2.4 小数加法

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  小数加法

  

  [Figure 1]

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  Lindsey made $415.85 babysitting over the summer. She went to the bank to deposit her money into a savings account. At the bank she learned that she can expect to earn 2%  interest  on her money in one year. She does the calculations and finds that 2% of 415.85 is 8.317. Assuming Lindsey doesn’t add any more money to her savings account, how can she figure out how much money will be in her account after one year?
  ::Lindsey在夏天做了415.85美元的保姆。她去银行把钱存入一个储蓄账户。 在银行,她学会了在一年的时间里可以指望挣到2%的利息。 她做了计算,发现415.85的2%是8.317美元。 假设Lindsey不再给储蓄账户增加钱,她又如何知道一年后她的账户会有多少钱?

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  In this concept, you will learn how to add decimals with and without  rounding .
  ::在此概念中,您将学习如何在四舍五入和不四舍五入的情况下添加小数。

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  Adding Decimals
  ::添加十进数

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  The  decimal system  lets us represent numbers that are less than 1 and other numbers that exist between  whole numbers . In a decimal number, the  decimal point  divides the whole part of the number from the fractional part of the number.
  ::小数点制让我们可以代表小于1的数字和整个数字之间存在的其他数字。在小数点数中,小数点将数字的整个部分与数字的分数部分分开。

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  To add decimal numbers:
  ::要添加小数数 :

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  1. Arrange the numbers vertically so that the decimal points line up.
     ::垂直排列数字,使小数点排成一行。
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  2. Add zeros for any missing spaces so that each number has the same number of digits to the right of the decimal point.
     ::为任何缺失的空格添加零,使每个数字在小数点右侧的位数相同。
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  3. Add from right to left, carrying just as if you were adding whole numbers.
     ::从右到左添加, 随身携带就像您正在添加整个数字一样 。
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  4. Insert a decimal point into your answer so that it lines up with the decimal points of the original numbers.
     ::在您的答复中插入小数点,使其与原始数字的小数点一致。

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  Here is an example.
  ::举一个例子。

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  Find the  sum  of 1.87 and 2.3.
  ::查找1.87和2.3之和。

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  First, arrange the numbers vertically so that the decimal points line up.  
  ::首先,垂直排列数字,使小数点排成一行。

  $$\begin{array}{r}\begin{array}{rc}& 1.87\\ & \underset{\_}{+2.3}\end{array}\end{array}$$

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  Next, add a  zero  after the 3 in the second number so that both numbers have two digits to the right of the decimal point.
  ::接下来,在第二个数字的3后面加上0,这样两个数字在小数点右侧有两个位数。

  $$\begin{array}{r}1.87\\ \underset{\_}{+2.30}\end{array}$$

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  Now, add from right to left. Start by adding 7 and 0 to make 7. Then 8 and 3 to make 11. At this point you will carry the 1 and add it along with the 1 and the 2 to make 4. Insert a decimal point into your answer directly under the decimal points in the original numbers.
  ::现在, 从右到左添加。 从增加 7 和 0 到 7 开始, 然后 8 和 3 到 11 开始 。 此时, 您将携带 1 和 1 和 2 并加上 4 。 在您的答复中, 在原数的小数点的小数点下直接插入一个小数点 。

  $$\begin{array}{r}1.87\\ \underset{\_}{+2.30}\\ 4.17\end{array}$$

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  The answer is  $\begin{array}{r}1.87+2.3=4.17\end{array}$ .
  ::答案是1.87+2.3=4.17。

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  Here is another example.
  ::下面是另一个例子。

