Saira has arranged a square of side x units, 8 rectangular strips of sides x units and width 1 unit, and 15 squares of side 1 unit to form a bigger rectangle. Find the length and breadth of the rectangle in terms of x.

Step-by-step solution

Idea: The rectangle’s area is the total area of all the pieces. Factorising that area into two linear factors gives the two sides.

1. Areas: square = x²; 8 strips = 8 × x = 8x; 15 unit squares = 15. Total area = x² + 8x + 15 sq. units.1 mark
2. Factorise x² + 8x + 15: need a + b = 8 and ab = 15 → a = 3, b = 5.½ mark
3. x² + 8x + 15 = x² + 3x + 5x + 15 = x(x + 3) + 5(x + 3) = (x + 5)(x + 3).1 mark
4. Length = x + 5 units, breadth = x + 3 units.½ mark
5. Drawing Saira’s rectangle: put the x × x square in a corner, 5 strips standing along its right side, 3 strips lying along its bottom, and the 15 unit squares in a 5 × 3 block in the remaining corner (see the diagram).

Check: (x + 5)(x + 3) = x² + 3x + 5x + 15 = x² + 8x + 15 ✓. With x = 10: 15 × 13 = 195 = 100 + 80 + 15 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Writing the area of the 8 strips as 8 instead of 8x: each strip is x × 1.
• Choosing 1 and 15 or 4 and 4 (wrong product or sum) for splitting 8x.
• Giving sides like x + 8 and x + 15 by reading the coefficients directly instead of factorising.

How this can come in the exam

Try one yourself

One x × x square, 6 strips x × 1 and 8 unit squares form a rectangle. Find its length and breadth.

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