节: 混合数或分数的商估计 | 3.14混合数或分数的商估计

章节大纲

Lucy's mom is helping to sew leotards for Lucy's dance team. She has $\begin{aligned} 8 \frac{1}{5} \end{aligned}$ yards of the main leotard fabric. Lucy looks at the pattern for the leotards and notices that her mom will need $\begin{aligned} \frac{5}{6} \end{aligned}$ yard of fabric for each leotard. How can Lucy estimate the number of leotards that her mom will be able to make with the fabric she has?
::Lucy的妈妈正在帮助Lucy的舞蹈团队缝制皮鞋。她有85码的皮鞋主要布料。 Lucy查看了皮鞋的型式,并发现她妈妈需要5-6码的布料来供每只皮鞋。Lucy如何估计她妈妈能够用她的布料制造的皮鞋数量?

In this concept, you will learn to estimate quotients of fractions and mixed numbers.
::在这个概念中,你将学会估计分数和混合数字的商数。

Estimating Quotients of Fractions and Mixed Numbers
::分数和混合数字估计数字

People use estimation all the time in their daily lives. Estimation is often used before determining an exact value, or when an exact value is not necessary. You can estimate to find the approximate quotient of fractions and mixed numbers.
::人们在日常生活中使用了所有时间的估算。在确定准确值之前,或者在不需要准确值时,往往使用估算法。您可以估算出分数和混合数字的大致商数。

Estimating quotients of fractions is similar to estimating products of fractions. To estimate the quotient of two fractions, you will first rewrite the problem as a multiplication problem. Then, you will use the fraction benchmarks of 0, $\begin{aligned} \frac{1}{2} \end{aligned}$ and 1 to estimate the answer.
::分数的估测商数与估计分数的产值相似。要估算两个分数的商数,您将首先将问题重写为乘法问题。然后,您将使用0、1、2和1的分数基准来估计答案。

Here are the steps for estimating the quotients of fractions using benchmarks.
::以下是使用基准估算分数商数的步骤。

Rewrite the division problem as a multiplication problem by finding the reciprocal of the divisor .
::将分裂问题重写为乘法问题,

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 the estimated quotient.
::乘以大致值以获得估计商数。

Here is an example.
::举一个例子。

Estimate $\begin{aligned} \frac{7}{8} \div \frac{1}{3} \end{aligned}$ .
::估计数 7 8 + 1 3

First, note that the reciprocal of the divisor, $\begin{aligned} \frac{1}{3} \end{aligned}$ , is $\begin{aligned} \frac{3}{1} \end{aligned}$ . Next, rewrite the division problem as a multiplication problem by multiplying the dividend by the reciprocal of the divisor.
::首先,请注意,divisor的对等( 13 ) 是 3 1 。其次, 将分割问题改写成乘法问题, 将红利乘以divisor的对等。

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

Now, approximate the value of each individual fraction.
::现在,大约每个分数的值。

$\begin{aligned} \frac{7}{8} \end{aligned}$ is approximately 1
::7 8 大约是 1

$\begin{aligned} \frac{3}{1} \end{aligned}$ is equal to 3
::1等于3 1等于3

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

$\begin{aligned} 1 \cdot 3 = 3 \end{aligned}$

The answer is $\begin{aligned} \frac{7}{8} \div \frac{1}{3} \end{aligned}$ is approximately 3.
::答案是7 8 1 3 大约是3 。

The answer of 3 makes sense if you think about what the division problem is asking. $\begin{aligned} \frac{7}{8} \div \frac{1}{3} \end{aligned}$ means how many thirds can fit in $\begin{aligned} \frac{7}{8} \end{aligned}$ . Since $\begin{aligned} \frac{7}{8} \end{aligned}$ is almost 1, approximately 3 thirds make up $\begin{aligned} \frac{7}{8} \end{aligned}$ .
::3的答案是有道理的,如果你想想分裂问题正在问什么的话。 7 8 13 意味着有多少三分之一可以适应 7 8。因为7 8 几乎是 1, 大约是 3 /3 等于 7 8 。

You can also estimate quotients of mixed numbers. The steps are similar, but you will be approximating the value of each mixed number using the fraction benchmarks first.
::您也可以估计混合数字的商数。步骤相似,但您将首先使用分数基准,接近每个混合数字的价值。

Here are the steps for estimating the quotients of mixed numbers using benchmarks.
::以下是使用基准估算混合数字商数的步骤。

