for 3 loaves but she only wants to make 2 loaves. Her plan is to use $\begin{aligned} \frac{2}{3} \end{aligned}$ of all the ingredients listed in the recipe. The recipe calls for $\begin{aligned} 4 \frac{1}{2} \end{aligned}$ tablespoons of olive oil. Cynthia wants to figure out how much olive oil she should use to make only 2 loaves of bread.
::她的食谱是3个面包,但她只想做2个面包。她的计划是使用食谱中列出的所有成分中的23个。食谱要求412个橄榄油汤匙。辛西娅想弄清楚她应该用多少橄榄油来做2个面包。

Cynthia needs to find $\begin{aligned} \frac{2}{3} \end{aligned}$ of $\begin{aligned} 4 \frac{1}{2} \end{aligned}$ . She will need to multiply.
::Cynthia需要找到412中的23

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

First, Cynthia should rewrite the mixed number as an improper fraction.
::首先,辛西娅应该把混合数字改写为不适当的分数。

$\begin{aligned} 4 \frac{1}{2} = \frac{9}{2} \end{aligned}$

Next, she should multiply the numerators and multiply the denominators.
::接下来,她应该乘以分子数,乘以分母数。

$\begin{aligned} \frac{2}{3} \cdot \frac{9}{2} = \frac{18}{6} \end{aligned}$

Now, she should simplify her answer.
::现在,她应该简化她的答案

$\begin{aligned} \frac{18}{6} = 3 \end{aligned}$

The answer is Cynthia should use 3 tablespoons of olive oil to make 2 loaves of bread.
::答案是辛西娅应该用三勺橄榄油做两片面包

Example 2
::例2

Deirdre claims that it takes her $\begin{aligned} 6 \frac{3}{4} \end{aligned}$ hours to complete her homework every night. Carlos thinks he can finish his homework in $\begin{aligned} \frac{2}{3} \end{aligned}$ that time. How long does Carlos think it will take him to complete his homework?
::Deirdre声称,她每晚要花6、3、4个小时才能完成功课。 Carlos认为他可以在23小时内完成功课。 Carlos认为他要花多久才能完成功课?

To figure out how long Carlos thinks it will take him to complete his homework, you need to find $\begin{aligned} \frac{2}{3} \end{aligned}$ of $\begin{aligned} 6 \frac{3}{4} \end{aligned}$ . The word “of” indicates multiplication, so you will need to multiply.
::要弄清楚卡洛斯认为他要花多久才能完成功课,你就需要在6 3 4中找到2 3。“of”一词表示乘法,所以你需要乘法。

$\begin{aligned} \frac{2}{3} \cdot 6 \frac{3}{4} \end{aligned}$

First, rewrite the mixed number as an improper fraction.
::首先,重写混合数字,作为不适当的部分。

$\begin{aligned} 6 \frac{3}{4} = \frac{27}{4} \end{aligned}$

Next, multiply the numerators and multiply the denominators.
::接下去,乘乘乘乘数数,乘乘乘分母。

$\begin{aligned} \frac{2}{3} \cdot \frac{27}{4} = \frac{54}{12} \end{aligned}$

Now, simplify your answer.
::现在,简化你的答案。

$\begin{aligned} \frac{54}{12} = 4 \frac{6}{12} = 4 \frac{1}{2} \end{aligned}$

The answer is Carlos thinks he can complete his homework in $\begin{aligned} 4 \frac{1}{2} \end{aligned}$ hours.
::答案是卡洛斯认为他可以在412小时内完成功课

Example 3
::例3

Find the product of $\begin{aligned} \frac{1}{3} \cdot \frac{5}{6} \end{aligned}$ .
::查找1,3,5,6的产物。

First, multiply the numerators and multiply the denominators.
::首先,乘以分子数,乘以分母数。

$\begin{aligned} \frac{1}{3} \cdot \frac{5}{6} = \frac{1 \cdot 5}{3 \cdot 6} = \frac{5}{18} \end{aligned}$

Next, check to see if you need to simplify your answer. In this case, $\begin{aligned} \frac{5}{18} \end{aligned}$ is in simplest form .
::下一步, 请检查是否需要简化答案。在此情况下, 5 18 是最简单的。

The answer is $\begin{aligned} \frac{1}{3} \cdot \frac{5}{6} = \frac{5}{18} \end{aligned}$ .
::答案是 1 3 5 6 = 5 18 。

Example 4
::例4

Find the product of $\begin{aligned} \frac{18}{20} \cdot \frac{4}{9} \end{aligned}$ .
::查找18 20 4 9 的产物。

