课程信息 | 7.11混合数除以分数的除法

7.11混合数除以分数的除法

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Audrey is checking to see how much dog food she has left. There are $\begin{aligned} 3 \frac{3}{4} \end{aligned}$ pounds of dog food left in the bag. Her dog eats about a half pound of food per day. How many days does Audrey have left before she runs out of food?
::奥黛丽正在检查她还剩多少狗食。袋子里还有3,3,4磅的狗食。她的狗每天吃大约半磅的食物。奥黛丽在食物用完之前还剩下多少天?

In this concept, you will learn how to divide a mixed number by a fraction.
::在这个概念中,你将学会如何将混合数除以一个小数。

Dividing Mixed Numbers by Fractions
::按分数排列的混合数字

You can divide a number by a fraction by multiplying the number by the reciprocal of the divisor . A mixed number consists of a whole number and a fraction, or "a whole and a part." When dividing a mixed number by a fraction, the quotient is the number of times the fraction divides into the mixed number.
::您可以通过将数字乘以对等数字来将数字除以一个分数。混合数字由整数和一个分数组成,或“整数和一部分”。当混合数字除以一个分数时,商数是分数乘以混合数的倍数。

Here is a division problem.
::这是一个分裂问题。

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

In this problem, you are trying to figure out how many sets or groups of one-third can be made from one and one-half.
::在这个问题上,你试图弄清楚三分之一的一组或几组从一个半可以制成。

Here is a drawing of one and one-half.
::这是一张1.5的图。

To figure out how many groups of one–third can be made from this quantity, you would have to divide these boxes up again into thirds. Dividing by a fraction using diagrams does not always provide clear answers.
::要想弄清楚从这个数量中可以得出多少组的三分之一,你就必须将这些盒子再分成三/3。使用图表进行分解并不总能提供清晰的答案。

Instead, use these rules for dividing mixed numbers and fractions.
::相反,使用这些规则来分解混合数字和分数。

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

Change the division to its inverse, multiplication, and change the divisor to its reciprocal.
::将分数改变为反向、倍增,并将分数改变为对等。

Multiply and simplify to find the quotient.
::乘以和简化以找到商数。

Let's apply this information to the division problem.
::让我们应用这个信息来解决分裂问题。

\begin{aligned} 1 \frac{1}{2} \div \frac{1}{3} = \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。

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

Next, rewrite the problem.
::下一个,重写问题。

\begin{aligned} \frac{3}{2} \div \frac{1}{3} \end{aligned}

Then, change the division to multiplication and change the divisor to its reciprocal. Flip the numerator and the denominator of a fraction to find the reciprocal. The reciprocal of $\begin{aligned} \frac{1}{3} \end{aligned}$ is $\begin{aligned} \frac{3}{1} \end{aligned}$ .
::然后,将分数改变为乘法,并将分数改变为对等。翻转分子和分母分母以找到对等。对等为13是31。

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

Next, multiply and simplify the fractions.
::下一步, 乘数并简化分数。

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

The quotient is $\begin{aligned} 4 \frac{1}{2} \end{aligned}$ .
::商数是412

Examples
::实例实例实例实例

Example 1
::例1

Earlier, you were given a problem about Audrey and her dog.
::早些时候,你得到一个问题关于奥德丽和她的狗。

Audrey's dog eats about a half pound of food per day and she has $\begin{aligned} 3 \frac{3}{4} \end{aligned}$ pounds of dog food left. Divide $\begin{aligned} 3 \frac{3}{4} \end{aligned}$ by $\begin{aligned} \frac{1}{2} \end{aligned}$ to find out how many days' worth of food she has.
::奥德丽的狗每天吃大约半磅的食物,她还有334磅的狗食物。 334磅乘以12分之3,34磅,看看她有多少天的食物价值。

First, write a division problem for the problem.
::首先,写一个分裂问题。

\begin{aligned} 3 \frac{3}{4} \div \frac{1}{2} \end{aligned}

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

\begin{aligned} 3 \frac{3}{4} = \frac{15}{4} \\ \frac{15}{4} \div \frac{1}{2} \end{aligned}

Then, change the operation to multiplication and the divisor to its reciprocal.
::然后,将操作改变为乘法和分判为对等。

\begin{aligned} \frac{15}{4} \div \frac{1}{2} = \frac{15}{4} \times \frac{2}{1} \end{aligned}

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

\begin{aligned} \frac{15}{4^{2}} \times \frac{2^{1}}{1} = \frac{15}{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{15}{2} = 7 \frac{1}{2} \end{aligned}

The answer is $\begin{aligned} 7 \frac{1}{2} \end{aligned}$ .
::答案是712

Audrey will run out of dog food after 7 days.
::7天后,欧蕊会用完狗食

Example 2
::例2

Divide the mixed number: $\begin{aligned} 6 \frac{2}{3} \div \frac{3}{4} \end{aligned}$ . Answer in simplest form .
::混合数除以: 6 2 3 + 3 4 。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} 6 \frac{2}{3} = \frac{20}{3} \end{aligned}

Next, rewrite the problem.
::下一个,重写问题。

\begin{aligned} \frac{20}{3} \div \frac{3}{4} \end{aligned}

Then, change the division to multiplication and the divisor to its reciprocal.
::然后,将分数改为乘法,将分数改为对等。

$\begin{aligned} \frac{20}{3} \div \frac{3}{4} = \frac{20}{3} \times \frac{4}{3} \end{aligned}$

