**Q) If the length of a rectangle is reduced by 5 cm and its breadth is increased by 2 cm, then the area of the rectangle is reduced by 80 cm². However, if we increase the length by 10 cm and decrease the breadth by 5 cm, its area is increased by 50 cm². Find the length and breadth of the rectangle.**

**Ans:**

**STEP BY STEP SOLUTION**

Let’s consider the length of the rectangle be X and breadth be Y

Since the area of a rectangle is given by, A = L x B

∴ Area, A = X . Y

**Step 1:** By given first condition, new length, L’ = ( X – 5) and new breadth, B’ = (Y + 2)

∴ New area, A’ = (X – 5) ( Y + 2)

Given that new area is reduced by 80 cm

∴ A – A’ = 80

∴ X Y – (X – 5) (Y + 2) = 80

∴ X Y – X Y + 5 Y – 2X + 10 = 80

∴ – 2 X + 5 Y = 70 ……………… (i)

**Step 2:** By given second condition, new length, L” = ( X + 10) and new breadth, B” = (Y – 5)

∴ New area, A” = (X + 10) ( Y – 5)

Given that new area is increased by 50 cm

∴ A” – A = 50

∴ (X + 10) (Y – 5) – X Y = 50

∴ X Y + 10 Y – 5 X – 50 – X Y = 50

∴ – 5 X + 10 Y = 50 ……………… (ii)

**Step 3:** Let’s multiply equation (i) by 2 and subtract equation (ii), we get:

2 (- 2 X + 5 Y) – ( – 5 X + 10 Y ) = 2 (70) – 50

∴ – 4 X + 10 Y + 5 X – 10 Y = 140 – 50

∴ X = 90

**Step 4:** By substituting value of X in equation (i), we get:

– 2 X + 5 Y = 70

∴ – 2 (90) + 5 Y = 70

∴ 5 Y = 70 + 180 = 250

∴ Y = = 50

^{ }**Therefore, length is 90 cm and breadth is 50 cm of the given rectangle.**

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