RBSE Solutions Class 6 Maths Chapter 7 Vedic Mathematics Exercise 7.7

Question 1. Multiply (Using formula of Nikhilam).
(i) 12 x 13
(ii) 11 x 19
(iii) 13 x 15
(iv) 8 x 7
(v) 6 x 9
(vi) 8 x 12
(vii) 102 x 104
(viii) 106 x 107
(ix) 112 x 109
(x) 91 x 98
(xi) 96 x 94
(xii) 98 x 104
(xiii) 85 x 93
Answer:
To multiply using the Nikhilam formula for numbers close to a base (like 10 or 100), we find their deviations from the base and combine sums and products of these deviations.

(i) Calculation for \( 12 \times 13 \):

For \( 12 \times 13 \), we use base 10. The deviations are \( +2 \) for 12 and \( +3 \) for 13.
The right side of the answer is the product of deviations: \( (+2) \times (+3) = 6 \).
The left side is the sum of one number and the other's deviation: \( 12 + 3 = 15 \) (or \( 13 + 2 = 15 \)).
Combining these, we get \( 15 | 6 \). The final product is 156.
(ii) Calculation for \( 11 \times 19 \):

Using base 10, the deviations are \( +1 \) for 11 and \( +9 \) for 19.
The right side is the product of deviations: \( (+1) \times (+9) = 9 \).
The left side is the sum: \( 11 + 9 = 20 \) (or \( 19 + 1 = 20 \)).
Combining these, we get \( 20 | 9 \). The final product is 209.
(iii) Calculation for \( 13 \times 15 \):

For \( 13 \times 15 \), using base 10, the deviations are \( +3 \) and \( +5 \).
The left side is \( 13 + 5 = 18 \) (or \( 15 + 3 = 18 \)).
The right side is \( (+3) \times (+5) = 15 \).
We have \( 18 | 15 \). Since the base is 10 (one zero), only one digit can be on the right. We keep 5 and carry over 1 to the left side.
The left side becomes \( 18 + 1 = 19 \).
Combining \( 19 \) and \( 5 \) gives the final product as 195.
(iv) Calculation for \( 8 \times 7 \):

Using base 10, the deviations are \( -2 \) for 8 and \( -3 \) for 7.
The right side is the product of deviations: \( (-2) \times (-3) = +6 \).
The left side is the sum: \( 8 + (-3) = 5 \) (or \( 7 + (-2) = 5 \)).
Combining these, we get \( 5 | 6 \). The final product is 56.
(v) Calculation for \( 6 \times 9 \):

Using base 10, the deviations are \( -4 \) for 6 and \( -1 \) for 9.
The right side is the product of deviations: \( (-4) \times (-1) = +4 \).
The left side is the sum: \( 6 + (-1) = 5 \) (or \( 9 + (-4) = 5 \)).
Combining these, we get \( 5 | 4 \). The final product is 54.
(vi) Calculation for \( 8 \times 12 \):

Using base 10, the deviations are \( -2 \) for 8 and \( +2 \) for 12.
The left side is the sum: \( 8 + 2 = 10 \) (or \( 12 - 2 = 10 \)).
The right side is the product of deviations: \( (-2) \times (+2) = -4 \).
We have \( 10 | -4 \). Since the right side is negative, we borrow 1 from the left side (which becomes \( 10 - 1 = 9 \)). The borrowed 1 is multiplied by the base (10) and added to the right side: \( 10 + (-4) = 6 \).
Combining \( 9 \) and \( 6 \) gives the final product as 96.
(vii) Calculation for \( 102 \times 104 \):

Using base 100, the deviations are \( +02 \) for 102 and \( +04 \) for 104. We use two digits for deviation as base 100 has two zeros.
The left side is \( 102 + 4 = 106 \) (or \( 104 + 2 = 106 \)).
The right side is \( (+02) \times (+04) = 08 \).
Combining these, we get \( 106 | 08 \). The final product is 10608.
(viii) Calculation for \( 106 \times 107 \):

Using base 100, the deviations are \( +06 \) for 106 and \( +07 \) for 107.
The left side is \( 106 + 7 = 113 \) (or \( 107 + 6 = 113 \)).
The right side is \( (+06) \times (+07) = 42 \).
Combining these, we get \( 113 | 42 \). The final product is 11342.
(ix) Calculation for \( 112 \times 109 \):

Using base 100, the deviations are \( +12 \) for 112 and \( +09 \) for 109.
The left side is \( 112 + 9 = 121 \) (or \( 109 + 12 = 121 \)).
The right side is \( (+12) \times (+09) = 108 \).
We have \( 121 | 108 \). Since the base is 100 (two zeros), only two digits can be on the right. We keep 08 and carry over 1 to the left side.
The left side becomes \( 121 + 1 = 122 \).
Combining \( 122 \) and \( 08 \) gives the final product as 12208.
(x) Calculation for \( 91 \times 98 \):

Using base 100, the deviations are \( -09 \) for 91 and \( -02 \) for 98.
The left side is \( 91 - 2 = 89 \) (or \( 98 - 9 = 89 \)).
The right side is \( (-09) \times (-02) = +18 \).
Combining these, we get \( 89 | 18 \). The final product is 8918.
(xi) Calculation for \( 96 \times 94 \):

Using base 100, the deviations are \( -04 \) for 96 and \( -06 \) for 94.
The left side is \( 96 - 6 = 90 \) (or \( 94 - 4 = 90 \)).
The right side is \( (-04) \times (-06) = +24 \).
Combining these, we get \( 90 | 24 \). The final product is 9024.
(xii) Calculation for \( 98 \times 104 \):

Using base 100, the deviations are \( -02 \) for 98 and \( +04 \) for 104.
The left side is \( 98 + 4 = 102 \) (or \( 104 - 2 = 102 \)).
The right side is \( (-02) \times (+04) = -08 \).
We have \( 102 | -08 \). Since the right side is negative, we borrow 1 from the left side (which becomes \( 102 - 1 = 101 \)). The borrowed 1 is multiplied by the base (100) and added to the right side: \( 100 + (-08) = 92 \).
Combining \( 101 \) and \( 92 \) gives the final product as 10192.
(xiii) Calculation for \( 85 \times 93 \):

Using base 100, the deviations are \( -15 \) for 85 and \( -07 \) for 93.
The left side is \( 85 - 7 = 78 \) (or \( 93 - 15 = 78 \)).
The right side is \( (-15) \times (-07) = 105 \).
We have \( 78 | 105 \). Since the base is 100 (two zeros), only two digits can be on the right. We keep 05 and carry over 1 to the left side.
The left side becomes \( 78 + 1 = 79 \).
Combining \( 79 \) and \( 05 \) gives the final product as 7905.
In simple words: The Nikhilam method helps multiply numbers by using a base like 10 or 100. You find how much each number is more or less than this base. Then, you sum one number with the other's "difference" for the left part and multiply the "differences" for the right part. If the right part has too many digits or is negative, you make adjustments by carrying over or borrowing from the left side. This makes big multiplications simpler.
🎯 Exam Tip: Always clearly identify your chosen base (10, 100, etc.) and ensure the number of digits on the right side matches the number of zeros in your base. Remember to carry over or borrow when needed.