Course: 3.11混合数或分数积估计

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混合数或分数积估计
This year Michelle's family is hosting a big Thanksgiving dinner at her house. Michelle is in charge of making the mashed potatoes. Because they are having such a crowd over for dinner, Michelle's mom tells her she will need to make 4 times the recipe. Michelle looks at the recipe and notices it calls for $\begin{aligned} 1 \frac{3}{8} \end{aligned}$ pounds of potatoes. How can Michelle quickly approximate how many pounds of potatoes she will need to buy in order to make 4 times the recipe? Will a 5 pound bag of potatoes be enough?
::今年,米歇尔的家人正在她家举办一个大型的感恩节晚宴。米歇尔负责做土豆泥。米歇尔的妈妈告诉她,她需要做食谱的4倍。米歇尔看食谱,注意到它需要13.8磅土豆。米歇尔如何能快速地估计她需要买多少磅土豆才能做4倍食谱? 5磅土豆袋够不够吗?

In this concept, you will learn how to estimate the products of fractions and mixed numbers.
::在这个概念中,你将学会如何估计分数和混合数字的产物。

Estimating Products of Fractions and Mixed Numbers
::分数和混合数字的估算产品

You can estimate to find the approximate product of fractions and mixed numbers. Estimating can allow you to get a quick sense for how big or small a product is. It can also allow you to see if your exact answer is reasonable.
::估计可以找到分数和混合数字的近似产物。估计可以让您快速了解产品大小。也可以让您查看准确答案是否合理。

Just like when estimating sums and differences, you can use the fraction benchmarks of 0, $\begin{aligned} \frac{1}{2} \end{aligned}$ , and 1 to help you to estimate products.
::就像估算总和和差异时一样,你可以使用零、12和1的分数基准来帮助您估算产品。

Here are the steps for estimating the products of fractions and mixed numbers using benchmarks.
::以下是使用基准估计分数和混合数字产品的步骤。

Approximate the value of each fraction or mixed number using the benchmarks of 0, $\begin{aligned} \frac{1}{2} \end{aligned}$ , and 1.
::使用0、1、2和1的基准,接近每个分数或混合数的数值。

Multiply the approximated values to get an estimated product. If necessary, convert mixed numbers to improper fractions first.
::乘以大致值以获得估计产品。如有必要, 先将混合数字转换为不适当的分数。

Here is an example.
::举一个例子。

Estimate the product of $\begin{aligned} \frac{19}{20} \cdot \frac{6}{7} \end{aligned}$ .
::估计产品为19 20 6 7。

First, approximate the value of each individual fraction.
::首先,大致相当于每一分数的数值。

$\begin{aligned} \frac{19}{20} \end{aligned}$ is approximately 1
::19个20个大概是1个

$\begin{aligned} \frac{6}{7} \end{aligned}$ is approximately 1
::6 7 约 1 , 6 7 约 1

Next, multiply the approximated values.
::下一步,乘以接近值。

$\begin{aligned} 1 \cdot 1 = 1 \end{aligned}$

The answer is $\begin{aligned} \frac{19}{20} \cdot \frac{6}{7} \end{aligned}$ is approximately 1.
::答案是19 20 6 7 大约是 1。

Here is another example that involves mixed numbers.
::下面是涉及混合数字的另一个例子。

Estimate the product of $\begin{aligned} 3 \frac{6}{8} \cdot 5 \frac{1}{10} \end{aligned}$ .
::估计产品为3 6 8 5 1 10。

First, approximate the value of each individual mixed number.
::首先,大约每个混杂数字的价值。

$\begin{aligned} 3 \frac{6}{8} \end{aligned}$ is approximately 4
::3 6 8 大约是 4

$\begin{aligned} 5 \frac{1}{10} \end{aligned}$ is approximately 5
::5 1 10 大约 5

Next, multiply the approximated values.
::下一步,乘以接近值。

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

The answer is $\begin{aligned} 3 \frac{6}{8} \cdot 5 \frac{1}{10} \end{aligned}$ is approximately 20.
::答案是3 6 8 5 1 10 大约是20。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Michelle and her upcoming Thanksgiving dinner.
::早些时候,你得到一个问题关于米歇尔和她即将到来的感恩节晚宴。

Michelle is in charge of making the mashed potatoes. The original recipe calls for $\begin{aligned} 1 \frac{3}{8} \end{aligned}$ pounds of potatoes, but Michelle is supposed to make 4 times the recipe in order to have enough potatoes for everyone. Michelle wants to estimate how many pounds of potatoes she will need and wants to know if a 5 pound bag of potatoes will be enough.
::米歇尔负责做土豆泥。最初的食谱要求13.8磅土豆,但米歇尔应该做4倍的食谱,以便每个人都有足够的土豆。米歇尔想估计她需要多少磅土豆,并想知道一袋5磅土豆是否足够。

Michelle is trying to estimate the product of $\begin{aligned} 1 \frac{3}{8} \cdot 4 \end{aligned}$ .
::Michelle正试图估算1,3,8,4的产物。

First, she should approximate the value of the mixed number and convert it to an improper fraction.
::首先,她应接近混合数字的价值,将其转换成不适当的部分。

$\begin{aligned} 1 \frac{3}{8} \end{aligned}$ is approximately $\begin{aligned} 1 \frac{1}{2} = \frac{3}{2} \end{aligned}$
::1 3 8 大约 1 1 2 = 3 2

Next, she should multiply the approximated improper fraction by 4. Remember that 4 can be rewritten as $\begin{aligned} \frac{4}{1} \end{aligned}$ .
::接下来,她应该把大约不适当的部分乘以4. 记住4可以改写为4 1。

