A boat goes 30 km downstream in 2 hours and returns upstream in 3 hours. What is the speed of the boat in still water?

Prabhakar, let us solve this carefully step by step. Given Data Distance downstream \(= 30 \text{ km}\) Time taken downstream \(= 2 \text{ hours}\) Distance upstream \(= 30 \text{ km}\) Time taken upstream \(= 3 \text{ hours}\) Relevant Concept Speed means the distance travelled in unit time. It is given by the formula: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \] In problems of boats and streams: Downstream speed \(=\) speed of boat in still water \(+\) speed of stream Upstream speed \(=\) speed of boat in still water \(-\) speed of stream This applies because when the boat moves downstream , the current of water helps the boat, and when it moves upstream , the current opposes the boat. Solution Step 1: Find the downstream speed \[ \text{Downstream speed} = \frac{30}{2} = 15 \text{ km/h} \] Step 2: Find the upstream speed \[ \text{Upstream speed} = \frac{30}{3} = 10 \text{ km/h} \] Step 3: Use the formula for speed in still water Let the speed of the boat in still water be \(b\) km/h and the speed of the stream be \(s\) km/h. \[ b + s = 15 \] \[ b - s = 10 \] Add the two equations: \[ (b+s) + (b-s) = 15 + 10 \] \[ 2b = 25 \] \[ b = \frac{25}{2} = 12.5 \] Final Answer Therefore, the speed of the boat in still water is \[ \boxed{12.5 \text{ km/h}} \] Summary of Key Points Downstream speed \(= \frac{30}{2} = 15 \text{ km/h}\) Upstream speed \(= \frac{30}{3} = 10 \text{ km/h}\) Speed of boat in still water \(= \frac{\text{downstream speed} + \text{upstream speed}}{2}\) So, speed in still water \(= \frac{15+10}{2} = 12.5 \text{ km/h}\)

A car travels 150 km in 3 hours. What is its average speed?

It’s okay — this one is simpler than it looks. We need the average speed. Average speed means: \[ \text{Average speed} = \frac{\text{distance}}{\text{time}} \] Given: Distance \(= 150\) km Time \(= 3\) hours Step 1: Put the numbers into the formula. \[ \text{Average speed} = \frac{150}{3} \] Step 2: Divide \(150\) by \(3\). \[ \frac{150}{3} = 50 \] So the car travels \(50\) km in \(1\) hour, like covering \(150\) km by splitting it equally across \(3\) hours. Final answer: \(50\) km/h The one thing to remember is: \( \text{speed} = \frac{\text{distance}}{\text{time}} \).

A person covers the first half of a distance at 40 km/h and the second half at 60 km/h. What is the average speed for the whole journey?

Understanding the Problem We need to find the average speed for the whole journey when: The first half of the distance is covered at \(40\) km/h The second half of the distance is covered at \(60\) km/h Given Data Speed for first half \(= 40\) km/h Speed for second half \(= 60\) km/h Formula to Use Average speed is given by: \[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \] When equal distances are covered at two different speeds \(a\) and \(b\), the average speed is: \[ \text{Average Speed} = \frac{2ab}{a+b} \] Step-by-Step Solution Step 1: Let the total distance be \(2d\) Then each half of the distance is \(d\). Step 2: Find time taken for each half Time \(=\frac{\text{Distance}}{\text{Speed}}\) Time for first half: \[ t_1 = \frac{d}{40} \] Time for second half: \[ t_2 = \frac{d}{60} \] Step 3: Find total time \[ t = \frac{d}{40} + \frac{d}{60} \] Taking LCM \(= 120\): \[ t = \frac{3d}{120} + \frac{2d}{120} = \frac{5d}{120} = \frac{d}{24} \] Step 4: Find average speed \[ \text{Average Speed} = \frac{2d}{d/24} \] \[ \text{Average Speed} = 2d \times \frac{24}{d} = 48 \] Final Answer The average speed for the whole journey is \(48\) km/h. Quick Summary Average speed is not the simple average of \(40\) and \(60\). Because the distances are equal , we use the harmonic mean formula. \[ \text{Average Speed} = \frac{2 \times 40 \times 60}{40 + 60} = \frac{4800}{100} = 48 \text{ km/h} \]

Find the missing number in the series: 3, 6, 12, 24, ?

Finding the Missing Number in the Series We need to find the next number in the series: \[ 3,\ 6,\ 12,\ 24,\ ? \] Understanding the Pattern Let us look at how each number is related to the previous one. \(3 \times 2 = 6\) \(6 \times 2 = 12\) \(12 \times 2 = 24\) So, each number is obtained by multiplying the previous number by \(2\) . Step-by-Step Solution Step 1: Identify the rule. The pattern is: \[ \text{Next term} = \text{Previous term} \times 2 \] Step 2: Apply the rule to the last given term. The last number is \(24\). \[ 24 \times 2 = 48 \] Final Answer The missing number is \(48\) . Summary Given series: \(3, 6, 12, 24, ?\) Pattern used: multiply by \(2\) Missing number: \(48\)

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Prabhakar, let us solve this carefully step by step. Given Data Distance downstream \(= 30 \text{ km}\) Time taken downstream \(= 2 \text{ hours}\) Distance upstream \(= 30 \text{ km}\) Time taken upstream \(= 3 \text{ hours}\) Relevant Concept Speed means the distance travelled in unit time. It is given by the formula…
It’s okay — this one is simpler than it looks. We need the average speed. Average speed means: \[ \text{Average speed} = \frac{\text{distance}}{\text{time}} \] Given: Distance \(= 150\) km Time \(= 3\) hours Step 1: Put the numbers into the formula. \[ \text{Average speed} = \frac{150}{3} \] Step 2: Divide \(150\) by \…
Understanding the Problem We need to find the average speed for the whole journey when: The first half of the distance is covered at \(40\) km/h The second half of the distance is covered at \(60\) km/h Given Data Speed for first half \(= 40\) km/h Speed for second half \(= 60\) km/h Formula to Use Average speed is giv…
Finding the Missing Number in the Series We need to find the next number in the series: \[ 3,\ 6,\ 12,\ 24,\ ? \] Understanding the Pattern Let us look at how each number is related to the previous one. \(3 \times 2 = 6\) \(6 \times 2 = 12\) \(12 \times 2 = 24\) So, each number is obtained by multiplying the previous n…