NCERT Solutions Ganita Prakash (Part 1) Chapter 6 .4 This Way or That Way, All Ways Lead to the Bay — In-text Questions

Book page 1526 Updated on2026-09-05

Q1.

Use this formula to find the number of circles in Step 15.

Answer

Number of circles at Step k = k² + 2k
At k = 15: 15² + 2 × 15
= 225 + 30
= 255 circles

Cross-check with the other three methods, all at k = 15:

Method	Expression	Value at k = 15
1	(k + 1)² – 1	256 – 1 = 255
2	k² + 2 × k	225 + 30 = 255
3	k × (k + 1) + k	240 + 15 = 255
4	k × (k + 2)	15 × 17 = 255

Why it happens: the four expressions look different because they describe four different ways of seeing the picture, but each simplifies to k² + 2k. Equal expressions must give equal values at every k — that is what “equal expressions” means.

Q2.

Consider the pattern made of square tiles in the picture below.

Answer

Each figure is a square frame of tiles — a large square with a smaller square hole cut out of the middle.

Step	Outer square	Hole	Tiles
1	3 × 3	1 × 1	9 – 1 = 8
2	4 × 4	2 × 2	16 – 4 = 12
3	5 × 5	3 × 3	25 – 9 = 16

Step 3: a 5 × 5 square of tiles with a 3 × 3 hole — 25 – 9 = 16 tiles.

Tip: at Step n the hole is n × n and the outer square is (n + 2) × (n + 2), because the frame is one tile thick all round.

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