Section: 带分数积 | 7.6带分数积

Section outline

Maria can run $\begin{aligned} 7 \frac{1}{3} \end{aligned}$ miles an hour. She decides to see how far she can run in $\begin{aligned} 2 \frac{1}{2} \end{aligned}$ hours. If she maintains her speed, how many miles did Maria run?
::玛利亚每小时能跑713英里她决定看她能在212小时内跑多远如果她保持速度玛利亚跑多少英里?

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

Multiplying Mixed Numbers
::乘数混合数字

A mixed number consists of a whole number and a fraction.
::混合数包括一个整数和一个分数。

Here is multiplication problem involving a whole number and a mixed number.
::这是一个乘法问题,涉及一个整数和一个混合数。

$\begin{aligned} 6 \times 1 \frac{1}{4} = \underline{} \end{aligned}$

Multiplication is a shortcut for repeated addition. The product of this expression is the total quantity of 6 groups of $\begin{aligned} 1 \frac{1}{4} \end{aligned}$ . You can multiply 6 wholes and 6 parts and then find the sum of both. Or you can think of $\begin{aligned} 1 \frac{1}{4} \end{aligned}$ in terms of parts. Convert $\begin{aligned} 1 \frac{1}{4} \end{aligned}$ to an improper fraction . Remember that to convert a mixed number to an improper fraction, multiply the denominator by the whole number. Then, add the numerator to the product and write that sum over the original denominator.
::乘法是重复添加的捷径。此表达式的产物是 6 组的总量 14 。您可以乘以 6 个整体和 6 个部分, 然后找到两者的总和。或者您可以按部件来想 14 个整体和 6 个部分。将 1 14 个部分转换为不适当的分数。记住, 要将混合数转换成不适当的分数, 将分母乘以整数。然后, 将数字添加到产品中, 并在原始分母上写这个总和。

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

Here is the problem again with the improper fraction.
::问题又在这里与不适当的分数。

$\begin{aligned} 6 \times 1 \frac{1}{4} = 6 \times \frac{5}{4} \end{aligned}$

Convert 6 into a fraction over one.
::将 6 转换成分数 1 以上 1 。

$\begin{aligned} \frac{6}{1} \times \frac{5}{4} \end{aligned}$

Then you can either multiply the fraction and simplify or simplify first and then multiply. Here you can cross simplify the fractions first.
::然后您可以乘以分数, 并简化或先简化, 然后再乘。您可以在此交叉简化分数。

$\begin{aligned} \frac{6}{1} \times \frac{5}{4} = \frac{3}{1} \times \frac{5}{2} = \frac{15}{2} \end{aligned}$
Finally, convert the improper fraction to a mixed number.

$\begin{aligned} \frac{15}{2} = 7 \frac{1}{2} \end{aligned}$

The product is $\begin{aligned} 7 \frac{1}{2} \end{aligned}$ .
::产品是712

Here is another mixed number multiplication problem.
::这是另一个混合数字乘法问题。

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

A fraction is a "part" and a mixed number consists of "wholes and a part." When multiplying a fraction and a mixed number, you are looking for "a part of a whole and a part." The product of this expression is half of $\begin{aligned} 2 \frac{1}{4} \end{aligned}$ .
::分数是“ part” , 混合数由“ 整数和一部分” 组成。当乘以一个分数和混合数时, 您正在寻找“ 整数和一部分的一部分” 。此表达式的产物为 2, 14 的半。

The first step is to convert the mixed number to an improper fraction.
::第一步是将混合数字转换成不适当的部分。

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

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

The fractions are in simplest form .
::分数以最简单的形式出现。

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

$\begin{aligned} \frac{1}{2} \times \frac{9}{4} = \frac{9}{8} \end{aligned}$

Next, simplify. Convert the improper fraction to a mixed number.
::接下来,简化一下,把不适当的分数转换成混合数。

