Section: 混合数与分数和的估计 | 3.5混合数与分数和的估计

Section outline

Megan and her mom planted a vegetable garden this summer. They noticed animals getting into the garden so they decided to put fencing around the perimeter of the garden. The garden is a rectangular shape. Megan's mom told her that the length of the garden is $\begin{aligned} 5 \frac{1}{6} \end{aligned}$ feet and the width of the garden is $\begin{aligned} 7 \frac{7}{8} \end{aligned}$ feet. The fencing is $6 a foot. How can Megan approximate the perimeter of the garden in order to get a quick estimate of how much the fencing will cost?
::梅根和她的妈妈今年夏天种植了一个蔬菜花园,他们注意到动物进入花园,因此决定在花园四周设置栅栏,花园是长方形的。梅根的妈妈告诉她,花园长516英尺,花园宽778英尺。栅栏是6美元一英尺。梅根如何接近花园四周,以便快速估计围栏的成本?

In this concept, you will learn to estimate the sums of fractions and mixed numbers.
::在此概念中,您将学会估算分数和混合数字的总和。

Estimating Sums of Fractions and Mixed Numbers
::分数和混合数字的估算总和

Estimation is a method for finding an approximate solution to a problem. You can estimate a solution when you don't need an exact answer or when you want to check if your exact answer is reasonable.
::估计是找到问题近似解决办法的一种方法。当您不需要确切答案或想要检查您准确答案是否合理时, 您可以估计一个解决方案。

When working with fractions and mixed numbers, you can use the fraction benchmarks of $\begin{aligned} 0, \frac{1}{2}, \end{aligned}$ and $\begin{aligned} 1 \end{aligned}$ to help you to estimate sums.
::当使用分数和混合数字时,您可以使用 0、12和1 的分数基准来帮助您估算数字。

Here are the steps for estimating the sums of fractions and mixed numbers using benchmarks.
::以下是使用基准估算分数和混合数字的合计数的步骤。

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

Add the approximated values to get an estimated sum .
::加上大致数值以获得估计总和。

Here is an example.
::举一个例子。

Estimate the sum of $\begin{aligned} \frac{5}{9} + \frac{1}{77} \end{aligned}$ .
::估计总和为5 9+1 77。

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

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

$\begin{aligned} \frac{1}{77} \end{aligned}$ is approximately
::1 77 大概是77个

Next, add the approximated values.
::下加近似值。

$\begin{aligned} \frac{1}{2} + 0 = \frac{1}{2} \end{aligned}$

The answer is $\begin{aligned} \frac{5}{9} + \frac{1}{77} \end{aligned}$ is approximately $\begin{aligned} \frac{1}{2} \end{aligned}$ .
::答案是5 9+1 77大约是12。

Here is another example.
::下面是另一个例子。

Estimate the sum of $\begin{aligned} 3 \frac{6}{7} + 1 \frac{4}{9} \end{aligned}$ .
::估计数为3 6 7+1 4 9。

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

$\begin{aligned} 3 \frac{6}{7} \end{aligned}$ is approximately $\begin{aligned} 4 \end{aligned}$
::3 6 7 大约是 4

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

Next, add the approximated values.
::下加近似值。

$\begin{aligned} 4 + 1 \frac{1}{2} = 5 \frac{1}{2} \end{aligned}$
The answer is $\begin{aligned} 3 \frac{6}{7} + 1 \frac{4}{9} \end{aligned}$ is approximately $\begin{aligned} 5 \frac{1}{2} \end{aligned}$ .

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Megan and her vegetable garden.
::早些时候,你得到一个问题梅根和她的菜园。

Her garden is $\begin{aligned} 5 \frac{1}{6} \end{aligned}$ feet by $\begin{aligned} 7 \frac{7}{8} \end{aligned}$ feet. Megan wants to estimate the perimeter of the garden so she can estimate how much fencing she will need to buy to fence it in. The fencing is $6 a foot so Megan wants to use her perimeter estimate to estimate the cost of the fencing.
::梅根想估计花园的周界,以便她能估计需要购买多少围栏才能将其围成栅栏。栅栏是每英尺6美元,梅根想用其周界估计来估计围栏的费用。

First, Megan should write an expression for the perimeter of her garden. The garden is a rectangle so it has four sides. The perimeter is the sum of the lengths of the four sides.
::首先,梅根应该在花园周围写一个表达词。花园是一个矩形,它有四面。周界是四面的长度之和。

$\begin{aligned} Perimeter = 5 \frac{1}{6} + 7 \frac{7}{8} + 5 \frac{1}{6} + 7 \frac{7}{8} \end{aligned}$

Next, Megan should approximate the value of each individual mixed number.
::接下来,梅根应该接近每个混合数字的价值。

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

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

Next, add the approximated values.
::下加近似值。

$\begin{aligned} Perimeter \approx 5 + 8 + 5 + 8 = 26 feet \end{aligned}$

The perimeter of the garden is approximately 26 feet. If the fencing costs $6 per foot, it will cost approximately $\begin{aligned} 26 \times 6 = $ 156 \end{aligned}$ to fence in the garden.
::花园周边大约26英尺,如果围栏每英尺6美元,花园围栏大约为26×6=156美元。

The answer is the perimeter of Megan's garden is approximately 26 feet. It will cost approximately $156 to put a fence around it.
::答案是梅根花园的周围大约26英尺周围有一道栅栏

Example 2
::例2

Estimate the sum of $\begin{aligned} \frac{18}{20} + 5 \frac{9}{10} \end{aligned}$ .
::估计总数为18 20+ 5 9 10。

First, approximate the value of each individual fraction and mixed number.
::首先,大约每个个别部分和混合数字的价值。

$\begin{aligned} \frac{18}{20} \end{aligned}$ is approximately $\begin{aligned} 1 \end{aligned}$
::18 20 大约是 1

$\begin{aligned} 5 \frac{9}{10} \end{aligned}$ is approximately $\begin{aligned} 6 \end{aligned}$
::5 9 10 约 6

Next, add the approximated values.
::下加近似值。

$\begin{aligned} 1 + 6 = 7 \end{aligned}$
The answer is $\begin{aligned} \frac{18}{20} + 5 \frac{9}{10} \end{aligned}$ is approximately $\begin{aligned} 7 \end{aligned}$ .

