Multiplying Decimals by Decimals: Fifth Grade Mathematics
Multiplying a decimal by another decimal involves finding a part of a part, where the product contains the combined total of decimal places from both factors.
Visualizing Multiplication of Decimals with Area Models
When multiplying two decimals both less than 1, such as 0.7 * 0.4, an area model visually demonstrates why the product becomes smaller than either factor.
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| AREA MODEL FOR 0.7 * 0.4 ON A 10x10 GRID |
| |
| - Shade 7 vertical columns representing 0.7 (7 tenths). |
| - Shade 4 horizontal rows representing 0.4 (4 tenths). |
| - The overlapping region forms a rectangle of 7 * 4 = 28 squares. |
| - Since each square is 0.01 (one hundredth), the product is 0.28. |
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ASCII Grid of Overlap (Partial Representation):
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1 [ X ][ X ][ X ][ X ][ X ][ X ][ X ][ ][ ][ ]
0.2 [ X ][ X ][ X ][ X ][ X ][ X ][ X ][ ][ ][ ]
0.3 [ X ][ X ][ X ][ X ][ X ][ X ][ X ][ ][ ][ ]
0.4 [ X ][ X ][ X ][ X ][ X ][ X ][ X ][ ][ ][ ]
0.5 [ ][ ][ ][ ][ ][ ][ ][ ][ ][ ]
...
Overlap region contains 4 rows * 7 columns = 28 squares = 0.28
The Standard Algorithm for Decimal Multiplication
Follow these three steps to multiply any two decimals:
1. Align the digits to the right as if multiplying whole numbers. Do NOT line up the decimal points!
2. Multiply the numbers using the standard multiplication algorithm.
3. Count the total number of decimal places in BOTH factors. Count that same number of places from right to left in the product and place the decimal point.
Multiplying 3.45 * 2.6:
3 . 4 5 <--- 2 decimal places
* 2 . 6 <--- 1 decimal place
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2 0 7 0 <--- 345 * 6
+ 6 9 0 0 <--- 345 * 20
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8 . 9 7 0 <--- 3 total decimal places (2 + 1 = 3)
Final Product: 8.97 (trailing zero can be dropped)
Why You Do Not Line Up Decimals When Multiplying
Lining up decimals is required for addition and subtraction because you can only combine identical place values. In multiplication, you are multiplying every place value of one factor by every place value of the other factor. Lining up decimals creates unnecessary placeholder zeros that make multiplication unwieldy.
Exercises and Solutions
Exercise 1: Standard Decimal Multiplication. Calculate 4.82 * 0.75 using the standard algorithm. Count the decimal places in each factor and explain how the total dictates decimal placement.
Solution 1: Multiply 482 by 75 as whole numbers. First partial product: 482 * 5 = 2,410. Second partial product: 482 * 70 = 33,740. Add the partial products: 2,410 + 33,740 = 36,150. Count the decimal places in the factors: 4.82 has 2 decimal places, and 0.75 has 2 decimal places, giving a total of 2 + 2 = 4 decimal places. Start at the right of 36150 and count 4 places to the left: 3.6150. Simplifying by dropping the trailing zero gives 3.615.
Exercise 2: Finding Area with Decimal Dimensions. A rectangular craft tile measures 2.4 meters long and 1.35 meters wide. Calculate the area of the tile in square meters.
Solution 2: Multiply length by width: 2.4 * 1.35. Multiply 135 by 24 ignoring decimals: 135 * 4 = 540; 135 * 20 = 2,700; sum of partial products = 540 + 2,700 = 3,240. Count decimal places: 1.35 has 2 decimal places and 2.4 has 1 decimal place, totaling 3 decimal places. Move 3 places to the left in 3240: 3.240. The area of the tile is 3.24 square meters.