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

The product is $\begin{aligned} 1 \frac{1}{8} \end{aligned}$ .
::产品是11.8。

Here is a multiplication problem with two mixed numbers.
::这是一个乘法问题,有两个混合数字。

The product of this expression is a whole and a part of another whole and a part. The key is to follow the same steps to find the solution.
::这种表达方式的产物是另一个整体和部分的完整和一部分,关键是遵循同样的步骤找到解决办法。

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

Simplify if possible
::可能时简化

Multiply
::乘乘乘乘

Check that your answer is in simplest form.
::请检查您的答案是否以最简单的形式出现。

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

First, convert each mixed number to an improper fraction.
::首先,将每个混合数字转换成不适当的部分。

$\begin{aligned} 2 \frac{1}{4} & = \frac{9}{4} \\ 1 \frac{1}{2} & = \frac{3}{2} \end{aligned}$

Rewrite the problem.
::重写问题。

$\begin{aligned} \frac{9}{4} \times \frac{3}{2} = \underline{} \end{aligned}$

The fractions cannot be simplified so at this point.
::分数无法简化。

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

$\begin{aligned} \frac{9}{4} \times \frac{3}{2} = \frac{27}{8} \end{aligned}$

Finally, check if the fraction is in simplest form. Convert the improper fraction to a mixed number.
::最后,检查分数是否最简单。将不适当的分数转换成混合数。

$\begin{aligned} \frac{27}{8} = 3 \frac{3}{8} \end{aligned}$

The product is $\begin{aligned} 3 \frac{3}{8} \end{aligned}$ .
::产品为3,3,8。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Maria's run.
::之前,你对Maria的跑步有问题

Maria can run $\begin{aligned} 7 \frac{1}{3} \end{aligned}$ miles per hour and decides to run for $\begin{aligned} 2 \frac{1}{2} \end{aligned}$ hours. Multiply her speed times the number of hours to find the total number of miles run.
::Maria每小时可以跑713英里, 决定跑212小时。乘以她的速度乘以小时数, 才能找到行驶总英里数。

$\begin{aligned} 7 \frac{1}{3} \times 2 \frac{1}{2} = \underline{} \end{aligned}$

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

$\begin{aligned} 7 \frac{1}{3} & = \frac{22}{3} \\ 2 \frac{1}{2} & = \frac{3}{2} \\ \frac{22}{3} & \times \frac{3}{2} \end{aligned}$

Then, simplify the fractions. You can simply 2 and 22 with the GCF of 2 and 3 and 3 with the GCF 3.
::然后,简化分数。如果全球合作框架为2,3,3,3,2,22,则可以简单2,22。全球合作框架为2,3,3,3。

$\begin{aligned} \frac{22}{3} & \times \frac{3}{2} = \frac{11}{1} \times \frac{1}{1} \end{aligned}$

Next, multiply.
::下一个,乘数。

$\begin{aligned} 11 \times 1 = 11 \end{aligned}$

Maria ran 11 miles.
::玛丽亚跑了11英里

Example 2
::例2

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

$\begin{aligned} \frac{1}{3} \times 2 \frac{1}{5} = \underline{} \end{aligned}$

First, convert the mixed number to an improper fraction.
::首先,将混合数字转换成不适当的部分。

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

$\begin{aligned} \frac{1}{3} \times \frac{11}{5} \end{aligned}$

The fractions are in simplest form.
::分数以最简单的形式出现。

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

$\begin{aligned} \frac{1}{3} \times \frac{11}{5} = \frac{11}{15} \end{aligned}$

The product is $\begin{aligned} \frac{11}{15} \end{aligned}$ .
::产品为11 15。

Example 3
::例3

Find the product: $\begin{aligned} 4 \times 2 \frac{1}{2} = \underline{} \end{aligned}$ . Answer in simplest form.
::查找产品: 4 × 2 1 2 = _. 回答最简单。

First, convert the whole number and mixed number to fractions.
::首先,将整数和混合数转换成分数。

