Dividing Decimals by Decimals - Sixth Grade Mathematics

Dividing a decimal by another decimal requires transforming the divisor into a whole number by shifting the decimal point, preserving numerical equivalence through powers of ten scaling.

The Divisor Transformation Principle

In long division, dividing directly by a decimal divisor (such as dividing by 0.25) is mathematically awkward because long division algorithms require counting in whole-unit groups. To resolve this, we transform the divisor into an integer by multiplying both the divisor and the dividend by an identical power of 10.

The Fundamental Equivalence Principle:
Any division problem a / b can be written as a fraction: a / b.
Multiplying both numerator and denominator by 10, 100, or 1000
creates an equivalent fraction with an identical quotient!

        4.5 / 0.15  =  (4.5 * 100) / (0.15 * 100)  =  450 / 15  =  30

The Operational Protocol

Dividing decimals follows a four-step shifting protocol.

+-------------------------------------------------------------------------+
|                  DIVIDING DECIMALS BY DECIMALS PROTOCOL                 |
|                                                                         |
|   Step 1: Look at the DIVISOR (outside the bracket). Count how many     |
|           decimal places it contains (n places).                        |
|   Step 2: Move the decimal point n places to the right to make the      |
|           divisor a whole number.                                       |
|   Step 3: Move the decimal point in the DIVIDEND (inside the bracket)   |
|           the EXACT SAME number of places (n places) to the right.      |
|           Annex zeros if needed to complete the shift.                  |
|   Step 4: Bring the new decimal point STRAIGHT UP into the quotient,    |
|           and divide as with whole numbers.                             |
+-------------------------------------------------------------------------+

Worked Example: Evaluate 7.35 / 0.3
Step 1: The divisor 0.3 has 1 decimal place.
Step 2: Shift divisor decimal 1 place right: 0.3 becomes 3.
Step 3: Shift dividend decimal 1 place right: 7.35 becomes 73.5.
Step 4: The problem is now: 73.5 / 3.
        Bring decimal point straight up:
        - 3 into 7 goes 2 times (remainder 1).
        - Bring down 3 to make 13: 3 into 13 goes 4 times (remainder 1).
        - Bring down 5 to make 15: 3 into 15 goes 5 times (remainder 0).
Result: 24.5.

Annexing Zeros in the Dividend During the Shift

When the divisor has more decimal places than the dividend, shifting the decimal point requires annexing zeros to the dividend.

Evaluate: 12 / 0.04
Step 1: Divisor 0.04 has 2 decimal places.
Step 2: Shift divisor 2 places right to make 4.
Step 3: The dividend 12 has an implicit decimal: 12.00.
        Shift decimal 2 places right: 12 becomes 1,200.
Step 4: Divide: 1,200 / 4 = 300.
Conclusion: 12 / 0.04 = 300.

Why Does Dividing by a Small Decimal Produce a Huge Quotient?

Notice that 12 / 0.04 produced 300. In physical terms, 0.04 represents 4 pennies (four hundredths of a dollar). The question asks: "How many 4-cent increments are in 12 whole dollars?" Since each dollar contains 25 groups of 4 cents, 12 dollars contains 12 * 25 = 300 groups. Dividing by a decimal smaller than 1 always produces a quotient larger than the dividend.

Exercises

Exercise 1. Calculate each quotient by shifting decimal points: (a) 4.8 / 0.6; (b) 1.44 / 0.12; (c) 6.3 / 0.09.

Exercise 2. Calculate the quotient: 0.075 / 0.25, showing all shifting and division steps.

Exercise 3. A ribbon of length 15.75 feet is cut into decorative strips measuring 0.75 feet each. How many decorative strips can be produced?

Solutions

Solution to Exercise 1. For (a) 4.8 / 0.6: shift both 1 place right to get 48 / 6 = 8. For (b) 1.44 / 0.12: shift both 2 places right to get 144 / 12 = 12. For (c) 6.3 / 0.09: shift both 2 places right; 6.3 becomes 630, and 0.09 becomes 9; 630 / 9 = 70.

Solution to Exercise 2. In 0.075 / 0.25, the divisor 0.25 has 2 decimal places. Shift both numbers 2 places to the right: the divisor becomes 25, and the dividend 0.075 becomes 7.5. The problem is now 7.5 / 25. Place the decimal straight up above the decimal point. 25 into 7 goes 0 times. 25 into 75 goes 3 times (3 * 25 = 75; remainder 0). The quotient is 0.3.

Solution to Exercise 3. To find the number of strips, divide total ribbon by strip length: 15.75 / 0.75. The divisor 0.75 has 2 decimal places. Shift both decimal points 2 places to the right, transforming the problem into 1,575 / 75. Divide using long division: 75 into 157 goes 2 times (2 * 75 = 150; 157 - 150 = 7). Bring down 5 to make 75: 75 into 75 goes 1 time (1 * 75 = 75; remainder 0). Exactly 21 decorative strips can be produced.