Question: 400 grams of sugar solution has 30% sugar in it. How much sugar should be added to make 50% in the solution?

Solution:
Amount of sugar = 400 × (30/100) = 120 grams
let,
x gm sugar to be added

According to the question,
(120 + x)/(400 + x) = 50%
⇒ (120 + x)/(400 + x) = 50/100
⇒ (120 + x)/(400 + x) = 1/2
⇒ 2 × (120 + x) = (400 + x)
⇒ 240 + 2x = 400 + x
⇒ 2x - x = 400 - 240
⇒ x = 160 grams

Step 1: Let the original quantities be

• Milk = $5x$ liters

• Water = $3x$ liters

(According to the original ratio 5 : 3)

Step 2: Add 4 liters of water

• New water quantity = $3x + 4$ liters

• Milk remains $5x$ liters

• New ratio = 5 : 4

$$\frac{\text{Milk}}{\text{Water}} = \frac{5}{4}$$ $$\frac{5x}{3x + 4} = \frac{5}{4}$$

Step 3: Solve for $x$

$$5x \cdot 4 = 5 \cdot (3x + 4)$$ $$20x = 15x + 20$$ $$20x - 15x = 20$$ $$5x = 20$$ $$x = 4$$

Step 4: Find the original amount of milk

$$\text{Milk} = 5x = 5 \times 4 = 20 \text{ liters}$$

Answer: Original amount of milk = 20 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