Question

A bus starts running with an initial speed of 33 km/hr and its speed increases every hour by a certain amount. If it takes 7 hours to cover a distance of 315 km, then what will be the hourly increment (in km/hr) in the speed of the bus?
A) 1
B) 2
C) 3
D) 4

Answer 1

Hint: It is given that the speed of the bus increases every hour by a certain amount, this means all the speeds in 7 hours will have a common difference i.e. the incremented speed and hence will be in Arithmetic Progression (AP).
The sum of these speeds will be equal to the total distance covered by the bus.
Formula for sum of an AP is:
 $ S = \dfrac{n}{2}[2a + (n - 1)d] $ where,
S = Sum
n = Total number of terms
a = Initial value
d = Common difference

Complete step-by-step answer:
If the speed increases every hour by a certain amount, then the speeds for 7 hours will be in AP because all of them will have a common difference i.e. the increment in speed and their sum will be equal to the total distance covered.
Let the increment in speed be x.
Then,
Initial value (a) = 33 [given initial speed]
Total number of terms (n) = 7 [for 7 hours, we will have 7 different speeds]
Common difference (d) = x [speed increases every hour by a certain amount x]
Sum (S) = 315 [total distance covered]
Applying the formula for sum of an AP:
 $ \Rightarrow S = \dfrac{n}{2}[2a + (n - 1)d] $
Substituting the values, we get:
 $
\Rightarrow 315 = \dfrac{7}{2}[2 \times 33 + (7 - 1)x] \\
\Rightarrow 315 = \dfrac{7}{2}[66 + 6x] \\
  \dfrac{{315 \times 2}}{7} = 66 + 6x \\
\Rightarrow 90 - 66 = 6x \\
\Rightarrow 6x = 24 \\
\Rightarrow x = \dfrac{{24}}{6} \\
  $
x = 4
Therefore, the hourly increment in the speed of the bus is 4 km/hr

Note: Alternately we could have used the formula:
Average X N = Sum ________ (a)
For an AP the average is half of the sum of its first and last values.
The speeds for 7 hours will be:
33
33 + x
33 + 2x
..
..
..
33 + 6x
Average = $ \left( {\dfrac{{33 + 33 + 6x}}{2}} \right) $
                = $ \left( {\dfrac{{66 + 6x}}{2}} \right) $
                = 33 + 3x
N = 7 (7 hours)
Sum = total distance = 315
Substituting these in (a), we get:
(33 + 3x) X 7 = 315
231 + 21x = 315
21x = 315 – 231
21x = 84
 $ x = \dfrac{{84}}{{21}} $
x = 4