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 1546 Updated on2026-09-05

Q1.

By expanding the expressions, verify that all three expressions are equivalent. If x = 8 and y = 3, find the area of the shaded region.

Answer

Method	Expression	Expanded
Anusha	x² – xy	x² – xy
Vaishnavi	x(x + 2y) – 3xy	x² + 2xy – 3xy = x² – xy
Aditya	2x((x – y)⁄2)	x(x – y) = x² – xy

All three reduce to x² – xy, so they are equivalent.

At x = 8, y = 3:
x² – xy = 64 – 24 = 40 square units
Check with Vaishnavi: 8 × 14 – 3 × 24 = 112 – 72 = 40 ✓
Check with Aditya: 8 × 5 = 40 ✓

Why it happens: the three students cut the same region in three different ways. Expanding strips away the differences in how the cutting was described and leaves the one thing they all measure.

Q2.

Write an expression for the area of the dashed region in the figure below. Use more than one method to arrive at the answer. Substitute p = 6, r = 3.5, and s = 9, and calculate the area.

Answer

The whole figure is a rectangle s wide and p tall. The shaded L-shape is a vertical arm of width r and a horizontal arm of height r; the dashed region is what is left in the top-left corner.

The dashed region is the rectangle left when an L of width r is taken from an s by p rectangle.

Method 1 — measure the leftover rectangle directly.

Width = s – r, height = p – r
Area = (s – r)(p – r) = ps – pr – sr + r²

Method 2 — whole rectangle minus the L. The L is a p × r arm plus an (s – r) × r foot.

Area of L = pr + (s – r)r = pr + sr – r²
Dashed area = ps – (pr + sr – r²)
= ps – pr – sr + r² ✓

Method 3 — cut off two strips and put the corner back. Remove the right strip pr and the bottom strip sr; the r × r corner has gone twice, so add it back once: ps – pr – sr + r² ✓ (the same bookkeeping as Identity 1B).

At p = 6, r = 3.5, s = 9:
(s – r)(p – r) = (9 – 3.5)(6 – 3.5)
= 5.5 × 2.5
= 13.75 square units

Check by the second method: 54 – 21 – 31.5 + 12.25 = 13.75 ✓

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