Multiplication using Base Method
"All from 9 and the last from 10" and "Vertically and Crosswise" techniques make it very easy to find the product of two numbers that are close to a base number.
Steps:
Find the Base and Differences: Identify the base number for the given numbers and write the difference of each number from the base. The difference can be a deficit or a surplus.
Multiply the Differences: Multiply the differences of the two numbers. This result forms the second part of the answer.
Cross Subtraction (or Addition): Subtract (or add) one number's difference from the other number. This result forms the first part of the answer.
Find 92 × 94
Step 1: Identify the base number for both 92 and 94, which is 100.
Subtract each number from the base:
92 - 100 = -8 and 94 - 100 = -6
Step 2: Multiply the differences to get the second part of the answer:
-8 × -6 = 48
Step 3: Cross subtraction: Subtract 6 from 92 (or 8 from 94) to get the first part of the answer:
92 - 6 = 86
Combine the first and second parts: 86 and 48
Therefore, 92 × 94 = 8648.
Find 978 × 996
Step 1: Identify the base number for both 978 and 996, which is 1000.
Subtract each number from the base:
978 - 1000 = -22 and 996 - 1000 = -4
Step 2: Multiply the differences to get the second part of the answer:
-22 × -4 = 88
Step 3: Cross subtraction: Subtract 4 from 978 (or 22 from 996) to get the first part of the answer:
978 - 4 = 974
Combine the first and second parts: 974 and 88
Therefore, 978 × 996 = 97488.
Find 112 × 118
Step 1: Identify the base number for both 112 and 118, which is 100.
Add the excess from each number to the base:
112 - 100 = +12 and 118 - 100 = +18
Step 2: Multiply the differences to get the second part of the answer:
12 × 18 = 216
Step 3: Add 12 and 118 (or 18 and 112) to get the first part of the answer:
112 + 18 = 130
Combine the first and second parts: 130 and 216
Therefore, 112 × 118 = 13216.
Find 88 × 96
Step 1: Identify the base number for both 88 and 96, which is 100.
Subtract each number from the base:
88 - 100 = -12 and 96 - 100 = -4
Step 2: Multiply the differences to get the second part of the answer:
-12 × -4 = 48
Step 3: Cross subtraction: Subtract 4 from 88 (or 12 from 96) to get the first part of the answer:
88 - 4 = 84
Combine the first and second parts: 84 and 48
Therefore, 88 × 96 = 8448.
Algebraic Proof: (x – y)2
Formula: (x – y)2 = x2 – 2xy + y2
Proof:
= x (x – 2y) + y2
= x (x – y – y) + y2
= Base (Number – Deficiency) + (Deficiency)2
Algebraic Proof: (x + y)2
Formula: (x + y)2 = x2 + 2xy + y2
Proof:
= x (x + 2y) + y2
= x (x + y + y) + y2
= Base (Number + Surplus) + (Surplus)2
