课程： 1.12乘法或除法方程的心算

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乘法或除法方程的心算
Olivia loves to play cards with her friends. She knows that normally a deck of cards has 52 cards. Unfortunately, her baby brother was playing with her deck of cards and she thinks some of the cards have gone missing. When she deals out all of the cards to herself and her 4 friends, each person ends up with 10 cards. How can Olivia figure out how many cards are in her deck? Is she missing any cards?
::Olivia喜欢和朋友玩牌。她知道牌牌牌通常有52张牌。不幸的是,她的弟弟玩牌牌牌,她认为有些牌牌不见了。当她自己和4个朋友把牌牌都给自己和4个朋友时,每个人都有10张牌。Olivia怎么弄清楚她的牌牌有多少?她有没有牌?

In this concept, you will learn how to solve single variable multiplication and division equations using mental math.
::在此概念中, 您将学习如何使用心理数学解析单一变量乘法和分裂方程式。

Solving Multiplication and Division Problems Using Mental Math
::使用心理数学解决乘数和分部问题

An equation is a mathematical sentence that says that two mathematical expressions are equal. A variable equation is an equation that contains a variable.
::等式是一个数学句子, 它表示两个数学表达式是相等的。变量方程式是包含变量的方程式。

Here are some examples of variable equations.
::以下是可变方程式的一些例子。

$\begin{aligned} 15 t = 45 \end{aligned}$

::15吨=45

$\begin{aligned} 12 = \frac{x}{3} \end{aligned}$

::12 = x 3

$\begin{aligned} \frac{20}{x} + 1 = 5 \end{aligned}$

::20 x + 1 = 5

When you solve an equation you are looking for the value of the variable that makes both sides of the equation equal. This value of the variable that makes both sides of the equation equal is called the solution .
::当您解决一个方程式时,您正在寻找使方程式两侧相等的变量值。使方程式两侧相等的变量值被称为解决方案。

Some variable equations are simple enough that you are able to determine the solution to the equation in your head using mental math.
::一些变量方程式非常简单,您能够用心理数学来决定您脑中方程式的解决方案。

Here is an example that involves multiplication.
::这里的例子涉及乘法。

Use mental math to solve the following equation.
::使用心理数学解析以下方程式。

$\begin{aligned} 9 p = 72 \end{aligned}$
::9 p = 72

First, remember that whenever you see a number right next to a variable it means the number is being multiplied by the variable. This equation says "9 times $\begin{aligned} p \end{aligned}$ equals 72". The number next to the variable is called the coefficient of the variable. Here, 9 is the coefficient of $\begin{aligned} p \end{aligned}$ .
::首先, 请记住, 当您在变量旁边看到数字时, 它意味着数字正被变量乘以变量。此方程式表示“ 9乘以 p 等于 72 ” 。变量旁边的数字被称为变量的系数。这里, 9 是 p 的系数。

Now, turn the equation into a question that you can ask yourself. Remember you are trying to figure out the value of $\begin{aligned} p \end{aligned}$ that will make both sides of the equation equal.
::现在, 将方程转换成一个问题, 你可以问自己。记住, 您正在尝试找出 p 的值, 使方程的两边都相等。

"9 times what number is equal to 72?"
::"9乘以多少数字等于72?"

Next, answer your question. You know that 9 times 8 is equal to 72. This means $\begin{aligned} p \end{aligned}$ must be 8.
::接下来,回答你的问题。你知道,9乘以8等于72,这意味着p必须是8。

The answer is $\begin{aligned} p = 8 \end{aligned}$ .
::答案是p=8。

You can check your solution by substituting that value for $\begin{aligned} p \end{aligned}$ back into the original equation and verifying that it makes both sides equal.
::您可以检查您的解决方案, 将 p 的值替换为 p 返回到原始方程, 并验证它是否使两边都相等。

Both sides of the equation are equal so your answer is correct!
::等式的两边都一样,所以你的回答是正确的!

