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Multiplication With Compensation Technique

Multiplication With Compensation Technique is a mathematical approach used to enhance the precision and accuracy of multiplication operations, especially when dealing with numbers that are close to each other in magnitude

Example: Let’s consider an example where we need to multiply two numbers: 0.12345 and 0.98765. When directly multiplying these two numbers using standard arithmetic, we get:

For Example:

0.12345 * 0.98765 = 0.1218693925

Now, let’s apply the Multiplication With Compensation Technique:

Calculate the product of the numbers without considering decimal points: 12345 * 98765 = 1218693925
Count the total number of decimal places in the original numbers (5 in each):
Insert the decimal point into the result, counting from the right: Result = 12186.93925

Compensation Multiplication Strategy

Multiplication With Compensation Technique strategy means converting the multiplier to a multiple of 10 and multiplying it by a given multiplicand. Further, subtract or add as many lots of the number as you need to maintain the balance. Compensation multiplication strategy works using the following equation.

Multiplicand Multiplier = Product

Example 1:

Multiply 27 → 13 using compensation multiplication strategy.

Solution:

Here, we will first change the multiplier 13 to a multiple of 10 by taking away 3. This gives:

13 → 10

Now, we will multiply the given multiplicand with a new multiplier. Accordingly, the new expression is:

27 → 10 = 270

Now, we will add (27 → 3 = 81) to the product (Note: It is because we have taken 3 from the multiplier to make the equation easier). Accordingly, the equation will be

270 + (27 3 = 81)

= 270 + 81

= 351

Therefore, 27 → 13 = 351

How do you use Multiplication With Compensation Technique?

Multiplication With Compensation Technique is a useful mathematical technique that involves adjusting numbers to simplify calculations. It can be particularly handy when you’re working with large numbers or mentally calculating products. By altering the numbers involved, you can make multiplication easier and more efficient.

For Example:

Example: Let’s say you want to calculate 37 x 18 mentally. Instead of directly multiplying these numbers, you can use compensation to simplify the process.

Start by rounding one of the numbers to a more convenient value. In this case, you can round 37 up to 40.
Next, determine the difference between the original number (37) and the rounded number (40), which is 3.
Now, multiply the other number (18) by the difference you calculated in step 2. So, 18 x 3 equals 54.
Add the result from step 3 to the rounded number (40 + 54), which equals 94.
Finally, adjust for the rounding you did in step 1. Since you rounded up, subtract the difference between the original number and the rounded number (3) from the result in step 4. So, 94 – 3 equals 91.

Multiplicative Compensation Method

This Multiplication With Compensation Technique strategy makes complex numbers easier to multiply. Let’s look at these two compensation strategy examples to understand this better.

Example 1: 76×5

Let’s multiply 76 by 10 and compensate by dividing the answer by 2.

76×5=76×102=7602=380

Example 2: 208×25

Let’s multiply 208 by 100 and compensate by dividing the answer by 4.

208×25=208×1004=208004=5200

Division Compensation Method

This Multiplication With Compensation Technique strategy makes complex numbers easier to multiply. We can understand it with the help of the following examples.

Example 1: 96÷24

To simplify the division, we divide both numbers by the same number.

Here, 962=48

242=12

So, 9624=4812

Now, for further division, we will need to know multiplication tables up to 12.

Since 12×4=48,

4812=4

Example 2: 245÷5

Instead of dividing by 5, we can multiply by 2 and then compensate by dividing the result by 10.

2455=24510×2=49010=49

Here, 245×2=490

Now, we divide 490 by 10 to get the final answer.

49010=49.