Calculating Average Speed for a Journey in Parts

The question asks us to find the average speed of Akhil's journey, which is covered in two distinct parts with different speeds.

Average speed is defined as the total distance covered divided by the total time taken for the journey.

Let's break down the journey into two parts:

Part 1 of the Journey

• Distance covered (\(d_1\)): 12 km
• Speed during this part (\(v_1\)): 16 km/h

To find the time taken for this part (\(t_1\)), we use the formula: Time = Distance / Speed.

So, \(t_1 = \frac{d_1}{v_1} = \frac{12 \text{ km}}{16 \text{ km/h}}\)

Simplifying the fraction:

\(t_1 = \frac{12}{16} = \frac{3}{4}\) hours

Part 2 of the Journey

• Distance covered (\(d_2\)): 6 km
• Speed during this part (\(v_2\)): 20 km/h

To find the time taken for this part (\(t_2\)), we use the same formula:

\(t_2 = \frac{d_2}{v_2} = \frac{6 \text{ km}}{20 \text{ km/h}}\)

Simplifying the fraction:

\(t_2 = \frac{6}{20} = \frac{3}{10}\) hours

Calculating Total Distance and Total Time

The total distance (\(D\)) for the entire journey is the sum of the distances of the two parts:

\(D = d_1 + d_2 = 12 \text{ km} + 6 \text{ km} = 18 \text{ km}\)

The total time (\(T\)) for the entire journey is the sum of the times taken for the two parts:

\(T = t_1 + t_2 = \frac{3}{4} \text{ hours} + \frac{3}{10} \text{ hours}\)

To add these fractions, we need a common denominator. The least common multiple (LCM) of 4 and 10 is 20.

\(T = \frac{3 \times 5}{4 \times 5} + \frac{3 \times 2}{10 \times 2} = \frac{15}{20} + \frac{6}{20}\)

\(T = \frac{15 + 6}{20} = \frac{21}{20}\) hours

Calculating Average Speed

Now we can calculate the average speed using the total distance and total time:

Average Speed = \(\frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{T}\)

Average Speed = \(\frac{18 \text{ km}}{\frac{21}{20} \text{ hours}}\)

To divide by a fraction, we multiply by its reciprocal:

Average Speed = \(18 \times \frac{20}{21} \text{ km/h}\)

We can simplify this by dividing 18 and 21 by their common factor, 3:

Average Speed = \(\frac{18 \div 3}{21 \div 3} \times 20 = \frac{6}{7} \times 20\)

Average Speed = \(\frac{6 \times 20}{7} = \frac{120}{7}\) km/h

Converting to a Mixed Number

The question asks for the answer in km/h, and the options are given as mixed numbers. Let's convert \(\frac{120}{7}\) to a mixed number.

Divide 120 by 7:

\(120 \div 7\)

\(120 = 17 \times 7 + 1\)

So, \(\frac{120}{7} = 17 \frac{1}{7}\) km/h.

The average speed of Akhil is \(17\frac{1}{7}\) km/h.

Revision Table: Speed, Distance, Time Formulas

Additional Information: Why Average Speed Isn't Just the Average of Speeds

It's important to note that average speed is not simply the average of the speeds (i.e., \(\frac{16 + 20}{2} = 18\)). This is because Akhil spends different amounts of time traveling at each speed. Average speed accounts for both the distance covered and the time taken at each segment of the journey. The formula Total Distance / Total Time correctly weights the speeds by the duration they were maintained.