Dividing Whole Numbers by Decimals: Fifth Grade Mathematics

Dividing a whole number by a decimal requires transforming the divisor into a whole number by multiplying both the dividend and divisor by an identical power of 10.

The Divisor Transformation Principle

You cannot easily divide by a fractional unit like 0.4. However, you can multiply both the divisor and dividend by 10 (or 100, or 1,000) to create an equivalent problem with a whole-number divisor.

+--------------------------------------------------------------------------+
|                  TRANSFORMING THE DIVISOR TO A WHOLE NUMBER              |
|                                                                          |
|       Problem:  24 / 0.6                                                 |
|                                                                          |
|       Step 1: Count decimal places in the divisor:                       |
|               0.6 has 1 decimal place.                                   |
|       Step 2: Multiply both divisor and dividend by 10^1 (10):           |
|               Divisor:  0.6 * 10 = 6                                     |
|               Dividend: 24 * 10  = 240                                   |
|       Step 3: Solve the equivalent whole-number problem:                 |
|               240 / 6 = 40                                               |
|                                                                          |
|       Therefore, 24 / 0.6 = 40.                                          |
+--------------------------------------------------------------------------+

Why Multiplying Both Numbers Works

Dividing 24 by 0.6 is equivalent to the fraction 24 / 0.6. Multiplying both the numerator and denominator by 10 creates the equivalent fraction 240 / 6. Multiplying both terms by the same factor preserves the exact ratio and value of the quotient.

Fraction Form Proof:
      24        24 * 10       240
    ------  =  ---------  =  -----  =  40
     0.6       0.6 * 10        6

Step-by-Step Long Division Setup

When using the division bracket:
1. Move the decimal point in the divisor all the way to the right to make it a whole number.

2. Move the decimal point in the dividend the same number of places to the right, adding zeros as necessary.

3. Place the decimal point straight up into the quotient and divide.

Dividing 35 / 0.25:

   Step 1: Move divisor decimal 2 places right: 0.25 becomes 25.
   Step 2: Place decimal at end of 35 (35.) and move 2 places right:
           35. becomes 3,500.
   Step 3: Divide 3,500 by 25:
           3,500 / 25 = 140.

Exercises and Solutions

Exercise 1: Shifting and Annexing Zeros. Solve 48 / 0.8. Explain why the quotient is larger than the original whole-number dividend.

Solution 1: The divisor 0.8 has 1 decimal place. Multiply both the divisor and dividend by 10: 0.8 becomes 8, and 48 becomes 480. Now divide 480 by 8: 480 / 8 = 60. The quotient is 60. The quotient is larger than 48 because dividing by a number less than 1 asks how many fractional parts (each 8 tenths in size) fit inside 48 wholes. Since each group is smaller than a whole, more groups fit inside the dividend.

Exercise 2: Board Cutting Word Problem. A woodworker has a timber board that is 18 feet long. How many small shelving brackets measuring 0.75 feet each can be cut from the board?

Solution 2: Divide 18 by 0.75. The divisor 0.75 has 2 decimal places. Multiply both the divisor and dividend by 100: 0.75 becomes 75, and 18 becomes 1,800. Divide 1,800 by 75: 75 goes into 180 two times (2 * 75 = 150; 180 - 150 = 30). Bring down 0 to make 300. 75 goes into 300 four times (4 * 75 = 300; 300 - 300 = 0). The quotient is 24. Exactly 24 shelving brackets can be cut from the board.