Next, multiply and simply the fractions.
::下一个,乘数和简单的分数。

\begin{aligned} \frac{20 \times 4}{3 \times 3} = \frac{80}{9} = 8 \frac{8}{9} \end{aligned}

The quotient is $\begin{aligned} 8 \frac{8}{9} \end{aligned}$ .
::商数是8 8 9

Example 3
::例3

Divide the mixed numbers: $\begin{aligned} 2 \frac{1}{3} \div \frac{1}{4} = \underline{} \end{aligned}$ . Answer in simplest form.
::除以混合数字: 2,1,3,1,4 =_。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} 2 \frac{1}{3} = \frac{7}{3} \end{aligned}

\begin{aligned} 2 \frac{1}{3} \div \frac{1}{4} = \frac{7}{3} \div \frac{1}{4} \end{aligned}

Then, change the division to multiplication and the divisor to its reciprocal.
::然后,将分数改为乘法,将分数改为对等。

\begin{aligned} \frac{7}{3} \div \frac{1}{4} = \frac{7}{3} \times \frac{4}{1} \end{aligned}

Next, multiply and simplify the fractions.
::下一步, 乘数并简化分数。

\begin{aligned} \frac{7 \times 4}{3 \times 1} = \frac{28}{3} = 9 \frac{1}{3} \end{aligned}

The quotient is $\begin{aligned} 9 \frac{1}{3} \end{aligned}$ .
::商数是913

Example 4
::例4

Divide the mixed numbers: $\begin{aligned} 4 \frac{1}{2} \div \frac{1}{3} = \underline{} \end{aligned}$ . Answer in simplest form.
::除以混合数字: 4, 1, 2, 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} 4 \frac{1}{2} = \frac{9}{2} \end{aligned}

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

Then, change the division to multiplication and the divisor to its reciprocal.
::然后,将分数改为乘法,将分数改为对等。

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

Next, multiply and simplify the fractions.
::下一步, 乘数并简化分数。

\begin{aligned} \frac{9 \times 3}{2 \times 1} \times = \frac{27}{2} = 13 \frac{1}{2} \end{aligned}

The quotient is $\begin{aligned} 13 \frac{1}{2} \end{aligned}$ .
::商数是1312

Example 5
::例5

Divide the mixed numbers: $\begin{aligned} 5 \frac{2}{3} \div \frac{1}{2} = \underline{} \end{aligned}$ . Answer in simplest form.
::除以混合数字: 5 2 3 + 1 2 = _。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} 5 \frac{2}{3} = \frac{17}{3} \end{aligned}

\begin{aligned} 5 \frac{2}{3} \div \frac{1}{2} = \frac{17}{3} \div \frac{1}{2} \end{aligned}

Then, change the division to multiplication and the divisor to its reciprocal.
::然后,将分数改为乘法,将分数改为对等。

\begin{aligned} \frac{17}{3} \div \frac{1}{2} = \frac{17}{3} \times \frac{2}{1} \end{aligned}

Next, multiply and simplify the fractions.
::下一步, 乘数并简化分数。

\begin{aligned} \frac{17 \times 2}{3 \times 1} = \frac{34}{3} = 11 \frac{1}{3} \end{aligned}

The quotient is $\begin{aligned} 11 \frac{1}{3} \end{aligned}$ .

::商数是1113

Review
::审查审查审查审查

Divide the mixed number. Answer in simplest form.
::除以混合数字。回答最简单。

$\begin{aligned} 1 \frac{1}{2} \div \frac{1}{4} = \underline{} \end{aligned}$
$\begin{aligned} 1 \frac{1}{4} \div \frac{1}{5} = \underline{} \end{aligned}$
$\begin{aligned} 1 \frac{1}{2} \div \frac{1}{3} = \underline{} \end{aligned}$
$\begin{aligned} 2 \frac{1}{2} \div \frac{1}{4} = \underline{} \end{aligned}$
$\begin{aligned} 2 \frac{1}{2} \div \frac{1}{3} = \underline{} \end{aligned}$
$\begin{aligned} 3 \frac{1}{4} \div \frac{1}{3} = \underline{} \end{aligned}$
$\begin{aligned} 3 \frac{1}{2} \div \frac{1}{2} = \underline{} \end{aligned}$
$\begin{aligned} 4 \frac{1}{3} \div \frac{1}{5} = \underline{} \end{aligned}$
$\begin{aligned} 4 \frac{1}{2} \div \frac{1}{2} = \underline{} \end{aligned}$
$\begin{aligned} 5 \frac{1}{3} \div \frac{1}{4} = \underline{} \end{aligned}$
$\begin{aligned} 2 \frac{1}{2} \div \frac{1}{8} = \underline{} \end{aligned}$
$\begin{aligned} 1 \frac{1}{3} \div \frac{1}{9} = \underline{} \end{aligned}$
$\begin{aligned} 2 \frac{1}{3} \div \frac{1}{7} = \underline{} \end{aligned}$
$\begin{aligned} 2 \frac{1}{2} \div \frac{2}{3} = \underline{} \end{aligned}$
$\begin{aligned} 4 \frac{1}{4} \div \frac{1}{5} = \underline{} \end{aligned}$

7.11混合数除以分数的除法

Dividing Mixed Numbers by Fractions ::按分数排列的混合数字

Examples ::实例实例实例实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::审查审查审查审查

Review (Answers) ::审查(答复)

Dividing Mixed Numbers by Fractions
::按分数排列的混合数字

Examples
::实例实例实例实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::审查审查审查审查

Review (Answers)
::审查(答复)