Question: A train running at a certain speed crosses a 496-meter-long platform in 56 seconds. If the length of the train is 560 meters, how long will it take to cross a bridge that is 100 meters in length?

Solution:
We know,
When a train crosses any object, it covers a distance equal to the sum of the object's length and its own length.

So, when crossing a platform, the distance covered by the train = (496 + 560) meters = 1056 meters
And
when crossing a bridge, the distance covered by the train = (100 + 560) meters = 660 meters

Now,
The train covers 1056 meters in = 56 seconds
∴ It covers 1 meter in = (56/1056) seconds
∴ It covers 660 meters in = (56 × 660)/1056 seconds = 35 seconds

Question: A jar contains a mixture of oil and water in the ratio 7 : 5. If 9 liters of the mixture is removed and replaced with the same amount of water, the new ratio becomes 7 : 9 . What was the initial quantity of oil in the jar?

Solution:
ধরি,
শুরুতে জারের মধ্যে তেলের পরিমাণ = 7x লিটার 
পানির পরিমাণ = 5x লিটার 
∴ মোট অংশ = 12x

9 লিটার মিশ্রণ ফেলে দিলে,
ফেলে দেওয়া মিশ্রণে তেলের পরিমাণ = 9 এর (7x/12x) = 21/4 লিটার 
এবং পানির পরিমাণ = 9 এর (5x/12x) = 15/4 লিটার 

বাকি মিশ্রণে,
তেলের পরিমাণ = 7x - (21/4) = (28x - 21)/4
পানির পরিমাণ = 5x - (15/4) = (20x - 15)/4

মিশ্রণে 9 লিটার পানি যোগ করা হলে পানির নতুন পরিমাণ = {(20x - 15)/4} + 9 = (20x - 15 + 36)/4 = (20x + 21)/4

প্রশ্নমতে,
{(28x - 21)/4}/{(20x + 21)/4} = 7/9
⇒ (28x - 21)/(20x + 21) = 7/9
⇒ 9(28x - 21) = 7(20x + 21)
⇒ 252x - 189 = 140x + 147
⇒ 252x - 140x = 189 + 147
⇒ 112x = 336
⇒ x = 336/112
⇒ x = 3

∴ শুরুতে মিশ্রণে তেলের পরিমাণ ছিলো = (7 × 3) লিটার = 21 লিটার

Step 1: Let the shares of P, Q, R, S be proportional to 7 : 3 : 5 : 2

Let the common multiple be $x$. Then:

• P = $7x$

• Q = $3x$

• R = $5x$

• S = $2x$

Step 2: Use the information given about R and S

$$R - S = 2000$$ $$5x - 2x = 2000$$ $$3x = 2000$$ $$x = \frac{2000}{3} \approx 666.67$$

Step 3: Find Q's share

$$Q = 3x = 3 \times 666.67 = 2000$$

Answer: Q's share = Tk. 2000