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

Then, cross simplify the fractions.
::然后,交叉简化分数。

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

Next, multiply the fractions.
::接下来,乘以分数。

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

Note that a fraction with a number over the denominator of 1 is a whole number.
::请注意,一分数超过1分母的分数是一个整数。

The product is 10.
::产品是10。

Example 4
::例4

Find the product: $\begin{aligned} \frac{1}{6} \times 1 \frac{1}{3} = \underline{} \end{aligned}$ . Answer in simplest form.
::查找产品: 1 6 × 1 1 3 = _. 回答最简单。

First, convert the mixed number to an improper fraction.
::首先,将混合数字转换成不适当的部分。

$\begin{aligned} \frac{1}{6} \times \frac{4}{3} \end{aligned}$

Then, cross simplify the fractions.
::然后,交叉简化分数。

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

Next, multiply the fractions.
::接下来,乘以分数。

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

The fraction is in simplest form.
::分数以最简单的形式出现。

The product is $\begin{aligned} \frac{2}{9} \end{aligned}$ .
::产品为2 9。

Example 5
::例5

Find the product: $\begin{aligned} 4 \frac{1}{3} \times 1 \frac{1}{2} = \underline{} \end{aligned}$ . Answer in simplest form.
::查找产品: 4, 1, 3 × 1, 1, 2 = _. 回答最简单。

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

$\begin{aligned} 4 \frac{1}{3} & = \frac{13}{3} \\ 1 \frac{1}{2} & = \frac{3}{2} \end{aligned}$

$\begin{aligned} 7 \frac{1}{3} \times 2 \frac{1}{2} = \underline{} \end{aligned}$

Then, cross simplify the fractions.
::然后,交叉简化分数。

$\begin{aligned} \frac{13}{3} \times \frac{3}{2} = \frac{13}{1} \times \frac{1}{2} \end{aligned}$

Next, multiply the fractions.
::接下来,乘以分数。

$\begin{aligned} \frac{13}{1} \times \frac{1}{2} = \frac{13}{2} \end{aligned}$

Finally, convert the improper fraction to a mixed number.
::最后,将不当的分数转换成混合数。

$\begin{aligned} \frac{13}{2} = 7 \frac{1}{2} \end{aligned}$

The product is $\begin{aligned} 7 \frac{1}{2} \end{aligned}$ .

::产品是712

Review
::回顾

Find the product in simplest form.
::以最简单的形式寻找产品。

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

$\begin{aligned} 8 \times 2 \frac{1}{2} = \underline{} \end{aligned}$

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

$\begin{aligned} 5 \times 3 \frac{1}{3} = \underline{} \end{aligned}$

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

$\begin{aligned} 7 \times 4 \frac{1}{2} = \underline{} \end{aligned}$

$\begin{aligned} 9 \times 2 \frac{1}{5} = \underline{} \end{aligned}$

$\begin{aligned} 6 \times 4 \frac{1}{2} = \underline{} \end{aligned}$

$\begin{aligned} 8 \times 2 \frac{1}{4} = \underline{} \end{aligned}$

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

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

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

$\begin{aligned} \frac{1}{4} \times 6 \frac{2}{3} = \underline{} \end{aligned}$

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

$\begin{aligned} \frac{1}{5} \times 5 \frac{1}{3} = \underline{} \end{aligned}$

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

$\begin{aligned} \frac{4}{7} \times 2 \frac{1}{7} = \underline{} \end{aligned}$

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

$\begin{aligned} 5 \frac{1}{2} \times 3 \frac{1}{4} = \underline{} \end{aligned}$

$\begin{aligned} 1 \frac{4}{5} \times 3 \frac{1}{4} = \underline{} \end{aligned}$

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

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

$\begin{aligned} \frac{1}{8} \times 8 \frac{1}{3} = \underline{} \end{aligned}$

$\begin{aligned} \frac{4}{7} \times 2 \frac{1}{3} = \underline{} \end{aligned}$

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

Multiplying Mixed Numbers ::乘数混合数字

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 Mixed Numbers
::乘数混合数字

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