Question

Q-12
Mansi travels 300 kms to her native partly by train and partly by bus. She takes
4 hours, if she travels 60 kms by train and the remaining by bus. If she travels 100 kms
by train and the remaining by bus, she takes 10 minutes longer. Find the average
speed of the train and the bus separately.

Answer 1

Given:

Mansi travels 300 km to her native partly by train and partly by bus.

She takes 4 hours if she travels 60 km by train and the remaining by bus.

If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer

To find:

The average  speed of the train and the bus separately.

Solution:

Let's assume,

"$\bold{S_T}$" → speed of the train

"$\bold{S_B}$" → speed of the bus

So, according to the first part of the question, we can form an equation as,

$\frac{60}{S_T} + \frac{300 - 60}{S_B} = 4$

$\implies \frac{60}{S_T} + \frac{240}{S_B} = 4$

$\implies 60S_B + 240S_T = 4S_BS_T$

$\implies 15S_B + 60S_T = S_BS_T$ ..... (i)

Again, according to the second part of the question, we can form an equation as,

$\frac{100}{S_T} + \frac{300 - 100}{S_B} = 4\frac{10}{60}$

$\implies \frac{100}{S_T} + \frac{200}{S_B} = 4\frac{1}{6}$

$\implies \frac{100}{S_T} + \frac{200}{S_B} = \frac{25}{6}$

$\implies 100S_B + 200S_T = \frac{25}{6} S_BS_T$

substituting the value of $S_T \:\&\: S_B$ from (i)

$\implies 100S_B + 200S_T = \frac{25}{6} [ 15S_B + 60S_T ]$

$\implies 600S_B + 1200S_T = 375S_B + 1500S_T$

$\implies 225S_B = 300S_T$

$\implies \bold{S_B = \frac{4}{3} S_T}$ .... (ii)

Now, substituting from (ii) in (i), we get

$(15\times \frac{4}{3} S_T) + 60S_T = \frac{4}{3} S_TS_T$

$\implies 20S_T + 60S_T = \frac{4}{3} S_T^2$

$\implies 80S_T = \frac{4}{3} S_T^2$

$\implies S_ T = \frac{80\times 3}{4}$

$\implies \bold{S_ T =60\:km/hr}$

Substituting the value of $S_T$ in (ii), we get

∴ $\bold{S_B} = \frac{4}{3} \times 60= 4 \times 20 = \bold{80\:km/hr}$

Thus,

The speed of the train is → 60 km/hr

The speed of the bus is → 80 km/hr

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Answer 2

$\huge\bf\purple{\mid{\fbox{\underline{\underline{Answer}}}}\mid}$

Let the speed of train and bus be u km/h and v km/h respectively.

According to the question,

$\frac{60}{u} + \frac{240}{v} = 4$....(i)

$\frac{100}{u} + \frac{200}{v} = 4 + \frac{10}{60} = \frac{25}{6}$....(ii)

Let $\frac{1}{u} = p\: and \frac{1}{v} = q$

The given equations reduce to:

60p + 240q = 4 ....(iii)

100p + 200q = $\frac{25}{6}$

600p + 1200q = 25....(iv)

Multiplying equation (iii) by 10, we obtain:

600p + 2400q = 40....(v)

Subtracting equation (iv) from equation (v), we obtain:

1200q = 15

q = $\frac{15}{1200} = \frac{1}{80}$

Substituting the value of q in equation (iii), we obtain:

60p + 3 = 4

60p = 1

p = $\frac{1}{60}$

:. p = $\frac{1}{u} = \frac{1}{60}$, q = $\frac{1}{v} = \frac{1}{80}$

u = 60 km/h , v = 80 km/h

Thus, the speed of train and the speed of bus are 60 km/h and 80 km/h respectively.

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