During decimal multiplication, we know that the placement of the decimal point in the product is decided by the sum of number of digits after the decimal point in both the factors being multiplied.
I know that's what we do but i don't understand why this rule works?What's the reasoning behind it?
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Multiplying and dividing decimals, multiply two or more decimals.
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w w w . m o n t e r e y i n s t i t u t e . o r g
34.56
x3.002
is the same as 34x3 + 34x.002 + .56x3+.56x.002
so now note. the final decimal place will be decided by the last term above, .56x.002, or .00112 (five digits), which is the sum of the digits of the two numbers being multiplied.
The placement of the decimal point in the product during decimal multiplication is determined by the rule you mentioned, which states that the number of decimal places in the product is equal to the sum of the number of decimal places in both factors.
To understand why this rule works, let's break down the reasoning behind it using an example.
Consider the following multiplication: 2.5 x 0.4
Step 1: Disregard the decimal points and perform the multiplication as if you were multiplying whole numbers:
25 x 4 = 100
Step 2: Count the number of decimal places in both factors. In this example, the first factor (2.5) has one decimal place, and the second factor (0.4) has one decimal place as well.
Step 3: Add the numbers of decimal places together. In this case, 1 + 1 = 2.
Step 4: Place the decimal point in the product, starting from the right, according to the sum of decimal places obtained in the previous step. In our example, we will place the decimal point two places from the right.
Therefore, the product of 2.5 x 0.4 is 1.00.
Now, let's understand the reasoning behind this rule:
When we multiply two decimals, we can think of them as fractions. For example, 2.5 is equivalent to 2 1/2 (or 5/2) and 0.4 is equivalent to 4/10 (or 2/5).
To multiply these fractions, we multiply their numerators together and their denominators together:
(5/2) x (2/5) = 10/10 = 1
Now, when we convert this fraction back to a decimal, we get 1.00.
And here's the key: In order to convert this fraction back to a decimal, we need to ensure that the number of decimal places in the product matches the sum of the decimal places in the original factors. Otherwise, the fractional part of the product may not correctly represent the decimal value.
Hence, the rule is devised such that the placement of the decimal point in the product is determined by the sum of the decimal places in both factors. This ensures that the decimal portion of the product aligns correctly with the decimal values being multiplied.