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2.14: Estimation to Check Decimal Division

Difficulty Level: At Grade Created by: CK-12
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Let’s Think About It

Steve’s dad is looking to buy a new car. He is considering a car that would cost $21,045.68 if he paid for it over 48 months. Before deciding on the car, Steve’s dad wants to figure out what his monthly payments would be. He doesn’t want a car payment of more than $400 a month. How can Steve help his dad to determine whether or not he should buy this car?

In this concept, you will learn to estimate decimal quotients by dividing leading digits.

Guidance

Recall that estimation is a process by which you find an approximate solution. When you are working with decimal numbers with many digits, one way to estimate their quotient is to divide only their two leading digits.

Here are the steps for dividing decimal numbers using leading digits.

  1. Identify the two leading (left-most) digits of each number, preserving the location of the decimal point. Note that if 0 is the only digit to the left of a decimal point, it does not count as one of the leading digits.
  2. Divide the leading digits using your steps for decimal division.

Here is an example.

Estimate the quotient of \begin{align*}4.819 \div 1.245\end{align*}4.819÷1.245.

First, identify the two leading digits of each number.

  • The leading digits of 4.819 are 4.8
  • The leading digits of 1.245 are 1.2

Now, note that 4.8 is your dividend and 1.2 is your divisor.

Next, move the decimal points. You will move the decimal point of your divisor, 1.2, 1 place to the right to turn it into 12. Move the decimal point of your dividend, 4.8, 1 place as well to turn it into 48.

Now, divide as usual using long division.

\begin{align*}\overset{ \ \ \ \ \ \ \ \ 4}{12 \overline{ ) {48 \;}}}\end{align*}12)48¯¯¯¯¯¯¯¯        4

The answer is \begin{align*}4.819 \div 1.245\end{align*}4.819÷1.245 is approximately 4.

Guided Practice

Estimate the quotient by dividing the leading digits. If your answer is a repeating decimal, round it to the nearest tenth.

\begin{align*}6.4256 \div 2.2453\end{align*}6.4256÷2.2453

First, identify the two leading digits of each number.

  • The leading digits of 6.4256 are 6.4
  • The leading digits of 2.2453 are 2.2

Now, note that 6.4 is your dividend and 2.2 is your divisor.

Next, move the decimal points. You will move the decimal point of your divisor, 2.2, 1 place to the right to turn it into 22. Move the decimal point of your dividend, 6.4, 1 place as well to turn it into 64.

Now, divide as usual using long division.

\begin{align*}\begin{array}{rcl} && \overset{ \ \ \ \ \ \ \ \ \ \ 2.9090 \ldots}{22 \overline{ ) {64.0000 \;}}}\\ && \underline{\ - 44 \;\;\;\;\;\;\;\;\;}\\ && \quad \ \ 200 \\ && \underline{\ - 198\;\;\;\;\;\;\;\;}\\ && \quad \ \ \ \ \ \ 20 \\ && \underline{\ \ \ \ \ \ \ - 0\;\;\;\;\;\; }\\ && \quad \ \ \ \ \ \ 200 \\ && \underline{\ \ \ \ \ \ - 198\;\;}\\ && \quad \ \ \ \ \ \ \ \ \ 20 \end{array}\end{align*}22)64.0000¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯          2.9090 44  200 198      20       0      200      198         20
This time your answer is a repeating decimal. The \begin{align*}9090 \ldots\end{align*}9090 pattern will repeat over and over. You can round your answer to the nearest tenth.

The answer is \begin{align*}6.4256 \div 2.2453\end{align*} is approximately 2.9.

Examples

Example 1

Estimate the quotient of \begin{align*}15.934 \div 2.57\end{align*} using leading digits.

First, identify the two leading digits of each number.

  • The leading digits of 15.934 are 15
  • The leading digits of 2.57 are 2.5

Now, note that 15 is your dividend and 2.5 is your divisor.

Next, move the decimal points. You will move the decimal point of your divisor, 2.5, 1 place to the right to turn it into 25. Move the decimal point of your dividend, 15, 1 place as well to turn it into 150.

Now, divide as usual using long division.

\begin{align*} \overset{ \ \ \ \ \ \ \ \ \ \ \ 6 }{25 \overline{ ) {150 \;}}}\end{align*}

The answer is \begin{align*}15.934 \div 2.57\end{align*} is approximately 6.

Example 2

Estimate the quotient of \begin{align*}4.368 \div 3.12\end{align*} using leading digits. Round your answer to the nearest hundredth.

First, identify the two leading digits of each number.

  • The leading digits of 4.368 are 4.3
  • The leading digits of 3.12 are 3.1

Now, note that 4.3 is your dividend and 3.1 is your divisor.

Next, move the decimal points. You will move the decimal point of your divisor, 3.1, 1 place to the right to turn it into 31. Move the decimal point of your dividend, 4.3, 1 place as well to turn it into 43.

Now, divide as usual using long division.

\begin{align*}\begin{array}{rcl} && \overset{ \ \ \ \ \ \ \ \ \ \ 1.387 \ldots}{31 \overline{ ) {43.000 \;}}}\\ && \underline{\ - 31 \;\;\;\;\;\;\;}\\ && \quad \ \ 120 \\ && \underline{\ \ \ \ - 93\;\;\;\;}\\ && \quad \ \ \ \ \ 270 \\ && \underline{\ \ \ \ \ - 248\;\;}\\ && \quad \ \ \ \ \ \ \ \ 220 \\ && \underline{\ \ \ \ \ \ \ \ - 217\;\;}\\ && \quad \ \ \ \ \ \ \ \ \ \ \ \ \ 30 \\ \end{array}\end{align*}Because the question said to round your answer to the nearest hundredth, you can stop the long division once you’ve reached the thousandths place. \begin{align*}1.387 \ldots\end{align*} rounds to 1.39.