Here is an example that involves division.
::这里的例子涉及分裂。

Use mental math to solve the following equation.
::使用心理数学解析以下方程式。

$\begin{aligned} \frac{x}{3} = 4 \end{aligned}$
::x 3= 4x 3= 4

First, remember that a fraction bar means division. $\begin{aligned} \frac{x}{3} \end{aligned}$ is the same as $\begin{aligned} x \div 3 \end{aligned}$ .
::首先,请记住分数条表示除法。 x 3 与 x { { 3 相同。

Now, turn the equation into a question that you can ask yourself.
::现在,把方程式变成一个问题你可以问自己。

"What number divided by 3 is equal to 4?"
::"什么数字除以3等于4?"

Next, answer your question. You know that 12 divided by 3 is equal to 4. This means $\begin{aligned} x \end{aligned}$ must be 12.
::接下来,回答你的问题。你知道12除以3等于4,这意味着x必须是12。

The answer is $\begin{aligned} x = 12 \end{aligned}$ .
::答案是 x = 12 。

Make sure to check your solution.
::确保检查你的解决方案。

Because both sides of the equation are equal, you can be confident that your answer is correct.
::因为等式的两面是平等的, 你可以相信你的答案是正确的。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Olivia and her deck that might be missing some cards.
::之前有人给你一个奥利维亚和她的甲板的问题可能漏掉了一些卡片

When she deals out all the cards to herself and 4 of her friends, each person ends up with 10 cards. She wants to know how many cards are in her deck.
::当她把自己和四个朋友的牌都发出去时,每个人最后都有10张牌,她想知道她的牌里有多少张牌。

First, define your variable. You don't know how many cards are in the deck so this unknown quantity will be your variable.
::首先, 定义您的变量。您不知道甲板上有多少牌, 所以这个未知的数量将会成为您的变量。

Let $\begin{aligned} x \end{aligned}$ equal the number of cards in the deck.
::x 等于牌堆中的牌数。

Next, you know that when all the cards in the deck are divided evenly between 5 people, each person ends up with 10 cards. This scenario involves division because the total number of cards divided by the 5 people will equal the 10 cards each person got.
::接下来,你知道,当甲板上所有的牌在5人之间平均分配时, 每个人最后都会有10张牌。这个假设涉及分裂, 因为由5人除以的牌总数将等于每人得到的10张牌。

Now, turn the equation into a question that you can ask yourself.
::现在,把方程式变成一个问题你可以问自己。

"What number divided by 5 is equal to 10?"
::"什么数字除以5等于10等于10?"

Then, answer your question. You know that 50 divided by 5 is equal to 10. This means $\begin{aligned} x \end{aligned}$ must be 50.
::那么,回答你的问题。你知道50除以5等于10,这意味着x必须是50。

The answer is that Olivia's deck only has 50 cards. Since a deck of cards should have 52 cards, Olivia is missing 2 cards.
::答案是奥利维亚的甲板只有50张牌,因为牌牌牌应该有52张,奥利维亚少了2张牌。

Example 2
::例2

Alyssa sold $120 in raffle tickets. If each ticket costs $6, how many tickets did she sell? Write a variable equation and solve.
::Alyssa卖掉120美元的彩票。如果每张票要6美元,她卖了多少票?写一个变数方程和解答。

First, define your variable. You don't know how many tickets Alyssa sold so this unknown quantity will be your variable.
::首先,定义你的变量。你不知道Alyssa卖了多少张票,所以这个未知的数量就是你的变量。

Let $\begin{aligned} x \end{aligned}$ equal the number of tickets that Alyssa sold.
::x 等于 Alyssa 售出的票数。

Next, you know that Alyssa sold each ticket for $6 and made $120 total. This scenario involves multiplication because the number of tickets times the cost of each ticket will equal the amount of money Alyssa made.
::接下来,你知道,Alyssa将每张票卖出6美元,总价120美元。这一假设涉及倍增,因为每张票的票数乘以每张票的费用,等于Alyssa赚的钱。

Now, turn the equation into a question that you can ask yourself.
::现在,把方程式变成一个问题你可以问自己。

"6 times what number is equal to 120?"
::"6倍什么数字等于120?"

