$2$ . $6$ In covering a distance of $30 \mathrm{~km}$ , Arun takes $2 \mathrm{hrs}$ more than Anil. If Arun triples his speed, then he would take $1 \mathrm{hr}$ less than Anil. What is Arun's speed?

Full solution

2

6

In covering a distance of

30 \mathrm{~km}

, Arun takes

2 \mathrm{hrs}

more than Anil. If Arun triples his speed, then he would take

1 \mathrm{hr}

less than Anil. What is Arun's speed?

Equation $1$ : Let's call Arun's speed $A$ km/hr and Anil's speed $B$ km/hr. Since Arun takes $2$ hours more than Anil to cover $30$ km, we can write the equation: $\frac{30}{A} = \frac{30}{B} + 2$ .
Equation $2$ : Now, if Arun triples his speed, his new speed is $3A$ km/hr. The problem says that at this new speed, Arun would take $1$ hour less than Anil to cover the same distance. So we can write another equation: $\frac{30}{3A} = \frac{30}{B} - 1$ .
Simplify Equation $1$ : Let's simplify the first equation: $\frac{30}{A} - \frac{30}{B} = 2$ . Multiplying both sides by $AB$ to get rid of the fractions, we get: $30B - 30A = 2AB$ .
Simplify Equation $2$ : Now, let's simplify the second equation: $\frac{30}{3A} - \frac{30}{B} = -1$ . Multiplying both sides by $3AB$ gives us: $30B - 90A = -3AB$ .
Eliminate B: We now have two equations: $30B - 30A = 2AB$ and $30B - 90A = -3AB$ . Let's multiply the first equation by $3$ to help us eliminate B: $90B - 90A = 6AB$ .
Solve for $A$ : Subtract the second equation from the first: $(90B - 90A) - (30B - 90A) = 6AB - (-3AB)$ . This simplifies to $60B = 9AB$ .
Find $A$ : Divide both sides by $B$ to solve for $A$ : $60 = 9A$ . Now, divide both sides by $9$ to find $A$ : $A = \frac{60}{9}$ .