  $\begin{array}{r}4.5+2.137=\underset{\_}{}\end{array}$

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  First, arrange the numbers vertically so that the decimal points line up.
  ::首先,垂直排列数字,使小数点排成一行。

  $$\begin{array}{rl}& 4.5\\ & \underset{\_}{+2.137}\end{array}$$

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  Next, add two zeros after the 5 in the first number so that both numbers have three digits to the right of the decimal point.
  ::接下来,在第一个数字的5之后加上两个零,使两个数字在小数点右侧有三个位数。

  $$\begin{array}{r}4.500\\ \underset{\_}{+2.137}\end{array}$$

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  Now, add from right to left. Start by adding 0 and 7 to make 7. Then 0 and 3 to make 3. Then 5 and 1 to make 6. Finally add 4 and 2 to make 6. Insert a decimal point into your answer directly under the decimal points in the original numbers.
  ::现在, 从右到左添加。 从增加 0 和 7 到 7 开始 。 然后 0 和 3 到 3 。 然后 5 和 1 到 6 。 最后 添加 4 和 2 到 6 。 在您的答复中直接插入一个小数点, 在原数的小数点的小数点下插入一个小数点 。

  $$\begin{array}{r}4.500\\ \underset{\_}{+2.137}\\ 6.637\end{array}$$

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  The answer is  $\begin{array}{r}4.5+2.137=6.637\end{array}$ .
  ::答案是4.5+2.137=6.637。

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  Sometimes you will want to  round  numbers before adding them. Recall that there are four steps for rounding numbers.
  ::有时您想要在添加数字前先对数字进行四舍五入。 提醒注意四步后再对数字进行四舍五入 。

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  1. Identify the place you want to round to and find that digit within your number.
     ::识别您想要绕行的位置,并在您的电话号码中找到该位数 。
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  2. Look at the digit to the right of the place you want to round to.
     ::看着您想要绕到的地方右侧的位数 。
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  3. If the digit to the right is between 5 and 9, increase the digit in the place you want to round to by 1. If the digit to the right is between 0 and 4, keep the digit in the place you want to round to the same. NOTE: If the digit in the place you want to round to is a 9 and you need to increase it by 1, change it to zero and increase the digit to its left by 1. 
     ::如果右侧的位数在5到9之间, 则将您想要转动的位置的位数增加为 1。 如果右侧的位数在0到4之间, 将您想要转动的位置的位数保持同一位置。 注意: 如果您想要转动的位置的位数是9, 您需要将其增加1, 将其修改为零, 并将位数增加为1 。
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  4. Replace all digits after the place you are rounding to with zeros. If these digits occur to the right of the decimal point, it is not necessary to write the zeros.
     ::在四舍五入到零的位置之后,将所有数字替换为零。如果这些数字出现在小数点右侧,则不必写零。

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  Here is an example.
  ::举一个例子。

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  Round each number to the nearest hundredth and then find the sum.
  ::每一回合到最接近的一百,然后找到总和。

  $\begin{array}{r}9.199,11.353,10.648\end{array}$

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  First, round each number to the nearest hundredth. Remember that the hundredths place is the second digit to the right of the decimal point. Look at each digit in the hundredths place and then the digit to the right of that to decide if you will need to round up the digit in the hundredths place or keep it the same.
  ::首先, 将每个数字回合到最近的一百。 记住, 第一百位位是小数点右侧的第二位数。 看看第一百位中的每个位数, 然后将位数移到右边, 以便决定您是否需要将位数排到第一百位中, 或者保持原样 。

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  • 9.199 rounds to 9.20
    ::9.199发至9.20子弹
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  • 11.353 rounds to 11.35
    ::11.353发11.353发炮弹至11.35
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  • 10.648 rounds to 10.65
    ::10.648发至10.65子弹

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  Now, find the sum of the three rounded numbers. Start by arranging the numbers vertically so that the decimal points line up.
  ::现在, 找到三个四舍五入数字的总和。 从垂直排列数字开始, 让小数点排队 。

  $$\begin{array}{r}9.20\\ 11.35\\ \underset{\_}{+10.65}\end{array}$$

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  Next, add from right to left. Start by adding 0, 5, and 5 to make 10. Carry the 1 and add it with the 2, 3, and 6 to make 12. Once again carry the 1 and add it with the 9 and 1 to make 11. Carry the 1 one more time and add it with the 1 and 1 to make 3. Insert a decimal point into your answer directly under the decimal points in the original numbers.
  ::下一步, 从右到左。 开始增加 0, 5, 5 和 10 。 开始增加 0, 5 和 5 。 开始增加 0 。 开始增加 0 、 5 和 5 。 开始增加 0 。 开始增加 1 , 并加上 2 、 3 和 6 ; 12 ; 12 ; 再次增加 1 , 加上 9 和 1 。 11 开始增加 9 和 1 。 再增加 1 , 并加上 1 和 1 ; 开始增加 1 3 。 开始在您的答复中, 在原数小数小数点下直接插入一个小数点 。