Then, answer your question. You know that 6 times 2 is equal to 12, so 6 times 20 is equal to 120. This means $\begin{aligned} x \end{aligned}$ must be 20.
::那么,回答你的问题。你知道6乘以2等于12, 所以6乘以20等于120。这意味着x必须是20。

The answer is that Alyssa sold 20 tickets.
::答案是Alyssa卖了20张票

Example 3
::例3

Use mental math to solve the following equation.
::使用心理数学解析以下方程式。

$\begin{aligned} 5 y = 20 \end{aligned}$
::5y = 20

First, turn the equation into a question that you can ask yourself.
::首先,把方程式变成一个问题你可以问自己。

"5 times what number is equal to 20?"
::"5乘以什么数字等于20?"

Next, answer your question. You know that 5 times 4 is equal to 20. This means $\begin{aligned} y \end{aligned}$ must be 4.
::接下来,回答你的问题。你知道5乘以4等于20,这意味着你必须是4。

The answer is $\begin{aligned} y = 4 \end{aligned}$ .
::答案是y=4。

Now, check your solution.
::现在,检查你的解决方案。

The answer is correct.
::答案是正确的。

Example 4
::例4

Use mental math to solve the following equation.
::使用心理数学解析以下方程式。

$\begin{aligned} 6 g = 42 \end{aligned}$
::6克=42

First, turn the equation into a question that you can ask yourself.
::首先,把方程式变成一个问题你可以问自己。

"6 times what number is equal to 42?"
::"6倍什么数字等于42?"

Next, answer your question. You know that 6 times 7 is equal to 42. This means $\begin{aligned} g \end{aligned}$ must be 7.
::接下来,回答你的问题。你知道6乘以7等于42,这意味着g必须是7。

The answer is $\begin{aligned} g = 7 \end{aligned}$ .
::答案是g=7。

Now, check your solution.
::现在,检查你的解决方案。

The answer is correct.
::答案是正确的。

Example 5
::例5

Use mental math to solve the following equation.
::使用心理数学解析以下方程式。

$\begin{aligned} \frac{x}{7} = 2 \end{aligned}$
::x 7 = 2

First, turn the equation into a question that you can ask yourself.
::首先,把方程式变成一个问题你可以问自己。

"What number divided by 7 is equal to 2?"
::"什么数字除以7等于2?"

Next, answer your question. You know that 14 divided by 7 is equal to 2. This means $\begin{aligned} x \end{aligned}$ must be 14.
::接下来,回答你的问题。你知道14除以7等于2,这意味着x必须是14。

The answer is $\begin{aligned} x = 14 \end{aligned}$ .
::答案是 x = 14 。

Now, check your solution.
::现在,检查你的解决方案。

The answer is correct.
::答案是正确的。

Review
::回顾

Use mental math to solve each multiplication or division equation.
::使用心理数学解析每个乘法或除法方程式。

$\begin{aligned} 5 x = 25 \end{aligned}$
::5 x = 25

$\begin{aligned} 6 x = 48 \end{aligned}$
::6x=48

$\begin{aligned} 2 y = 18 \end{aligned}$
::2y = 18

$\begin{aligned} 3 y = 21 \end{aligned}$
::3y = 21

$\begin{aligned} 4 a = 16 \end{aligned}$
::4 a = 16

$\begin{aligned} 13 b = 26 \end{aligned}$
::13 b = 26

$\begin{aligned} 15 a = 30 \end{aligned}$
::15 a = 30

$\begin{aligned} 15 x = 45 \end{aligned}$
::15x=45

$\begin{aligned} \frac{x}{2} = 3 \end{aligned}$
::2 x 2 = 3

$\begin{aligned} \frac{x}{4} = 5 \end{aligned}$
::x 4= 5x 4= 5

$\begin{aligned} \frac{x}{3} = 11 \end{aligned}$
::x 3 = 11

$\begin{aligned} \frac{x}{5} = 12 \end{aligned}$
::5 x 5 = 12

$\begin{aligned} \frac{x}{7} = 8 \end{aligned}$
::7x7=8

$\begin{aligned} \frac{x}{8} = 9 \end{aligned}$
::8x8=9

$\begin{aligned} \frac{x}{3} = 12 \end{aligned}$
::x 3 = 12

Review (Answers)
::回顾(答复)

章节大纲

常规

乘法或除法方程的心算

Solving Multiplication and Division Problems Using Mental Math ::使用心理数学解决乘数和分部问题

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)

Solving Multiplication and Division Problems Using Mental Math
::使用心理数学解决乘数和分部问题

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)