The answer is \begin{align*}4.368 \div 3.12\end{align*} is approximately 1.39.

Example 3

Estimate the quotient of \begin{align*}6.16 \div 1.12\end{align*} using leading digits. Round your answer to the nearest hundredth.

First, identify the two leading digits of each number.

  • The leading digits of 6.16 are 6.1
  • The leading digits of 1.12 are 1.1

Now, note that 6.1 is your dividend and 1.1 is your divisor.

Next, move the decimal points. You will move the decimal point of your divisor, 1.1, 1 place to the right to turn it into 11. Move the decimal point of your dividend, 6.1, 1 place as well to turn it into 61.

Now, divide as usual using long division.

\begin{align*}\begin{array}{rcl} && \overset{ \ \ \ \ \ \ \ \ 5.545 \ldots}{11 \overline{ ) {61.0000 \;}}}\\ && \underline{\ - 55 \;\;\;\;\;\;\;}\\ && \quad \ \ \ \ 60 \\ && \underline{\ \ \ \ - 55\;\;\;\;}\\ && \quad \ \ \ \ \ \ 50 \\ && \underline{\ \ \ \ \ \ - 44\;\;\;}\\ && \quad \ \ \ \ \ \ \ \ \ 60 \\ && \underline{\ \ \ \ \ \ \ \ \ - 55\;\;}\\ && \quad \ \ \ \ \ \ \ \ \ \ \ 50 \\ \end{array}\end{align*}

Because the question said to round your answer to the nearest hundredth, you can stop the long division once you’ve reached the thousandths place. \begin{align*}5.545 \ldots\end{align*} rounds to 5.55.

The answer is \begin{align*}6.16 \div 1.12\end{align*} is approximately 5.55.

Follow Up

Remember Steve and his dad who is looking to buy a new car? The car he is looking at would cost $21,045.68 and he would have 48 months to pay for it. Steve wants to figure out what his dad’s monthly payments would be to make sure they are not more than $400.

Steve needs to divide \begin{align*}$21,045.68 \div 48\end{align*}. Because Steve only wants to see if the monthly payment is above or below $400, he can estimate the answer by dividing the leading digits.

First, he should identify the two leading digits of each number.

  • The leading digits of 21,045.68 are 21,000 (remember that you have to maintain the placement of the decimal point!)
  • The leading digits of 48 are 48

Now, note that 21,000 is your dividend and 48 is your divisor. Because your divisor is already a whole number, you do not need to move any decimal points.

Now, divide as usual using long division.

\begin{align*}\begin{array}{rcl} && \overset{ \ \ \ \ \ \ \ \ 437.5}{48 \overline{ ) {21000.0 \;}}}\\ && \underline{\ - 192 \;\;\;\;\;\;\;}\\ && \quad \ \ \ \ 180 \\ && \underline{\ \ \ - 144 \;\;\;\;}\\ && \quad \ \ \ \ \ \ 360 \\ && \underline{\ \ \ \ \ - 336\;\;\;}\\ && \quad \ \ \ \ \ \ \ \ 240 \\ && \underline{\ \ \ \ \ \ \ \ - 240\;\;}\\ && \quad \ \ \ \ \ \ \ \ \ \ \ \ \ 0 \\ \end{array}\end{align*}

The answer is that the monthly payments will be approximately $437.50. Steve’s dad shouldn’t buy this car because the payments will be more than $400 a month.

Explore More

Estimate the quotients by dividing the leading digits. If your answer is a repeating decimal, round it to the nearest tenth.

  1. \begin{align*}4.992 \div .07123\end{align*} 
  2. \begin{align*}1.8921 \div 6.0341\end{align*}
  3. \begin{align*}26.2129 \div 1.3612\end{align*}
  4. \begin{align*}1.00765 \div .25\end{align*}
  5. \begin{align*}36.2129 \div 2.5612\end{align*}
  6. \begin{align*}.42129 \div .15612\end{align*}
  7. \begin{align*}6.4129 \div 2.2612\end{align*}
  8. \begin{align*}26.2129 \div 13.5612\end{align*}
  9. \begin{align*}8.2129 \div 2.2612\end{align*}
  10. \begin{align*}42.2129 \div 8.2612\end{align*}
  11. \begin{align*}16.2129 \div 4.1612\end{align*}
  12. \begin{align*}19.0029 \div 3.599\end{align*}
  13. \begin{align*}.45632 \div .09123\end{align*}
  14. \begin{align*}8.765 \div 1.098\end{align*}
  15. \begin{align*}.145632 \div .701023\end{align*}

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Vocabulary

Dividend

In a division problem, the dividend is the number or expression that is being divided.

divisor

In a division problem, the divisor is the number or expression that is being divided into the dividend. For example: In the expression 152 \div 6, 6 is the divisor and 152 is the dividend.

Estimation

Estimation is the process of finding an approximate answer to a problem.

Quotient

The quotient is the result after two amounts have been divided.

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Date Created:
Dec 02, 2015
Last Modified:
Mar 23, 2016
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