13. Two persons P and Q started a 300 meter race. Initially the speeds of P and Q are in the ratio 1 : 2. After ‘t’ seconds, P quadrupled (four times) his speed and as a result, both of them reached the endpoint at same instant of time. But after ‘t’ seconds, if P tripled his speed he would have lost the race by 5 seconds. What is the value of ‘t’? ఇద్దరు వ్యక్తులు P మరియు Qలు 300 మీటర్ల పరుగుపందెం ప్రారంభించారు. ప్రారంభంలో P మరియు Q ల వేగాల నిష్పత్తి 1:2. t సెకన్ల తర్వాత P తన వేగాన్ని నాల్గు రెట్లు చేయగా, ఇరువురు గమ్యస్థానాన్ని ఒకే సారి చేరారు. కాని t సెకెన్ల తర్వాత, P తన వేగాన్ని మూడు రెట్లు చేస్తే, 5 సెకన్ల తేడాతో, పరుగుపందెంను P ఓడిపోయి ఉండేవాడు. అప్పుడు t విలువ ఎంత ?

 

ఇది ఒక ఆపాదించదగిన Time-Speed problem. ఇది గమనికతో, సులభంగా సమీకరణాల ద్వారా పరిష్కరించవచ్చు.

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ఇవ్వబడినవి:

• Race length = 300 meters
• Initial speed ratio: P : Q = 1 : 2
  • Let P speed = x, then Q speed = 2x
• After t seconds, P increases speed.
• If P quadruples speed ⇒ P and Q reach together
• If P triples speed ⇒ P loses by 5 seconds

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Step 1: Distance covered in first t seconds

Let’s calculate distance covered by both in first t seconds:

• P covers = x × t meters
• Q covers = 2x × t meters

Hence, distance remaining after t seconds:

• P → 300 - xt
• Q → 300 - 2xt

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Step 2: Case 1 – P quadruples speed ⇒ reach same time

After t seconds, P’s speed = 4x
Let remaining time taken by both = T seconds (after t)

Then:

• Time for P to cover 300 - xt at 4x speed = (300 - xt)/(4x)
• Time for Q to cover 300 - 2xt at 2x speed = (300 - 2xt)/(2x)

Since they reach at same time:

\frac{300 - xt}{4x} = \frac{300 - 2xt}{2x}

Multiply both sides by 4x:

300 - xt = 2(300 - 2xt)

300 - xt = 600 - 4xt 

Bring variables together:

-xt + 4xt = 600 - 300  
\Rightarrow 3xt = 300  
\Rightarrow xt = 100 \quad \text{(Equation 1)}

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Step 3: Case 2 – P triples speed ⇒ loses by 5 sec

Now P’s speed after t seconds = 3x

Then time taken after t seconds:

• P: (300 - xt)/(3x)
• Q: (300 - 2xt)/(2x)

Given: P loses by 5 seconds ⇒ Q is faster ⇒

\frac{300 - 2xt}{2x} - \frac{300 - xt}{3x} = 5

Take LCM (6x):

\frac{3(300 - 2xt) - 2(300 - xt)}{6x} = 5

Expand numerator:

\frac{900 - 6xt - 600 + 2xt}{6x} = 5
\Rightarrow \frac{300 - 4xt}{6x} = 5

Cross-multiply:

300 - 4xt = 30x  
\Rightarrow 300 - 30x = 4xt \quad \text{(Equation 2)}

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Step 4: From Equation 1 → xt = 100 ⇒ x = 100/t

Substitute in Equation 2:

300 - 30(100/t) = 4 × 100  
\Rightarrow 300 - 3000/t = 400  
\Rightarrow -3000/t = 100  
\Rightarrow t = -3000 / 100 = -30 \quad \text{(discard negative)}

Take correct signs again:

From  
\[
300 - 30x = 400  
\Rightarrow -30x = 100  
\Rightarrow x = -10/3 \quad \text{(again negative)}

So something is wrong. Let's try again with correct sign:

**Actually:**

From before:
\[
300 - 4xt = 30x  
\Rightarrow 300 - 30x = 4xt \tag{✓}

From Eq (1): xt = 100 ⇒ So t = 100/x

Substitute in above:

300 - 30x = 4x × (100/x)  
\Rightarrow 300 - 30x = 400  
\Rightarrow -30x = 100  
\Rightarrow x = -10/3 \quad (\text{not possible})

Wait. That means we must have made a sign mistake.

Try again from this step:

\[
\frac{300 - xt}{3x} - \frac{300 - 2xt}{2x} = 5
\Rightarrow \frac{2(300 - xt) - 3(300 - 2xt)}{6x} = 5

Now expand:

\frac{600 - 2xt - 900 + 6xt}{6x} = 5  
\Rightarrow \frac{-300 + 4xt}{6x} = 5  
\Rightarrow -300 + 4xt = 30x  
\Rightarrow 4xt = 30x + 300
\Rightarrow t = (30x + 300) / 4x \tag{Eq 3}

But from Equation 1: xt = 100 ⇒ So t = 100/x

Equate:

100/x = (30x + 300)/(4x)  
\Rightarrow 100 = (30x + 300)/4  
\Rightarrow 400 = 30x + 300  
\Rightarrow 30x = 100  
\Rightarrow x = \frac{10}{3}
\Rightarrow t = \frac{100}{x} = \frac{100}{10/3} = \boxed{30 \text{ seconds}}

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✅ Final Answer:

\boxed{t = 30 \text{ seconds}}

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వివరణ తెలుగులో కావాలంటే చెప్పండి మధు!