Question

Multiply the following numbers by formula Nikhilam
(i) 11 $\times $ 12
(ii) 27 $\times $ 28

Answer 1

Hint: To solve this question, we should know the concept of multiplication by the Nikhilam Method. It is applied to multiply two numbers which are near to the same decimal bases. So, we will first find the deviation of the given number from their base. Then, we will multiply the deviation of the 2 given numbers which will give the last digits of multiplication and we will then add or subtract both the deviation from base to get the initial digits.

Complete step-by-step answer:

In this question, we have to apply the Nikhilam formula of multiplication to find the multiplication of the given two numbers. The concept of Nikhilam formula is to multiply two numbers which are near to the same decimal bases like 10, 100, 1000, etc. To find the multiplication we will first find deviation of both the numbers from their bases. Then, we will multiply the deviation of the 2 given numbers to obtain the last digits of multiplication. Multiplication of deviation will give the number of last digits as the same as the number of zeros in the decimal base. Then, to obtain the initial digits, we will add or subtract the deviation from the base.
(i) 11 $\times $ 12
In this, 11 and 12 are both nearest to the decimal base 10. So, we will take the decimal base equal to 10. So, we can write,
11 = 10 + 1 and 12 = 10 + 2
So, we can see that 1 and 2 are the deviation of 11 and 12 respectively. Now, we will multiply deviations to obtain the last digit of the number, that is, 1 $\times $ 2 = 2. So, the last digit of the number is 2. Here, we know that 11 and 12 are 1 and 2 more than 10 respectively. So, we will add deviation to the base to obtain the initial digit. So, we get,
10 + 1 + 2 =13
So, we get 13 as the first two digits of multiplication and 2 as the last digit of multiplication.
Therefore, 11 $\times $ 12 = 132.
(ii) 27 $\times $ 28
In this, 27 and 28 are both nearest to the decimal base 10. So, we will take the decimal base equal to 10. So, we can write,
27 = 10 + 17 and 28 = 10 + 18
So, we can see that 17 and 18 are the deviations of 27 and 28 respectively. Now, we will multiply deviations to obtain the last digit of the number, that is, 17 $\times $ 18 = 306. We know that the deviation gives numbers as much as zeroes in the decimal base. And we have the decimal base = 10. So, from the number 306 only 6 is the last digit of the multiplication. And 30 will carry forward to the initial digits. Now, we will add 17 and 18 to the decimal base to obtain the initial digits. So, we get,
10 + 17 + 18 = 45
Now, we will add the carry forward number that is 30 to 45. So, we get 30 + 45 = 75. So, the multiplication of 27 and 28 is 756.

Note: The possible mistake that the students can make is by not carrying forward the digits from the multiplication of deviation, which will result in wrong answers. Also, if deviation is less than the decimal base, then we will subtract the deviation from the decimal base.