Dividing 4-Digit Numbers by 1-Digit Divisors - Fourth Grade Mathematics
Dividing a four-digit number by a single-digit divisor is one of the crowning computational achievements of elementary school mathematics. Whether calculating the equal distribution of 5,424 medical kits among 6 regional clinics or determining the weekly savings needed to reach a $3,750 goal over 5 months, multi-digit division enables you to partition large quantities with absolute precision. In this chapter, you will master the classic step-by-step division routine—Divide, Multiply, Subtract, Bring Down—learning how place-value regrouping moves digits from thousands to hundreds, tens, and ones, while banishing the mystery of interior zeros in the quotient.
The Classic Four-Step Cycle
The standard long division algorithm operates in a repeating four-step rhythm across each place-value column from left to right.
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| THE FOUR-STEP DIVISION CYCLE |
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| [D] DIVIDE : How many times does the divisor fit into the current value?|
| [M] MULTIPLY : Multiply that quotient digit by the divisor. |
| [S] SUBTRACT : Subtract the product from the current value to find leftovers|
| [B] BRING DOWN : Bring down the next digit from the dividend. |
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| Memory Device : Does McDonalds Sell Burgers? |
| (Divide, Multiply, Subtract, Bring down!) |
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The Critical "Check" Step Before Bringing Down
There is an unspoken golden rule embedded between Subtract and Bring Down: the Check step! After you subtract, your difference must be strictly less than your divisor: Difference < Divisor. If your difference is equal to or greater than the divisor, your division estimate was too small! You must stop, erase that quotient digit, and increase it by 1 before bringing down the next digit.
Fully Annotated Step-by-Step Long Division
Let us execute the complete algorithm on a four-digit dividend: 4,738 divided by 3.
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| STEP-BY-STEP CALCULATION: 4,738 / 3 |
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| 1 , 5 7 9 R1 |
| 3 ) 4 , 7 3 8 |
| - 3 |
| --- |
| 1 7 |
| - 1 5 |
| ----- |
| 2 3 |
| - 2 1 |
| ----- |
| 2 8 |
| - 2 7 |
| ----- |
| 1 |
| |
| CYCLE 1 (Thousands): |
| D: 4 / 3 = 1 thousand. Write 1 above 4. |
| M: 1 x 3 = 3. |
| S: 4 - 3 = 1 thousand remaining. (1 < 3, Check passed!) |
| B: Bring down 7 hundreds to make 17 hundreds. |
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| CYCLE 2 (Hundreds): |
| D: 17 / 3 = 5 hundreds. Write 5 above 7. |
| M: 5 x 3 = 15. |
| S: 17 - 15 = 2 hundreds remaining. (2 < 3, Check passed!) |
| B: Bring down 3 tens to make 23 tens. |
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| CYCLE 3 (Tens): |
| D: 23 / 3 = 7 tens. Write 7 above 3. |
| M: 7 x 3 = 21. |
| S: 23 - 21 = 2 tens remaining. (2 < 3, Check passed!) |
| B: Bring down 8 ones to make 28 ones. |
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| CYCLE 4 (Ones): |
| D: 28 / 3 = 9 ones. Write 9 above 8. |
| M: 9 x 3 = 27. |
| S: 28 - 27 = 1 one remaining. (1 < 3, Check passed!) |
| Nothing left to bring down! |
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| Final Result: 1,579 R1 |
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The Peril of the Hidden Interior Zero
The most common mistake in multi-digit long division occurs when the divisor cannot fit into the number formed after bringing down a digit. Consider 6,180 divided by 6: Cycle 1: 6 / 6 = 1. 1 x 6 = 6. 6 - 6 = 0. Bring down 1. Now look closely: how many times does 6 fit into 1? Zero times! You MUST write a 0 in the quotient above the 1! If you forget to write 0 and immediately bring down the 8, your quotient will become 130 instead of 1,030. That drops the value by nearly 900! Every time you bring down a digit, you must record a digit in the quotient—even if that digit is 0.