$\begin{aligned} \frac{3}{2} \cdot \frac{4}{1} = \frac{12}{2} = 6 \end{aligned}$

The answer is that Michelle will need approximately 6 pounds of potatoes. A 5 pound bag of potatoes will not be enough.
::答案是米歇尔需要大约6磅土豆一袋5磅土豆还不够

Example 2
::例2

Estimate the product of $\begin{aligned} 4 \frac{5}{6} \times 2 \frac{1}{9} \end{aligned}$ .
::估计产值为 4 5 6 6 × 2 1 9 。

First, approximate the value of each individual mixed number.
::首先,大约每个混杂数字的价值。

$\begin{aligned} 4 \frac{5}{6} \end{aligned}$ is approximately 5
::4 5 6 大约5 5 6

$\begin{aligned} 2 \frac{1}{9} \end{aligned}$ is approximately 2
::2 9 大约2 9

Next, multiply the approximated values.
::下一步,乘以接近值。

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

The answer is $\begin{aligned} 4 \frac{5}{6} \times 2 \frac{1}{9} \end{aligned}$ is approximately 10.
::答案是 4 5 6 6 × 2 1 9 大约是 10。

Example 3
::例3

Estimate the product of $\begin{aligned} \frac{7}{12} \cdot \frac{10}{11} \end{aligned}$ .
::估计产品为7 12 10 11。

First, approximate the value of each individual fraction.
::首先,大致相当于每一分数的数值。

$\begin{aligned} \frac{7}{12} \end{aligned}$ is approximately $\begin{aligned} \frac{1}{2} \end{aligned}$
::7 7 12 大约是 1 2

$\begin{aligned} \frac{10}{11} \end{aligned}$ is approximately 1
::10个是11个大约1个

Next, multiply the approximated values.
::下一步,乘以接近值。

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

The answer is $\begin{aligned} \frac{7}{12} \cdot \frac{10}{11} \end{aligned}$ is approximately $\begin{aligned} \frac{1}{2} \end{aligned}$ .
::答案是7 12 10 11 大约是12。

Example 4
::例4

Estimate the product of $\begin{aligned} \frac{9}{10} \cdot \frac{5}{6} \end{aligned}$ .
::估计产品为9 10 5 6。

First, approximate the value of each individual fraction.
::首先,大致相当于每一分数的数值。

$\begin{aligned} \frac{9}{10} \end{aligned}$ is approximately 1
::9 10 大约是 1

$\begin{aligned} \frac{5}{6} \end{aligned}$ is approximately 1
::5,6是约1,5,6是约1

Next, multiply the approximated values.
::下一步,乘以接近值。

$\begin{aligned} 1 \cdot 1 = 1 \end{aligned}$

The answer is $\begin{aligned} \frac{9}{10} \cdot \frac{5}{6} \end{aligned}$ is approximately 1.
::答案是 9 10 5 6 大约是 1 。

Example 5
::例5

Estimate the product of $\begin{aligned} 4 \frac{1}{8} \cdot 2 \frac{11}{13} \end{aligned}$ .
::估计产品为4 1 8 2 11 13。

First, approximate the value of each individual mixed number.
::首先,大约每个混杂数字的价值。

$\begin{aligned} 4 \frac{1}{8} \end{aligned}$ is approximately 4
::4,1,8,大约4,4,8,大约4

$\begin{aligned} 2 \frac{11}{13} \end{aligned}$ is approximately 3
::2 11 13 约 3

Next, multiply the approximated values.
::下一步,乘以接近值。

$\begin{aligned} 4 \cdot 3 = 12 \end{aligned}$

The answer is $\begin{aligned} 4 \frac{1}{8} \cdot 2 \frac{11}{13} \end{aligned}$ is approximately 12.
::答案是 4 1 8 2 11 13 大约是 12。

Review
::回顾

Estimate each product.
::估计每种产品。

$\begin{aligned} \frac{1}{9} \cdot \frac{4}{5} \end{aligned}$

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

$\begin{aligned} \frac{18}{37} \cdot \frac{10}{11} \end{aligned}$

$\begin{aligned} \frac{13}{15} \cdot \frac{4}{9} \end{aligned}$

$\begin{aligned} 6 \frac{2}{3} \cdot 2 \frac{6}{11} \end{aligned}$

$\begin{aligned} 5 \frac{27}{29} \cdot 3 \frac{1}{18} \end{aligned}$

$\begin{aligned} 4 \frac{6}{7} \cdot 1 \frac{4}{7} \end{aligned}$

$\begin{aligned} 4 \frac{15}{16} \cdot 7 \frac{2}{21} \end{aligned}$

$\begin{aligned} 3 \frac{3}{8} \cdot 7 \frac{1}{15} \end{aligned}$

$\begin{aligned} 3 \frac{5}{6} \cdot 1 \frac{1}{21} \end{aligned}$

$\begin{aligned} 6 \frac{15}{16} \cdot 7 \frac{1}{9} \end{aligned}$

$\begin{aligned} 5 \frac{11}{12} \cdot 3 \frac{12}{80} \end{aligned}$

$\begin{aligned} 4 \frac{18}{60} \cdot 7 \frac{1}{21} \end{aligned}$

$\begin{aligned} 13 \frac{4}{5} \cdot 6 \frac{1}{10} \end{aligned}$

$\begin{aligned} 15 \frac{11}{13} \cdot 11 \frac{9}{10} \end{aligned}$

Review (Answers)
::回顾(答复)

Section outline

General

混合数或分数积估计

Estimating Products of Fractions and Mixed Numbers ::分数和混合数字的估算产品

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)

Estimating Products of Fractions and Mixed Numbers
::分数和混合数字的估算产品

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)