  $$\begin{array}{r}9.20\\ 11.35\\ \underset{\_}{+10.65}\\ 31.20\end{array}$$

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  The answer is  $\begin{array}{r}9.20+11.35+10.65=31.20\end{array}$ .
  ::答案是9.20+11.35+10.65=31.20。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about Lindsey and her savings account.
  ::早些时候,你得到一个问题 关于林赛和她的储蓄账户。

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  She deposited $415.85 into the account and expects to earn $8.317 in interest in one year. She wants to figure out how much money she will have in her account after one year.
  ::她将415.85美元存入账户,预计一年内将赚取8.317美元的利息,她想弄清楚她一年后在账户中将有多少钱。

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  First, round both numbers to the nearest cent, which is the nearest hundredth. $415.85 is already to the nearest hundredth, but $8.317 needs to be rounded.
  ::首先,将两个数字都打到最近的百分位数,即最近的百分位数。 415.85美元已经是最近的百分位数,但8.317美元需要四舍五入。

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  • 8.317 rounds to 8.32
    ::8.317发至8.32子弹

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  Now, find the sum of the two rounded numbers. Start by arranging the numbers vertically so that the decimal points line up.
  ::现在, 找到两个四舍五入数字的总和。 从垂直排列数字开始, 让小数点排队 。

  $$\begin{array}{r}415.85\\ \underset{\_}{+8.32}\end{array}$$

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  Next, add from right to left. Start by adding 5 and 2 to make 7. Then add 8 and 3 to make 11. Carry the 1 and add it to 5 and 8 to make 14. Carry the 1 and add it to 41 to make 42. Insert a decimal point into your answer directly under the decimal points in the original numbers.
  ::下一步, 从右到左增加。 开始增加 5 和 2 到 7 。 然后增加 8 和 3 到 11 。 然后增加 8 和 3 到 11 。 加上 8 到 11 。 加上 1 和 5 和 8 到 14 。 加上 1 和 41 。 在 回答中直接插入小数点, 在原数小数点下插入小数点 。

  $$\begin{array}{r}415.85\\ \underset{\_}{+8.32}\\ 424.17\end{array}$$

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  The answer is  $\begin{array}{r}\mathrm{\$}415.85+\mathrm{\$}8.32=\mathrm{\$}424.17\end{array}$ , so Lindsey can expect to have $424.17 in her account after one year.
  ::答案是415.85美元+8.32美元=424.17美元,因此Lindsey预计一年后账户有424.17美元。

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  Example 2
  ::例2

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  Round each value to the nearest tenth and then find the sum.
  ::将每一数值四舍五入至最接近的十分之一,然后找到总和。

  $\begin{array}{r}4.56,3.47,2.04,6.21\end{array}$

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  First, round each number to the nearest tenth. Remember that the tenths place is the first digit to the right of the decimal point. Look at each digit in the tenths place and then the digit to the right of that to decide if you will need to round up the digit in the tenths place or keep it the same.
  ::首先,将每个数字四舍五入到最接近的十分点。 记住, 十分位数是小数点右侧的第一个位数。 看看十位数中的每个位数, 然后将数字移到十位数右位数, 以决定您是否需要将位数四舍五入到十分位数, 或者保持原位数不变 。

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  • 4.56 rounds to 4.6
    ::4.56至4.6发子弹
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  • 3.47 rounds to 3.5
    ::3.47发至3.5发
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  • 2.04 rounds to 2.0
    ::2.04至2.0子弹
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  • 6.21 rounds to 6.2
    ::6.21发炮弹至6.2

