Question: A biker was riding at 60 km/h, but if he had gone at 80 km/h, he could’ve gone 100 km more in the same time. How far did he actually travel?

Solution:
Let, the actual distance travelled be x km.

Then,
x/60 = (x + 100)/80 
⇒ x/6 = (x + 100)/8
⇒ 6(x + 100) = 8x
⇒ 6x + 600 = 8x
⇒ 8x - 6x = 600
⇒ 2x = 600
⇒ x = 600/2
⇒ x = 300 km

Question: How much water should be added to 50 liters of pure milk to gain extra 12% profit when selling the mixture at the price of pure milk?

Solution:
Let’s assume,
Price of pure milk per liter = 100 Taka
So, the price of 50 liters of pure milk = 100 × 50 = 5000 Taka

Now assume,
Water added to the milk = x liters
Then the total quantity of the milk-water mixture = (50 + x) liters

Since the mixture is sold at the price of pure milk,
The selling price of (50 + x) liters = 100(50 + x) Taka

According to the question,
100(50 + x) = 5000 + 5000 এর 12%
⇒ 5000 + 100x = 5000 + {5000 × (12/100)}
⇒ 5000 + 100x = 5000 + 600
⇒ 5000 - 5000 + 100x = 600
⇒ 100x = 600
⇒ x = 600/100
⇒ x = 6

∴ Amount of water to be added = 6 liters

Shortcut:
Amount of water to be added = (Profit%/100) × Quantity of pure milk
= (12/100) × 50
= 6 liters

1. Let the train length = x meters.
   Distances covered while crossing the bridges:

   • First bridge: $x + 420$ meters

   • Second bridge: $x + 200$ meters

2. Time taken:

   • First bridge = 60 sec → speed = $(x + 420)/60$

   • Second bridge = 40 sec → speed = $(x + 200)/40$

3. Since speed is constant:

$$\frac{x + 420}{60} = \frac{x + 200}{40}$$

4. Solve for x:

$$40(x + 420) = 60(x + 200)$$ $$40x + 16800 = 60x + 12000$$ $$20x = 4800 \implies x = 240$$

Answer: The train is 240 meters long