Dividing Decimals by Decimals: Fifth Grade Mathematics
Dividing a decimal by another decimal requires shifting the decimal point in both numbers to create a whole-number divisor before executing the standard division algorithm.
The Two-Step Shift Algorithm
Whenever the divisor contains a decimal point, follow this two-step procedure:
1. Move the decimal point in the divisor to the right until the divisor is a whole number. Count how many places you moved it.
2. Move the decimal point in the dividend to the right by that exact same number of places.
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| THE TWO-STEP DECIMAL SHIFT RULE |
| |
| Problem: 1.75 / 0.5 |
| |
| Step 1: Shift divisor decimal 1 place right: |
| 0.5 ---> 5 |
| Step 2: Shift dividend decimal 1 place right: |
| 1.75 ---> 17.5 |
| |
| Equivalent Problem: 17.5 / 5 = 3.5 |
| |
| Notice: The dividend does NOT need to become a whole number! |
| Only the divisor must be a whole number. |
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When the Dividend Needs Annexed Zeros
If the divisor requires more decimal shifts than the dividend has decimal places, annex zeros to the dividend.
Dividing 3.6 / 0.08:
Divisor 0.08 has 2 decimal places. Shift 2 places right -> 8.
Dividend 3.6 must shift 2 places right:
Shift 1: 36.
Shift 2: 360. (Annex a zero!)
New Problem: 360 / 8 = 45.
Standard Division Setup and Execution
Once the divisor is a whole number, place the decimal point directly into the quotient above its new position in the dividend.
Solving 0.945 / 0.27:
Step 1: Divisor 0.27 shifts 2 places right -> 27.
Step 2: Dividend 0.945 shifts 2 places right -> 94.5.
Step 3: Set up long division:
3 . 5
+--------
27 | 9 4 . 5
- 8 1
-----
1 3 5
- 1 3 5
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0
Quotient: 3.5.
Exercises and Solutions
Exercise 1: Standard Decimal Division. Solve 4.92 / 1.2 using the decimal shift procedure. Show the shifted values, the placement of the decimal point, and the division steps.
Solution 1: Divisor 1.2 has 1 decimal place. Multiply both divisor and dividend by 10 to shift 1 place right: 1.2 becomes 12, and 4.92 becomes 49.2. Set up 49.2 divided by 12. Place the decimal point in the quotient directly above the decimal in 49.2. Divide: 12 goes into 49 four times (4 * 12 = 48; 49 - 48 = 1). Bring down 2 tenths to make 12. 12 goes into 12 one time (1 * 12 = 12; 12 - 12 = 0). The quotient is 4.1.
Exercise 2: Annexing Zeros in the Dividend. Calculate 8.5 / 0.25. Explain why a zero had to be annexed to the dividend during the shift.
Solution 2: Divisor 0.25 has 2 decimal places. To convert 0.25 into the whole number 25, you must shift the decimal point 2 places to the right (multiplying by 100). The dividend 8.5 has only 1 decimal place. Shifting 1 place right moves the decimal to 85. To complete the required 2-place shift, you must annex a zero to make 850. Divide 850 by 25: 25 goes into 85 three times (3 * 25 = 75; 85 - 75 = 10). Bring down 0 to make 100. 25 goes into 100 four times (4 * 25 = 100; 100 - 100 = 0). The final quotient is 34.