If the length of a rectangle is increased by 14 units and its breadth is decreased by 7 units then the area of the rectangle is increased by 35 square units. However, if we decrease its length by 7 units and increase breadth by 3 units. Its area is decreased by 105 square units. Find the length and breadth of the rectangle.
Answer:
Length=35 unitsBreadth=27 units
- Let us assume the length and breadth of the rectangle be x units and y units respectively.
- It is given that if the length of a rectangle is increased by 14 units and its breadth is decreased by 7 units then the area of the rectangle is increased by 35 square units.
Now, New length=(x+14) unitsNew breadth=(y−7) units ∴ - Similarly, if we decrease its length by 7 units and increase breadth by 3 units. Its area is decreased by 105 square units.
Now, \begin{aligned} & \text{New length} = (x - 7) \text{ units} \\ & \text{New breadth} = (y + 3) \text{ units } \\ \therefore \space & \text{New area} = (x - 7)(y + 3) \text{ square units} \\ \therefore \space & xy - (x - 7)(y + 3) = 105 \\ \implies & xy - (xy + 3 x - 7 y - 21) = 105 \\ \implies & xy - xy - 3 x + 7 y + 21 = 105 \\ \implies & 7 y - 3 x = 84 && \ldots (ii) \end{aligned} - On multiplying (i) by 3 we get:\begin{aligned} & 6 y - 3 x = 57 && \ldots (iii) \\ \end{aligned} Subtracting (iii) from (ii), we get:\begin{aligned} y = 27 \end{aligned}
- Putting y = 27 in (i), we get:\begin{aligned} & (2 \times 27) - x = 19 \\ \implies & x = 54 - 19 = 35 \end{aligned}
- Hence, \begin{aligned} & \text{Length} = x \text{ units} = 35 \text{ units} \\ & \text{Breadth} = y \text{ units} = 27 \text{ units} \end{aligned}