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  Now, find the sum of the four rounded numbers. Start by arranging the numbers vertically so that the decimal points line up.
  ::现在, 找到四个四舍五入数字的总和。 从垂直排列数字开始, 这样小数点会排队 。

  $$\begin{array}{r}4.6\\ 3.5\\ 2.0\\ \underset{\_}{+6.2}\end{array}$$

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  Next, add from right to left. Start by adding 6, 5, 0, and 2 to make 13. Carry the 1 and add it with the 4, 3, 2, and 6 to make 16. Insert a decimal point into your answer directly under the decimal points in the original numbers.
  ::接下来, 从右到左添加。 开始增加 6, 5, 0 和 2, 13 。 将 1 加上 4, 3, 2 和 6 加上 16 。 在您的答复中, 在原数小数点的小数点下直接插入一个小数点 。

  $$\begin{array}{r}4.6\\ 3.5\\ 2.0\\ \underset{\_}{+6.2}\\ 16.3\end{array}$$

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  The answer is  $\begin{array}{r}4.6+3.5+2.0+6.2=16.3\end{array}$ .
  ::答案是4.6+3.5+2.0+6.2=16.3。

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  Example 3
  ::例3

  $\begin{array}{r}4.56+1.2+37.89=\underset{\_}{}\end{array}$

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  First, arrange the numbers vertically so that the decimal points line up.
  ::首先,垂直排列数字,使小数点排成一行。

  $$\begin{array}{rl}& 4.56\\ & 1.2\\ & \underset{\_}{+37.89}\end{array}$$

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  Next, add one zero after the 2 in the second number so that all numbers have two digits to the right of the decimal point.
  ::接下来,在第二个数字的2后面加上1 0,这样所有数字在小数点右侧都有2位数。

  $$\begin{array}{r}4.56\\ 1.20\\ \underset{\_}{+37.89}\end{array}$$

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  Now, add from right to left. Start by adding 6, 0 and 9 to make 15. Carry the 1 and add it with the 5, 2, and 8 to make 16. Once again carry the 1 and add it with the 4, 1, and 7 to make 13. Carry the 1 one more time and add it with the 3 to make 4. Insert a decimal point into your answer directly under the decimal points in the original numbers.
  ::现在, 从右到左增加。 开始增加 6, 0和 9 以得出 15 。 携带 1 并加上 5, 2和 8 以得出 16 。 再次携带 1 并加上 4, 1 和 7 以得出 13 以 4 以 4 以 4 以 4 以 加 1 并 3 以 4 以 。 将小数点直接插入您的答复中, 在原数小数点的小数点下插入一个小数点 。

  $$\begin{array}{r}4.56\\ 1.20\\ \underset{\_}{+37.89}\\ 43.65\end{array}$$

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  The answer is  $\begin{array}{r}4.56+1.20+37.89=43.65\end{array}$ .
  ::答案是4.56+1.20+37.89=43.65。

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  Example 4
  ::例4

  $\begin{array}{r}14.2+56.78+123.4=\underset{\_}{}\end{array}$

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  First, arrange the numbers vertically so that the decimal points line up.
  ::首先,垂直排列数字,使小数点排成一行。

  $$\begin{array}{rl}& 14.2\\ & 56.78\\ & \underset{\_}{+123.4}\end{array}$$

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  Next, add one zero after the 2 in the first number and one zero after the 4 in the third number so that all numbers have two digits to the right of the decimal point.
  ::接下来,在第一个数字的2后面加上1 0,在第三个数字的4后面加上1 0,这样所有数字在小数点右侧都有2位数。

  $$\begin{array}{r}14.20\\ 56.78\\ \underset{\_}{+123.40}\end{array}$$

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  Now, add from right to left. Start by adding 0, 8, and 0 to make 8. Then add 2, 7, and 4 to make 13. Carry the 1 and add it with the 4, 6, and 3 to make 14. Once again carry the 1 and add it with the 1, 5, and 12 to make 19. Insert a decimal point into your answer directly under the decimal points in the original numbers.
  ::现在, 从右到左增加。 从增加 0, 8, 8 和 0 开始, 8 开始 。 然后增加 2, 7 和 4 开始 。 然后增加 2, 7 和 4 开始 。 13 加上 2, 7 和 4 、 7 和 4 。 将 1 和 4 、 6 和 3 加上 14 。 再一次将 1 和 1 、 5 和 12 加上 1 、 5 和 19 。 在您的答复中, 在小数点下插入一个小数点 。

  $$\begin{array}{r}14.20\\ 56.78\\ \underset{\_}{+123.40}\\ 194.38\end{array}$$

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  The answer is  $\begin{array}{r}14.20+56.78+123.40=194.38\end{array}$ .
  ::答案是14.20+56.78+123.40=194.38。

