Page 12 - Ganit Kaushal
P. 12
Rectangle
Some numbers can be arranged as rectangles using the dots. For example:
(a) The number 6 can be arranged as a rectangle with 2 rows and 3 columns as:
(b) The number 12 can be arranged as a rectangle with 3 rows and 4 columns as:
Square
Some numbers can be arranged as squares using the dots. For example:
(a) The number 4 can be arranged as:
(b) The number 9 can be arranged as:
Triangle
Some numbers can be arranged as triangles using the dots. Note
(a) The number 3 can be represented as: or The triangle should have at least
(b) The number 6 can be represented as: or two of its sides equal.
1.4 PATTERNS IN ARITHMETIC OPERATIONS OF NUMBERS
Many times, arithmetic operations like multiplication, addition, etc. of numbers follow a particular pattern.
Such patterns guide us in simplifying the calculations.
1. Multiplication of 1s by 1s.
(a) 11 × 11 = 121 (b) 111 × 111=12321
(c) 1111 × 1111 = 1234321 (d) 11111 × 11111 = 123454321, and so on.
2. Multiplication of 9s by 9s.
(a) 9 × 9 = 81 (b) 99 × 99 = 9801
(c) 999 × 999 = 998001 (d) 9999 × 9999 = 99980001, and so on.
3. Observe the following interesting algebraic pattern.
(a) 7 × 13 = 91 (b) 97 × 103 = 9991
(c) 997 × 1003 = 999991 (d) 9997 × 10003 = 99999991, and so on.
4. The following multiplication pattern of an even two-digit number, say 86, with an odd multiple of 5
simplifies our calculations to a large extent.
10
(a) 86 × 5 = 86 × = 43 × 10 = 430 × 1 = 430 (Pattern: 430 multiplied by 1, as 5 = 1 × 5)
2
30
(b) 86 × 15 = 86 × = 43 × 30 = 430 × 3 = 1290 (Pattern: 430 multiplied by 3, as 15 = 3 × 5)
2
(c) 86 × 25 = 430 × 5 = 2150 (Pattern: 430 multiplied by 5, as 25 = 5 × 5),
and so on.
Observing the above multiplication pattern, we can easily conclude the following:
86 × 65 = 430 ×13 = 5590; 86 × 125 = 430 × 25 = 10750, etc.
5. Observe the following interesting pattern of multiplication and addition operations.
(a) 1 × 8 + 1 = 9 (b) 12 × 8 + 2 = 98
(c) 123 × 8 + 3 = 987 (d) 1234 × 8 + 4 = 9876, and so on.
10 Mathematics-6