Approximate the value of each 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,与每个混合数字的数值相近。

Convert the approximated mixed numbers to improper fractions if necessary.
::必要时将混合数字转换为不适当的部分。

Rewrite the division problem as a multiplication problem by finding the reciprocal of the divisor.
::将分裂问题重写为乘法问题,

Multiply the approximated values to get the estimated quotient.
::乘以大致值以获得估计商数。

Here is an example.
::举一个例子。

Estimate $\begin{aligned} 2 \frac{4}{11} \div 1 \frac{7}{8} \end{aligned}$ .
::估计数 2 4 11 1 7 8

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

$\begin{aligned} 2 \frac{4}{11} \end{aligned}$ is approximately $\begin{aligned} 2 \frac{1}{2} \end{aligned}$
::2 4 11 大约是 2 1 2

$\begin{aligned} 1 \frac{7}{8} \end{aligned}$ is approximately 2
::1 7 8 大约是 2

Next, convert the approximated mixed numbers to improper fractions.
::接下来,将大约混合数转换成不适当的分数。

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

$\begin{aligned} 2 = 2 \end{aligned}$

Now, rewrite the division problem as a multiplication problem.
::现在,将分区问题重写为乘法问题。

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

Finally, multiply the approximated values.
::最后,乘以近似值。

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

The answer is $\begin{aligned} 2 \frac{4}{11} \div 1 \frac{7}{8} \end{aligned}$ is approximately $\begin{aligned} \frac{5}{4} \end{aligned}$ or $\begin{aligned} 1 \frac{1}{4} \end{aligned}$ .
::答案是 2 4 11 1 7 8 大约 5 4 或 1 14 。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Lucy's mom, who is helping to sew leotards for Lucy's dance team.
::之前,你得到一个问题关于露西的妈妈, 谁在帮助缝合皮带为露西的舞蹈团队。

Lucy's mom has $\begin{aligned} 8 \frac{1}{5} \end{aligned}$ yards of the main leotard fabric and needs $\begin{aligned} \frac{5}{6} \end{aligned}$ yard of this fabric for each leotard. Lucy wants to estimate how many leotards her mom will be able to make.
::Lucy的妈妈有815码的皮圈织物需要56码的这个织物

First, Lucy should approximate the value of each individual mixed number and fraction.
::首先,露西应该接近每个个别混合数和分数的价值。

$\begin{aligned} 8 \frac{1}{5} \end{aligned}$ is approximately 8
::8 1 5 大约 8

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

Because the approximated values are both whole numbers , now Lucy can just divide the whole numbers.
::因为接近的数值都是整数现在露西可以把整数分开了

$\begin{aligned} 8 \div 1 = 8 \end{aligned}$

The answer is Lucy's mom will be able to sew approximately 8 leotards with the fabric she has.
::答案是露西的妈妈能够用她拥有的织物缝合大约8只皮带

Example 2
::例2

Estimate $\begin{aligned} 7 \frac{7}{8} \div 4 \frac{5}{9} \end{aligned}$ .
::估计数 7 7 8 4 5 9

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

$\begin{aligned} 7 \frac{7}{8} \end{aligned}$ is approximately 8
::7 7 8 大约是 8

$\begin{aligned} 4 \frac{5}{9} \end{aligned}$ is approximately $\begin{aligned} 4 \frac{1}{2} \end{aligned}$
::4 5 9 大约是 4 5 9 大约是 4 1 2

Next, convert the approximated mixed numbers to improper fractions.
::接下来,将大约混合数转换成不适当的分数。

$\begin{aligned} 8 = 8 \end{aligned}$

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

Now, rewrite the division problem as a multiplication problem.
::现在,将分区问题重写为乘法问题。

$\begin{aligned} 8 \div \frac{9}{2} = 8 \cdot \frac{2}{9} \end{aligned}$

Finally, multiply the approximated values.
::最后,乘以近似值。

$\begin{aligned} \frac{8}{1} \cdot \frac{2}{9} = \frac{16}{9} \end{aligned}$

The answer is $\begin{aligned} 7 \frac{7}{8} \div 4 \frac{5}{9} \end{aligned}$ is approximately $\begin{aligned} \frac{16}{9} \end{aligned}$ or $\begin{aligned} 1 \frac{7}{9} \end{aligned}$ .
::答案是7 7 8 4 5 9 大约是 16 9 或 1 7 9 。

