1.13: Estimating Quotients

In this chapter, you will learn a shorter way to divide numbers that both end with zeros. First, do the following exercise and notice the pattern in the quotients.

Exercise 1

Divide. Check your work using the answer key at the end of the exercise.

  1. [latex]2 \enclose{longdiv}{6}[/latex]
  2. [latex]20 \enclose{longdiv}{60}[/latex]
  3. [latex]200 \enclose{longdiv}{600}[/latex]
  4. [latex]2000 \enclose{longdiv}{6000}[/latex]
  5. [latex]5 \enclose{longdiv}{25}[/latex]
  6. [latex]50 \enclose{longdiv}{250}[/latex]
  7. [latex]500 \enclose{longdiv}{2500}[/latex]
  8. [latex]5000 \enclose{longdiv}{25000}[/latex]
  9. [latex]14 \enclose{longdiv}{28}[/latex]
  10. [latex]140 \enclose{longdiv}{280}[/latex]
  11. [latex]1400 \enclose{longdiv}{2800}[/latex]
  12. [latex]14000 \enclose{longdiv}{28000}[/latex]

Exercise 1 Answers

  1. 3
  2. 3
  3. 3
  4. 3
  5. 5
  6. 5
  7. 5
  8. 5
  9. 2
  10. 2
  11. 2
  12. 2

Here is the shortcut:

When dividing numbers that both end with zeros, cross off the same number of zeros from the end of the divisor and the dividend. This is sometimes called cancelling zeros.

Example A

[latex]4800 \div 60 = 480\cancel{0} \div 6\cancel{0}[/latex]

[latex]\begin{array}{r}80\\ 6\enclose{longdiv}{480}\end{array}[/latex]

Example B

[latex]23000 \div 500 = 230\cancel{00} \div 5\cancel{00}[/latex]

[latex]\begin{array}{r}46\\ 5\enclose{longdiv}{230} \\ 20 \downarrow \\ \hline 30 \\ 30 \\ \hline 0\end{array}[/latex]

Example C

[latex]2\cancel{000}\enclose{longdiv}{680\cancel{000}}[/latex]

[latex]\begin{array}{r}340 \\ 2\enclose{longdiv}{680}\\ 6\downarrow \ \\\hline 08 \ \ \ \\ 8 \downarrow \\ \hline 00 \\ 0 \\ \hline 0\end{array}[/latex]

If you are interested in the facts of arithmetic that make this shortcut work, ask your instructor for an explanation.

Exercise 2

Quickly find the quotients. Remember to cancel the same number of zeros in the divisor and dividend.

  1. [latex]30\enclose{longdiv}{90}[/latex]
  2. [latex]40\enclose{longdiv}{1600}[/latex]
  3. [latex]300\enclose{longdiv}{1200}[/latex]
  4. [latex]400\enclose{longdiv}{20000}[/latex]
  5. [latex]500\enclose{longdiv}{35000}[/latex]
  6. [latex]700\enclose{longdiv}{42000}[/latex]
  7. [latex]60000\enclose{longdiv}{2400000}[/latex]
  8. [latex]800000\enclose{longdiv}{400000000}[/latex]

Exercise 2 Answers

  1. 3
  2. 40
  3. 4
  4. 50
  5. 70
  6. 60
  7. 40
  8. 500

Rounding Division Questions to Estimate

We round numbers and estimate to get a quick answer.

In division, round the divisor and dividend BEFORE you divide:

  • If the divisor only has one digit, do not round it,
  • Round the dividend to make the arithmetic easier for yourself.

Example D

Look at the two ways of rounding this question.

[latex]1796 \div 32 =[/latex]

The divisor (32) will round to 30. This dividend (1796) can be rounded to 1800 or to 2000. Let’s try each:

Round to 1800.

[latex]\begin{array}{r}60\ \ \ \ \\ 3\cancel{0}\enclose{longdiv}{180\cancel{0}}\end{array}[/latex]

Round to 2000.

