Page 84 - Ganit Kaushal

Instead, observe that this whole pattern can be divided into 40 40 40 40
six diagonals as shown in the adjoining figure (b). The two

diagonals from above and below consist of the same 50 50 50 50 50
numbers. They are shown by the same coloured arrows in 40 40 40 40
the next figure (b). 50 50 50 50 50
Therefore, adding them diagonally as shown in figure (b). 40 40 40 40

The sum of all numbers can be written as Figure (b)
= 2(40 + 50) + 2(2 × 40 + 2 × 50) + 2(3 × 40 + 2 × 50)

= 2 × 90 + 2 × (80 + 100) + 2(120 + 100) (Using BODMAS)
= 180 + 2 × 180 + 2 × 220

= 180 + 360 + 440
= 980

Quite easily done! Isn’t it?
Ex 5. In this example, dots in white colour are given in a particular pattern in the form of a square. Find the
sum of all the dots. (NCERT)

Sol. Here in this example, instead of adding them one by one, we can add all the
dots in a column-wise manner. There are eight columns in this picture. There are
two columns having 8 dots, 4 columns having 24 dots each, and two columns
having 16 dots each.
This way, we can do it quickly as follows.

Sum of all dots = 2 × 8 + 4 × 24 + 2 × 16 = 16 + 96 + 32 = 144
Quite easily done! Isn’t it?

Ex 6. In the picture, some numbers are placed in a pattern. Find out the sum 15 15 35 35 25 25
of the numbers in the pattern. Should we add them one by one, or can 25 15 15 35 35 25 25 15
15
25
we use a quicker way? (NCERT) 25 25 35 35 15 15
35 15 25 35
Sol. We observe that in diagonally opposite blue quadrilaterals, there are 35 35 25 25 35 35 15 15 35 35
25
25
15
15
the same numbers. The same pattern is followed in yellow hexagons 35 15 15 25 15 25 25 35
that are diagonally opposite. 15 15 35 35 25 25
25 25 35 35 15 15
Observing this pattern, the sum of all numbers can be written as 25 25 35 35 15 15
= (2 × 4 × 15 + 2 × 4 × 25 + 2 × 4 × 35)
+ (2 × 6 × 35 + 2 × 6 × 25 + 2 × 6 × 15)

+ (2 × 35 + 2 × 25 + 2 × 15)
= (120 + 200 + 280) + (420 + 300 + 180) + (70 + 50 + 30)

= 600 + 900 + 150
= 1650
Quite easily done! Isn’t it?

82 Mathematics-6