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  Example 5
  ::例5

  $\begin{array}{r}189.34+123.5+7.2=\underset{\_}{}\end{array}$

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  First, arrange the numbers vertically so that the decimal points line up.
  ::首先,垂直排列数字,使小数点排成一行。

  $$\begin{array}{rl}& 189.34\\ & 123.5\\ & \underset{\_}{+7.2}\end{array}$$

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  Next, add one zero after the 5 in the second number and one zero after the 2 in the third number so that all numbers have two digits to the right of the decimal point.
  ::接下来,在第二个数字的5后面加上一个零,在第三个数字的2后面加上一个零,以便所有数字在小数点右侧有两个位数。

  $$\begin{array}{r}189.34\\ 123.50\\ \underset{\_}{+7.20}\end{array}$$

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  Now, add from right to left. Start by adding 4, 0, and 0 to make 4. Then add 3, 5, and 2 to make 10. Carry the 1 and add it with the 9, 3, and 7 to make 20. Carry the 2 and add it with the 8 and 2 to make 12. Carry the 1 and add it with the 1 and 1 to make 3. Insert a decimal point into your answer directly under the decimal points in the original numbers.
  ::现在, 从右到左添加。 从增加 4, 0, 0 开始做 4, 4, 4, 4, 4, 10, 3, 5, 2 10, 10, 3, 5, 10, 10, 10, 3, 3, 7 10, 10, 3, 3, 3, 3, 7 和 9, 20, 20, 3, 2, 8, 2 12, 12, 12, 3, 3, 4, 4, 1 和 1, 3, 3, 3 开始增加 。 将一个小数点直接插入在您的回答中小数小数的小数下的小数点 。

  $$\begin{array}{r}189.34\\ 123.50\\ \underset{\_}{+7.20}\\ 320.04\end{array}$$

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  The answer is  $\begin{array}{r}189.34+123.50+7.20=320.04\end{array}$ .
  ::答案是189.34+123.50+7.20=320.04。

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  Review
  ::回顾

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  Find each sum. 
  ::找到每一笔钱

  1. $\begin{array}{r}29.451+711.36\end{array}$  
  2. $\begin{array}{r}0.956+0.9885\end{array}$  
  3. $\begin{array}{r}13.69+42.23\end{array}$  
  4. $\begin{array}{r}58.744+12.06+8.8\end{array}$  
  5. $\begin{array}{r}67.90+1.2+3.456\end{array}$  

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  Round to the nearest tenth, then find the sum. 
  ::圆到最近的十分之一,然后找出总和。

  6. $\begin{array}{r}88.95+23.451\end{array}$  
  7. $\begin{array}{r}690.026+831.163\end{array}$  
  8. $\begin{array}{r}3.27+7.818\end{array}$  
  9. $\begin{array}{r}0.99+0.909+0.048\end{array}$  
  10. $\begin{array}{r}5.67+3.45+1.23\end{array}$  

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  Round to the nearest hundredth, then find the sum.
  ::回合到最近的一百,然后找出总和。

  11. $\begin{array}{r}511.51109+99.0986\end{array}$  
  12. $\begin{array}{r}0.744+7.005+0.7071\end{array}$  
  13. $\begin{array}{r}32.368+303.409\end{array}$  
  14. $\begin{array}{r}1.0262+1.0242\end{array}$  
  15. $\begin{array}{r}1.309+3.4590\end{array}$  

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  Review (Answers)
  ::回顾(答复)

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  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 。

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  Resources
  ::资源