First, write the numerator and the denominator as a product.
::首先,将分子和分母写成产品。

$\begin{aligned} \frac{18}{20} \cdot \frac{4}{9} = \frac{18 \cdot 4}{20 \cdot 9} \end{aligned}$

Next, look for common factors along the diagonals in the numerator and the denominator. First look at 18 and 9. Both 18 and 9 have a factor of 9. Divide both 18 and 9 by 9.
::接下来,在分子和分母的对角线上寻找共同因素。首先看18和9,18和9的系数为9,18和9除以18和9除以9除以9。

$\begin{aligned} \frac{18 \cdot 4}{20 \cdot 9} = \frac{2 \cdot 4}{20 \cdot 1} \end{aligned}$

Now, look at 4 and 20. Both 4 and 20 have a factor of 4. Divide both 4 and 20 by 4.
::现在,看看4和20, 4和20都有4的因数 4, 4和20除以4, 4和20除以4。

$\begin{aligned} \frac{2 \cdot 4}{20 \cdot 1} = \frac{2 \cdot 1}{5 \cdot 1} \end{aligned}$

Now, multiply the numbers in the numerator and the numbers in the denominator.
::现在,乘以分子中的数字和分母中的数字。

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

The answer is $\begin{aligned} \frac{18}{20} \cdot \frac{4}{9} = \frac{2}{5} \end{aligned}$ .
::答案是18 20 4 9 = 2 5 。

Example 5
::例5

Find the product of $\begin{aligned} 2 \frac{1}{5} \cdot 3 \frac{1}{2} \end{aligned}$ .
::查找2,1,5,5,3,1,2的产物。

First, rewrite both mixed numbers as improper fractions.
::首先,改写两个混合数字,作为不适当的分数。

$\begin{aligned} 2 \frac{1}{5} = \frac{11}{5} \\ 3 \frac{1}{2} = \frac{7}{2} \end{aligned}$

Next, multiply the numerators and multiply the denominators.
::接下去,乘乘乘乘数数,乘乘乘分母。

$\begin{aligned} \frac{11}{5} \cdot \frac{7}{2} = \frac{77}{10} \end{aligned}$

Now, simplify your answer. You can rewrite $\begin{aligned} \frac{77}{10} \end{aligned}$ as a mixed number.
::现在,简化你的回答。你可以重写77 10作为一个混合数字。

$\begin{aligned} \frac{77}{10} = 7 \frac{7}{10} \end{aligned}$

The answer is $\begin{aligned} 2 \frac{1}{5} \cdot 3 \frac{1}{2} = 7 \frac{7}{10} \end{aligned}$ .
::答案是 2 1 5 3 1 2 = 7 7 10 。

Review
::回顾

Multiply.
::乘数

$\begin{aligned} \frac{1}{4} \cdot \frac{3}{7} \end{aligned}$

$\begin{aligned} \frac{5}{6} \cdot \frac{2}{3} \end{aligned}$

$\begin{aligned} \frac{3}{10} \cdot \frac{10}{12} \end{aligned}$

$\begin{aligned} \frac{4}{7} \cdot \frac{2}{3} \end{aligned}$

$\begin{aligned} \frac{1}{3} \cdot 2 \frac{2}{3} \end{aligned}$

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

$\begin{aligned} 2 \frac{3}{10} \cdot 2 \frac{1}{4} \end{aligned}$

$\begin{aligned} 7 \frac{1}{5} \cdot \frac{1}{11} \end{aligned}$

$\begin{aligned} 4 \frac{5}{8} \cdot 2 \end{aligned}$

$\begin{aligned} \frac{1}{7} \cdot \frac{1}{6} \end{aligned}$

$\begin{aligned} 3 \frac{5}{6} \cdot 1 \frac{2}{3} \end{aligned}$

$\begin{aligned} \frac{1}{5} \cdot \frac{7}{12} \end{aligned}$

$\begin{aligned} \frac{2}{3} \cdot \frac{9}{12} \cdot \frac{6}{7} \end{aligned}$

$\begin{aligned} \frac{1}{3} \cdot 1 \frac{4}{5} \cdot \frac{3}{4} \end{aligned}$

$\begin{aligned} \frac{4}{9} \cdot \frac{5}{8} \cdot \frac{3}{7} \end{aligned}$

$\begin{aligned} \frac{10}{12} \cdot (3 \frac{1}{5} \cdot \frac{7}{10}) \end{aligned}$

Review (Answers)
::回顾(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

Section outline

General

带分数积

Multiplying Fractions and Mixed Numbers ::乘数和混合数字

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)

Resources ::资源

Multiplying Fractions and Mixed Numbers
::乘数和混合数字

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)

Resources
::资源