Multiplying Decimals by Whole Numbers: Fifth Grade Mathematics

Multiplying a decimal by a whole number combines whole-number multiplication algorithms with place-value decimal counting.

Visualizing Decimal Multiplication as Repeated Addition

Multiplying a decimal by a whole number represents repeated addition of that decimal quantity. For example, 3 * 0.4 means 3 groups of 4 tenths: 0.4 + 0.4 + 0.4 = 1.2.

Number Line Representation of 4 * 0.3:

<---+---+---+---+---+---+---+---+---+---+---+---+---+--->
   0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
       [-- 0.3 --]
                   [-- 0.3 --]
                               [-- 0.3 --]
                                           [-- 0.3 --]
   Total Distance: 4 jumps of 0.3 = 1.2

The Standard Multiplication Algorithm

To multiply a decimal by a whole number:
1. Ignore the decimal point initially and multiply the numbers as if they were both whole numbers.

2. Count the total number of decimal places (digits to the right of the decimal point) in the decimal factor.

3. Place the decimal point in the product so it has that same number of decimal places, counting from right to left.

Multiplying 4.36 * 7:

   Step 1: Multiply as whole numbers:
                 2   4
                 4   3   6
              *          7
              ------------
               3  0  5   2

   Step 2: Count decimal places in factors:
           4.36 has 2 decimal places.
           7 has 0 decimal places.
           Total = 2 decimal places.

   Step 3: Place decimal point in product:
           Starting at the right of 3052, count 2 places left: 30.52.

Why Do We Count Decimal Places?

Counting decimal places is grounded in fraction multiplication. The number 4.36 is 436/100. Multiplying 436/100 by 7 gives (436 * 7) / 100 = 3052/100. Dividing 3,052 by 100 shifts the digits two places to the right, yielding 30.52.

Exercises and Solutions

Exercise 1: Whole Number and Multi-Digit Decimal. Calculate 14 * 3.25 using the standard algorithm. Show the whole-number multiplication and explain how to determine the final decimal placement.

Solution 1: Multiply 325 by 14 as whole numbers. First partial product: 325 * 4 = 1,300. Second partial product: 325 * 10 = 3,250. Add partial products: 1,300 + 3,250 = 4,550. Next, count the decimal places in the factors: 3.25 has 2 decimal places, and 14 has 0 decimal places, for a total of 2 decimal places. Count 2 places from the right in 4550: 45.50. The final product is 45.50 (or 45.5).

Exercise 2: Real-World Hourly Earnings. An employee earns $16.75 per hour at a part-time job. If the employee works 18 hours in one week, what are their total gross earnings?

Solution 2: Multiply 16.75 by 18. Multiply 1675 by 18 ignoring decimals: 1675 * 8 = 13,400; 1675 * 10 = 16,750; adding partial products gives 13,400 + 16,750 = 30,150. Count decimal places: 16.75 has 2 decimal places, while 18 has 0, requiring 2 decimal places in the answer. Starting at the right of 30150, move 2 places left to place the decimal: 301.50. The employee's total gross earnings are $301.50.