29.2: Using Long Division

Basic Math (Grade 6) · LibreTexts

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    Lesson

    Let's use long division.

    Exercise \(\PageIndex{1}\): Number Talk: Estimating Quotients

    Estimate these quotients mentally.

    \(500\div 7\)

    \(1,394\div 9\)

    Exercise \(\PageIndex{2}\): Lin Uses Long Division

    Lin has a method of calculating quotients that is different from Elena’s method and Andre’s method. Here is how she found the quotient of \(657\div 3\):

    clipboard_e7e4c0de84ad595f473cbb2ec7810c370.png
    Figure \(\PageIndex{1}\): Long division calculation of 657 divided by 3, 4 steps. Step 1, 1 row. First row: 3, long division symbol with 657 inside. Step 2, 4 rows. First row: 2, second row: 3, long division symbol with 657 inside, third row: minus 6, horizontal line, fourth row: 0 5. Step 3, 6 rows. First row: 2 1, second row: 3, long division symbol with 657 inside, third row: minus 6, horizontal line, fourth row: 5, fifth row: minus 3, horizontal line, sixth row: 2. Step 4, 8 rows. First row: 219, second row: 3, long division symbol with 657 inside, third row: minus 6, horizontal line, fourth row: 5, fifth row: minus 3, horizontal line, sixth row: 27, seventh row: minus 27, horizontal line, eighth row: 0.
    1. Discuss with your partner how Lin’s method is similar to and different from drawing base-ten diagrams or using the partial quotients method.
      • Lin subtracted \(3\cdot 2\) then \(3\cdot 1\), and lastly \(3\cdot 9\). Earlier, Andre subtracted \(3\cdot 200\) then \(3\cdot 10\), and lastly \(3\cdot 9\). Why did they have the same quotient?
      • In the third step, why do you think Lin wrote the 7 next to the remainder of 2 rather than adding 7 and 2 to get 9?
    2. Lin’s method is called long division. Use this method to find the following quotients. Check your answer by multiplying it by the divisor.
      1. \(846\div 3\)
      2. \(1,816\div 4\)
      3. \(768\div 12\)

    Exercise \(\PageIndex{3}\): Dividing Whole Numbers

    1. Find each quoteint.
    1. \(633\div 3\)
    2. \(1001\div 7\)
    3. \(2996\div 14\)
    1. Here is Priya's calculation of \(906\div 3\)
    clipboard_efa195e53b035f15cc21e449b40a8be63.png
    Figure \(\PageIndex{2}\): Long division of 906 divided by 3. 6 rows. First row: 320. Second row: 3, long division symbol with 906 inside. Third row: minus 9. Horizontal line. Fourth row: 0 6. Fifth row: minus 6. Horizontal line. Sixth row: 0.
    1. Priya wrote 320 for the value of \(906\div 3\). Check her answer by multiplying it by 3. What product do you get and what does it tell you about Priya’s answer?
    2. Describe Priya’s mistake, then show the correct calculation and answer.

    Summary

    Long division is another method for calculating quotients. It relies on place value to perform and record the division.

    When we use long division, we work from left to right and with one digit at a time, starting with the leftmost digit of the dividend. We remove the largest group possible each time, using the placement of the digit to indicate the size of each group. Here is an example of how to find \(948\div 3\) using long division.

    clipboard_e2769b77cb980b99501a7e13fe4d1c264.png
    Figure \(\PageIndex{3}\): Long division of 948 divided by 3. 8 rows. First row: 316. Second row: 3, long division symbol with 948 inside. Third row: minus 9, 3 groups of 3 hundreds. Horizontal line. Fourth row: 4. Fifth row: minus 3, 3 groups of 1 ten. Horizontal line. Sixth row: 18. Seventh row: minus 18, 3 groups of 6 ones. Horizontal line. Eight row: 0.
    • We start by dividing 9 hundreds into 3 groups, which means 3 hundreds in each group. Instead of writing 300, we simply write 3 in the hundreds place, knowing that it means 3 hundreds.
    • There are no remaining hundreds, so we work with the tens. We can make 3 groups of 1 ten in 4 tens, so we write 1 in the tens place above the 4 of 948. Subtracting 3 tens from 4 tens, we have a remainder of 1 ten.
    • We know that 1 ten is 10 ones. Combining these with the 8 ones from 948, we have 18 ones. We can make 3 groups of 6, so we write 6 in the ones place.

    In total, there are 3 groups of 3 hundreds, 1 ten, and 6 ones in 948, so \(948\div 3=316\).

    Glossary Entries

    Definition: Long Division

    Long division is a way to show the steps for dividing numbers in decimal form. It finds the quotient one digit at a time, from left to right.

    For example, here is the long division for \(57\div 4\).

    clipboard_e16caacb74b7f1b1bc6ac22b7041e343e.png
    Figure \(\PageIndex{4}\)

    Practice

    Exercise \(\PageIndex{4}\)

    Kiran is using long division to find \(696\div 12\).

    clipboard_e2ae401f77b30e15fb3400dd2a05e38fc.png
    Figure \(\PageIndex{5}\)

    He starts by dividing 69 by 12. In which decimal place should Kiran place the first digit of the quotient (5)?

    1. Hundreds
    2. Tens
    3. Ones
    4. Tenths

    Exercise \(\PageIndex{5}\)

    Here is a long-division calculation of \(917\div 7\).

    clipboard_e423981e43c2d10d578c9cc59f5b5bda9.png
    Figure \(\PageIndex{6}\): Example division problem 917 divided by 7. Inside the division bar is 917. 7 divides into 9, one time. Subtract 7 for a remainder of 2. Bring down the 1. 7 divides into 21, three times. Subtract 21 from 21 for a remainder of 0. Bring down the 7. 7 divides into 7, one time. Subtract 7 for a remainder of 0. The quotient is 131.
    1. There is a 7 under the 9 of 917. What does this 7 represent?
    2. What does the subtraction of 7 from 9 mean?
    3. Why is a 1 written next to the 2 from \(9-7\)?

    Exercise \(\PageIndex{6}\)

    Han's calculation of \(972\div 9\) is shown here.

    clipboard_e3c74feec14496c28b35791e68245b87f.png
    Figure \(\PageIndex{7}\): Division problem. Under the division house is 972. To the left of the house is 9. Above the house is 180. Below 972, row 1 is minus 9. horizontal line, row 2, 7. row 3, minus 0, horizontal line, Row 4, 72. Row 5 minus 72. horizontal line. Row 6. 0.
    1. Find \(180\cdot 9\).
    2. Use your calculation of \(180\cdot 9\) to explain how you know Han has made a mistake.
    3. Identify and correct Han’s mistake.

    Exercise \(\PageIndex{7}\)

    Find each quotient.

    clipboard_e498ab22349a6ffd8919f5105f7803c5b.png
    Figure \(\PageIndex{8}\)

    Exercise \(\PageIndex{8}\)

    One ounce of a yogurt contains of \(1.2\) grams of sugar. How many grams of sugar are in \(14.25\) ounces of yogurt?

    1. \(0.171\) grams
    2. \(1.71\) grams
    3. \(17.1\) grams
    4. \(171\) grams

    (From Unit 5.3.3)

    Exercise \(\PageIndex{9}\)

    The mass of one coin is 16.718 grams. The mass of a second coin is 27.22 grams. How much greater is the mass of the second coin than the first? Show your reasoning.

    (From Unit 5.2.3)

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