Speed and Distance Problems on Linear Equations: Class 10

Quick answer: Use time = distance ÷ speed and distance = speed × time. Two cars moving the same way close the gap at the difference of their speeds; moving towards each other, at the sum. For the two cars 100 km apart that meet in 5 hours (same direction) or 1 hour (towards each other): $u - v = 20$ and $u + v = 100$, so the speeds are 60 km/h and 40 km/h. Roohi's journey gives a train speed of 60 km/h and a bus speed of 80 km/h.

Syllabus note: Roohi's problem uses equations in $\frac{1}{u}$ and $\frac{1}{v}$, which are "equations reducible to a pair of linear equations". That section was removed from the rationalised NCERT Class 10 textbook, so treat Problem 2 as practice beyond the current board syllabus.

Relative speed

 Same direction:
 A (u) --->     B (v) --->

 Towards each other:
 A (u) --->   <--- B (v)

Problem 1: two cars on a 100 km highway

Points A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. Moving in the same direction, they meet in 5 hours. Moving towards each other, they meet in 1 hour. Find their speeds.

Step 1: Let the car from A travel at $u$ km/h and the car from B at $v$ km/h, with $u > v$.

Step 2: Same direction: in 5 hours, car A covers 100 km more than car B:

$5u - 5v = 100 \implies u - v = 20$

Step 3: Towards each other: in 1 hour, together they cover 100 km:

$u + v = 100$

Step 4: Add the two equations: $2u = 120$, so $u = 60$. Then $v = 100 - 60 = 40$.

Answer: The cars travel at 60 km/h and 40 km/h.

Check: In 5 hours the gap shrinks by $5 \times 20 = 100$ km, and in 1 hour they close $60 + 40 = 100$ km.

Problem 2: Roohi's 300 km journey by train and bus

Roohi travels 300 km to her home, partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the rest by bus. She takes 10 minutes longer if she travels 100 km by train and the rest by bus. Find the speeds of the train and the bus.

Step 1: Let the train's speed be $u$ km/h and the bus's speed $v$ km/h.

Step 2: First journey: 60 km by train and 240 km by bus in 4 hours:

$\frac{60}{u} + \frac{240}{v} = 4$

Step 3: Second journey: 100 km by train and 200 km by bus in 4 h 10 min. Since 10 minutes is $\frac{1}{6}$ hour, the time is $\frac{25}{6}$ hours:

$\frac{100}{u} + \frac{200}{v} = \frac{25}{6}$

Step 4: Put $p = \frac{1}{u}$ and $q = \frac{1}{v}$, and simplify:

$60p + 240q = 4 \implies 15p + 60q = 1$

$100p + 200q = \frac{25}{6} \implies 24p + 48q = 1$

Step 5: Eliminate $p$. Multiply the first equation by 8 and the second by 5:

$120p + 480q = 8$

$120p + 240q = 5$

Subtracting gives $240q = 3$, so $q = \frac{1}{80}$.

Step 6: Substitute in $15p + 60q = 1$: $15p + \frac{3}{4} = 1$, so $p = \frac{1}{60}$.

Answer: The train's speed is $u = 60$ km/h and the bus's speed is $v = 80$ km/h.

Check: $\frac{60}{60} + \frac{240}{80} = 1 + 3 = 4$ hours, and $\frac{100}{60} + \frac{200}{80} = \frac{5}{3} + \frac{5}{2} = \frac{25}{6}$ hours.

Summary table

Exam tips

• Define your variables with units before writing equations.
• Convert minutes to hours as a fraction: 10 minutes is $\frac{1}{6}$ hour, not 0.1 hour.
• Always check your speeds by substituting them back into the original conditions.

Common mistakes

• Adding the speeds when the cars move in the same direction. In that case the gap closes at the difference.
• Writing $60u + 240v = 4$. Time is distance divided by speed, so speeds go in the denominator.
• Writing 4 h 10 min as 4.1 hours. It is $4\frac{1}{6}$ hours.
• Stopping at $p$ and $q$ and forgetting to find $u = \frac{1}{p}$ and $v = \frac{1}{q}$.

Frequently Asked Questions

What is relative speed when two cars move in the same direction?

It is the difference of their speeds, because the faster car gains on the slower one at that rate.

What are the speeds of the two cars in the 100 km highway problem?

60 km/h and 40 km/h, from u − v = 20 and u + v = 100.

How do you convert 4 hours 10 minutes into hours?

10 minutes is 10/60 = 1/6 hour, so the total is 25/6 hours.

Why do we substitute p = 1/u and q = 1/v?

The equations have the speeds in the denominator. Replacing 1/u and 1/v with new letters turns them into ordinary linear equations.

Is Roohi's train and bus problem in the current syllabus?

It uses equations reducible to linear form, which were removed from the rationalised NCERT textbook, so it is extra practice rather than a core board question.