Example 3
::例3

Estimate the sum of $\begin{aligned} \frac{6}{7} + \frac{1}{2} \end{aligned}$ .
::估计数总和为6 7+1 2。

First, approximate the value of each individual fraction. $\begin{aligned} \frac{1}{2} \end{aligned}$ is already equal to a benchmark value so it does not need to be approximated.
::首先,大致相当于每一分数的数值。 1 2 已经等于基准值,因此不需要接近。

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

$\begin{aligned} \frac{1}{2} \end{aligned}$ is equal to $\begin{aligned} \frac{1}{2} \end{aligned}$
::1 2 等于 1 2

Next, add the approximated values.
::下加近似值。

$\begin{aligned} 1 + \frac{1}{2} = 1 \frac{1}{2} \end{aligned}$
The answer is $\begin{aligned} \frac{6}{7} + \frac{1}{2} \end{aligned}$ is approximately $\begin{aligned} 1 \frac{1}{2} \end{aligned}$ .

Example 4
::例4

Estimate the sum of $\begin{aligned} \frac{29}{30} + 7 \frac{8}{10} \end{aligned}$ .
::估计数总额为29 30+ 7 8 10。

First, approximate the value of each individual fraction and mixed number.
::首先,大约每个个别部分和混合数字的价值。

$\begin{aligned} \frac{29}{30} \end{aligned}$ is approximately $\begin{aligned} 1 \end{aligned}$
::29 30 约为 1

$\begin{aligned} 7 \frac{8}{10} \end{aligned}$ is approximately $\begin{aligned} 8 \end{aligned}$
::7 8 10 大约 8

Next, add the approximated values.
::下加近似值。

$\begin{aligned} 1 + 8 = 9 \end{aligned}$
The answer is $\begin{aligned} \frac{29}{30} + 7 \frac{8}{10} \end{aligned}$ is approximately $\begin{aligned} 9 \end{aligned}$ .

Example 5
::例5

Estimate the sum of $\begin{aligned} 1 \frac{1}{2} + 3 \frac{5}{6} \end{aligned}$ .
::估计总和为1,1、2+3 5 6。

First, approximate the value of each individual mixed number. $\begin{aligned} 1 \frac{1}{2} \end{aligned}$ is already equal to a benchmark value so it does not need to be approximated.
::首先,接近每个混杂数的数值。 1,12,已经等于基准值,因此不需要接近。

$\begin{aligned} 1 \frac{1}{2} \end{aligned}$ is equal to $\begin{aligned} 1 \frac{1}{2} \end{aligned}$
::1 2 等于 1 1 2

$\begin{aligned} 3 \frac{5}{6} \end{aligned}$ is approximately $\begin{aligned} 4 \end{aligned}$
::3 5 6 大约 4 5 6

Next, add the approximated values.
::下加近似值。

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

The answer is $\begin{aligned} 1 \frac{1}{2} + 3 \frac{5}{6} \end{aligned}$ is approximately $\begin{aligned} 5 \frac{1}{2} \end{aligned}$ .
::答案是 1 1 2 + 3 5 6 约为 5 1 2。

Review
::回顾

Estimate the sums.
::估计金额。

$\begin{aligned} \frac{1}{29} + \frac{4}{5} \end{aligned}$

$\begin{aligned} \frac{9}{11} + \frac{4}{10} \end{aligned}$

$\begin{aligned} \frac{2}{5} + \frac{12}{13} \end{aligned}$

$\begin{aligned} \frac{2}{71} + \frac{1}{29} \end{aligned}$

$\begin{aligned} \frac{1}{29} + \frac{4}{5} \end{aligned}$

$\begin{aligned} \frac{3}{20} + \frac{14}{15} \end{aligned}$

$\begin{aligned} \frac{6}{7} + \frac{1}{5} \end{aligned}$

$\begin{aligned} \frac{9}{18} + \frac{5}{6} \end{aligned}$

$\begin{aligned} \frac{12}{13} + \frac{1}{25} \end{aligned}$

$\begin{aligned} \frac{7}{9} + \frac{1}{30} \end{aligned}$

$\begin{aligned} 3 \frac{6}{7} + 2 \frac{10}{11} \end{aligned}$

$\begin{aligned} 8 \frac{1}{12} + 6 \frac{3}{7} \end{aligned}$

$\begin{aligned} 2 \frac{9}{10} + 3 \frac{1}{17} \end{aligned}$

$\begin{aligned} 1 \frac{2}{12} + \frac{44}{46} \end{aligned}$

$\begin{aligned} 8 \frac{1}{29} + 10 \frac{4}{5} \end{aligned}$

Review (Answers)
::回顾(答复)

Section outline

Estimating Sums of Fractions and Mixed Numbers ::分数和混合数字的估算总和

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 Sums of Fractions and Mixed Numbers
::分数和混合数字的估算总和

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)