Example 3
::例3

Estimate $\begin{aligned} \frac{9}{10} \div \frac{1}{13} \end{aligned}$ .
::估计数 9 10 + 1 13。

First, note that the reciprocal of the divisor, $\begin{aligned} \frac{1}{13} \end{aligned}$ , is $\begin{aligned} \frac{13}{1} \end{aligned}$ . Next, rewrite the division problem as a multiplication problem by multiplying the dividend by the reciprocal of the divisor.
::首先,请注意, 13, 13, 13, 13, 13, 13。其次, 将分割问题改写成乘法问题, 将红利乘以 disvisor 的对等。

$\begin{aligned} \frac{9}{10} \cdot \frac{13}{1} \end{aligned}$

Now, approximate the value of each individual fraction.
::现在,大约每个分数的值。

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

$\begin{aligned} \frac{13}{1} \end{aligned}$ is equal to 13
::13 1 等于 13

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

$\begin{aligned} 1 \cdot 13 = 13 \end{aligned}$

The answer is $\begin{aligned} \frac{9}{10} \div \frac{1}{13} \end{aligned}$ is approximately 13.
::答案是 9 10 1 13 大约是 13 。

Example 4
::例4

Estimate $\begin{aligned} 5 \frac{5}{6} \div 2 \frac{3}{4} \end{aligned}$ .
::估计数 5 5 6 6 2 3 4

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

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

$\begin{aligned} 2 \frac{3}{4} \end{aligned}$ is approximately 3
::2 3 4 大约3 3 4

Because the approximated values are both whole numbers, now you can just divide the whole numbers.
::因为大约的数值都是整数, 现在你可以把整数分开了。

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

The answer is $\begin{aligned} 5 \frac{5}{6} \div 2 \frac{3}{4} \end{aligned}$ is approximately 2.
::答案是5 5 6 2 3 4 大约2。

Example 5
::例5

Estimate $\begin{aligned} \frac{8}{9} \div \frac{1}{2} \end{aligned}$ .
::估计数 8 9 + 1 2

First, note that the reciprocal of the divisor, $\begin{aligned} \frac{1}{2} \end{aligned}$ , is $\begin{aligned} \frac{2}{1} \end{aligned}$ . Next, rewrite the division problem as a multiplication problem by multiplying the dividend by the reciprocal of the divisor.
::首先,请注意,二等分机的对等为 2 1 。其次,将分机问题改写成乘法问题,将股机的对等乘法乘以股机的对等法。

$\begin{aligned} \frac{8}{9} \cdot \frac{2}{1} \end{aligned}$

Now, approximate the value of each individual fraction.
::现在,大约每个分数的值。

$\begin{aligned} \frac{8}{9} \end{aligned}$ is approximately 1
::8个是9个大约1个

$\begin{aligned} \frac{2}{1} \end{aligned}$ is equal to 2
::二等二,二等二,二等二

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

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

The answer is $\begin{aligned} \frac{8}{9} \div \frac{1}{2} \end{aligned}$ is approximately 2.
::答案是 8 9 1 2 大约是 2 。

Review
::回顾

Estimate each quotient.
::估计每个商数。

$\begin{aligned} \frac{5}{6} \div \frac{30}{36} \end{aligned}$

$\begin{aligned} \frac{11}{12} \div \frac{6}{7} \end{aligned}$

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

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

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

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

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

$\begin{aligned} 9 \frac{1}{9} \div 2 \frac{4}{5} \end{aligned}$

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

$\begin{aligned} 8 \frac{2}{3} \div 3 \frac{1}{11} \end{aligned}$

$\begin{aligned} 9 \frac{2}{3} \div 2 \frac{1}{15} \end{aligned}$

$\begin{aligned} 16 \frac{1}{13} \div 2 \frac{1}{9} \end{aligned}$

$\begin{aligned} 26 \frac{1}{4} \div 13 \frac{2}{30} \end{aligned}$

$\begin{aligned} 44 \frac{3}{4} \div 15 \frac{1}{20} \end{aligned}$

$\begin{aligned} 26 \frac{1}{13} \div 2 \frac{1}{18} \end{aligned}$

Review (Answers)
::回顾(答复)

章节大纲

Estimating Quotients of Fractions and Mixed Numbers ::分数和混合数字估计数字

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)

Estimating Quotients of Fractions and Mixed Numbers
::分数和混合数字估计数字

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)