[latex]\begin{array}{l}\hspace{2.5em}66\text{ R}2\\3\cancel{0}\enclose{longdiv}{200\cancel{0}} \\\hspace{2em}18\downarrow \\\hline\hspace{2.5em}20\\\hspace{2.5em}18\\\hline\hspace{1.5cm}2\end{array}[/latex]

Rounding 1796 to 1800 is easier arithmetic because 18 \div 3 works out evenly, so [latex]180 \div 3[/latex] works out evenly. Both estimates are correct.

Example E

[latex]2\,688\div28=[/latex]

  • Round the divisor (28) to 30.
  • Round the dividend (2688) to 2700

Round to 2700:

[latex]\begin{array}{r}90\ \ \ \ \\\ 3\cancel{0}\enclose{longdiv}{270\cancel{0}}\end{array}[/latex]

Round to 3000:

[latex]\begin{array}{r}100\ \ \ \ \ \\ 3\cancel{0}\enclose{longdiv}{300\cancel{0}}\end{array}[/latex]

Both estimates are correct and both are easy to do.

Example F

[latex]2\,893\div47=[/latex]

  • Round the divisor (47) to 50.
  • Round the dividend (2893) to 2900 or 3000.

Which rounded dividend will be easier to divide by 50?

Answer: The 3000 because 5 goes evenly into 30.

[latex]\begin{array}{r}60\ \ \ \ \ \\5\cancel{0} \enclose{longdiv}{300\cancel{0}}\end{array}[/latex]

Exercise 3

Give an estimated quotient.  Show your rounding.

  1. [latex]365\enclose{longdiv}{27692}[/latex]
  2. [latex]23\enclose{longdiv}{34459}[/latex]
  3. [latex]45\enclose{longdiv}{4590}[/latex]
  4. [latex]16\enclose{longdiv}{6729}[/latex]
  5. [latex]56\enclose{longdiv}{4792}[/latex]
  6. [latex]75\enclose{longdiv}{7648}[/latex]
  7. [latex]81\enclose{longdiv}{4049}[/latex]
  8. [latex]68\enclose{longdiv}{5636}[/latex]
  9. [latex]19\enclose{longdiv}{1672}[/latex]
  10. [latex]200\enclose{longdiv}{20000}[/latex]
  11. [latex]557\enclose{longdiv}{41680}[/latex]

Exercise 3 Answers

  1. [latex]28\,000\div400=70[/latex]
  2. [latex]34\,000 \div 20 =1700[/latex]
  3. [latex]5\,000 \div 50 =100[/latex]
  4. [latex]7\,000 \div 20 = 350[/latex]
  5. [latex]4\,800 \div 60 = 80[/latex]
  6. [latex]8\,000 \div 80 = 100[/latex]
  7. [latex]4\,000 \div 80 = 50[/latex]
  8. [latex]5\,600 \div 70 = 80[/latex]
  9. [latex]2\,000 \div 20 = 100[/latex]
  10. [latex]20\,000 \div 200 = 100[/latex]
  11. [latex]42\,000 \div 600 = 70[/latex]

1.13: Practice Questions

  1. Give an estimated quotient. Show your work.
    1. [latex]98\enclose{longdiv}{8541}[/latex]
    2. [latex]27\enclose{longdiv}{2963}[/latex]
    3. [latex]241\enclose{longdiv}{26348}[/latex]
    4. [latex]55\enclose{longdiv}{3276}[/latex]
    5. [latex]24\enclose{longdiv}{1776}[/latex]
    6. [latex]59\enclose{longdiv}{11830}[/latex]

1.13: Practice Answers

  1. Give an estimated quotient. Show your work.
    1. [latex]8\,500\div 100 = 85[/latex]
    2. [latex]3\,000 \div 30 =100[/latex]
    3. [latex]26\,000 \div 200 =130[/latex]
    4. [latex]3\,000 \div 60 = 50[/latex]
    5. [latex]2\,000\div 20 = 100[/latex]
    6. [latex]12\,000 \div 60 = 200[/latex]

Attribution

This chapter has been adapted from Topic E: Estimating Quotients in Adult Literacy Fundamental Mathematics: Book 3 – 2nd Edition (BCcampus) by Wendy Tagami and Liz Girard (2023), which is under a CC BY 4.0 license.

License

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Intermediate PreAlgebra: Building Success Copyright © 2024 by Kim Tamblyn, TRU Open Press is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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