Chapter Practice Exercises
Exercise 1: Compute 7,425 / 5 using the standard long division algorithm. Show all four cycles of Divide, Multiply, Subtract, and Bring Down.
Exercise 2: Compute 8,164 / 4. Pay close attention to the hundreds place and clearly show whether an interior zero appears in the quotient.
Exercise 3: Compute 5,639 / 7. Determine the quotient and remainder, and show each step of your work.
Exercise 4: A factory manufactures 4,392 glass ornaments. The ornaments are packed 6 to a box. How many boxes can be completely filled? Are any ornaments left over?
Exercise 5: A student calculates 9,216 / 9 and writes an answer of 124. Without solving the whole problem from scratch, explain why the student's quotient is unreasonable using estimation, identify where the student made an error, and state the correct quotient.
Solutions and Step-by-Step Explanations
Solution 1: Executing 7,425 / 5: Cycle 1 (thousands): 7 / 5 = 1 (write 1); 1 x 5 = 5; 7 - 5 = 2. Bring down 4 to make 24. Cycle 2 (hundreds): 24 / 5 = 4 (write 4); 4 x 5 = 20; 24 - 20 = 4. Bring down 2 to make 42. Cycle 3 (tens): 42 / 5 = 8 (write 8); 8 x 5 = 40; 42 - 40 = 2. Bring down 5 to make 25. Cycle 4 (ones): 25 / 5 = 5 (write 5); 5 x 5 = 25; 25 - 25 = 0. The quotient is 1,485 with zero remainder.
Solution 2: Executing 8,164 / 4: Cycle 1 (thousands): 8 / 4 = 2 (write 2); 2 x 4 = 8; 8 - 8 = 0. Bring down 1. Cycle 2 (hundreds): 1 cannot be divided by 4 (fits 0 times). Write 0 in the hundreds place of the quotient! 0 x 4 = 0; 1 - 0 = 1. Bring down 6 to make 16. Cycle 3 (tens): 16 / 4 = 4 (write 4); 4 x 4 = 16; 16 - 16 = 0. Bring down 4. Cycle 4 (ones): 4 / 4 = 1 (write 1); 1 x 4 = 4; 4 - 4 = 0. The quotient is 2,041 with zero remainder. Notice the essential interior zero in the hundreds place.
Solution 3: Executing 5,639 / 7: The divisor 7 does not fit into 5 thousands. We group 5 thousands with 6 hundreds to make 56 hundreds. Cycle 1 (hundreds): 56 / 7 = 8 (write 8 above 6); 8 x 7 = 56; 56 - 56 = 0. Bring down 3. Cycle 2 (tens): 3 cannot be divided by 7 (fits 0 times). Write 0 in the tens place! 0 x 7 = 0; 3 - 0 = 3. Bring down 9 to make 39. Cycle 3 (ones): 39 / 7 = 5 (write 5 above 9); 5 x 7 = 35; 39 - 35 = 4. There are no further digits to bring down. The quotient is 805 with a remainder of 4 (805 R4).
Solution 4: Divide 4,392 by 6: Cycle 1: 43 / 6 = 7; 7 x 6 = 42; 43 - 42 = 1. Bring down 9 to make 19. Cycle 2: 19 / 6 = 3; 3 x 6 = 18; 19 - 18 = 1. Bring down 2 to make 12. Cycle 3: 12 / 6 = 2; 2 x 6 = 12; 12 - 12 = 0. The factory can fill exactly 732 boxes with zero ornaments left over.
Solution 5: Estimating 9,216 / 9: 9,000 / 9 = 1,000. The quotient must be slightly greater than 1,000! The student's answer of 124 is roughly ten times too small. The student made the classic interior zero omission: after computing 9 / 9 = 1, the student brought down 2. Because 9 does not fit into 2, the student should have recorded a 0 in the hundreds place before bringing down 1 to make 21 (21 / 9 = 2, with remainder 3; then 36 / 9 = 4). By skipping the 0, the digits shifted one column to the right, collapsing 1,024 into 124. The correct quotient is 1,024.