Problem 1 :
A person covers a certain distance at a certain speed. If he increases his speed by 33⅓ %, he takes 15 minutes less to cover the same distance. Find the time taken by him initially to cover the distance at the original speed.
Problem 2 :
A man traveled from the village to the post office at the rate of 25 k mph and walked back at the rate of 4 kmph. If the entire journey had taken 5 hours 48 minutes, find the distance of the post office from the village.
Problem 3 :
If a man walks at the rate of 5 km/hr, he misses a train by 7 minutes. However, if he walks at the rate of 6 km/hr, he reaches the station 5 minutes before the arrival of the train. Find the distance covered by him to reach the station.
Problem 4 :
A person has to cover a distance of 6 miles in 45 minutes. If he covers one-half of the distance in two-thirds of the total time. What must his speed be to cover the remaining distance in the remaining time ?
Problem 5 :
A is faster than B . A and B each walk 24 miles. The sum of their speeds is 7 miles per hour and the sum of their time taken is 14 hrs. Find A's speed and B's speed (in mph).
1. Answer :
Let the original speed be 100%.
Given : The speed is increased by 33⅓ %.
Then, the speed after increment is 133⅓ %.
Ratio of the speeds is
100 % : 133⅓ %
100 : 133⅓
100 : ⁴⁰⁰⁄₃
Divide both the terms by 100.
1 : ⁴⁄₃
So, ratio of the speeds is 1 : ⁴⁄₃.
If the ratio of the speeds is 1 : ⁴⁄₃, then the ratio of time taken to cover the same distance would be
1 : ¾
When the speed is increased by 33⅓%, ¾ of the original time is enough to cover the same distance.
That is, when the speed is increased by 33⅓ %, ¼ of the original time will be decreased.
The question says that when speed is increased by 33⅓%, time is decreased by 15 minutes.
So, we have
¼ of the original time = 15 minutes
Multiply both sides by 4.
4 ⋅ (¼ of the original time) = (15 minutes) ⋅ 4
Original time = 60 minutes
Original time = 1 hour
So, the time taken by the person initially is 1 hour.
2. Answer :
Here, the distance covered in both the ways is same.
So, the formula to find the average speed is
Substitute p = 25 and q = 4.
The average speed is ²⁰⁰⁄₂₉ km/hr.
Given :The entire journey had taken 5 hours 48 minutes
5 hour 48 min = 5⁴⁸⁄₆₀_{ }hours
5 hours 48 min = 5⅘ hours
5 hours 48 min = ²⁹⁄₅ hours
The formula to find the distance is
= Speed ⋅ Time
Then, the distance covered in ²⁹⁄₅ hours at the average speed ²⁰⁰⁄₂₉ kmph is
= ²⁰⁰⁄₂₉ ⋅ ²⁹⁄₅
= 40 km
So, the distance covered in the whole journey is 40 km.
(Whole journey : Village to post office + Post office to village)
Then the distance between the post office and village is
= ⁴⁰⁄₂
= 20
So, the distance of the post office from the village is 20 km.
3. Answer :
Let x be the distance to be covered by the person to reach the station.
The formula to find the time is
When the speed is 5 kmph, time is
= ˣ⁄₅ hrs
When the speed is 6 kmph, time is
= ˣ⁄₆ hrs
Let t be the actual time required to cover the distance x.
And also,
7 minutes = ⁷⁄₆₀ hrs
5 minutes = ⁵⁄₆₀ = ¹⁄₁₂ hrs
Given : If the man walks at the rate of 5 km/hr, he misses the train by 7 minutes.
That is, he takes 7 minutes more than actual time.
So, we have
t = ˣ⁄₅ - ⁷⁄₆₀ ----(1)
Given : If he walks at the rate of 6 km/hr, he reaches the station 5 minutes before.
That is, he takes 5 minutes less than actual time.
So, we have
t = ˣ⁄₆ + ¹⁄₁₂ ----(2)
From (1) and (2), we get
ˣ⁄₅ - ⁷⁄₆₀ = ˣ⁄₆ + ¹⁄₁₂
Solving for x :
¹²ˣ⁄₆₀ - ⁷⁄₆₀ = ²ˣ⁄₁₂ + ¹⁄₁₂
⁽¹²ˣ ⁻ ⁷⁾⁄₆₀ = ⁽²ˣ ⁺ ¹⁾⁄₁₂
L.C.M of (60, 12) is 60.
Multiply both sides by 60.
12x - 7 = 5(2x + 1)
12x - 7 = 10x + 5
Simplify.
2x = 12
Divide both sides by 2.
x = 6
So, the distance covered by him to reach the station is 6 kms.
4. Answer :
Given : Total distance is 6 miles and total time is 45 minutes. And also, he covers one-half of the distance in two-thirds of the total time.
One-half of the total distance 6 miles is
= ½ ⋅ 6
= 3 km
Two-thirds of the total time 45 minutes is
= ⅔ ⋅ 45
= 30 minutes
From the above calculations, we have
Remaining distance = 6 - 3 = 3 miles
Remaining time = 45 - 30 = 15 minutes
The formula to find the speed is
Substitute distance = 3 and time = 15.
= ³⁄₁₅
= ⅕ miles per minute
= ⅕ ⋅ 60 miles per hour
= 12 miles per hour.
So, the speed must be 12 miles per hour.
5. Answer :
Let x be the speed of A .
Then speed of B is
= 7 - x
The formula to find time is
Then, time taken by A is
= ²⁴⁄ₓ hrs
Time taken by B is
= ²⁴⁄₍₇ ₋ ₓ₎ hrs
Given : Sum of time taken is 14 hours.
So, we have
²⁴⁄ₓ + ²⁴⁄₍₇ ₋ ₓ₎ = 14
L.C.M of x and (7 - x) is x(7 - x).
Multiply both sides by x(7 - x).
24 ⋅ (7 - x) + 24 ⋅ x = 14 ⋅ x(7 - x)
168 - 24x + 24x = 98x - 14x^{2}
14x^{2 }- 98x + 168 = 0
Divide both sides by 14.
x^{2 }- 7x + 12 = 0
x^{2 }- 3x - 4x + 12 = 0
x(x - 3) - 4(x - 3) = 0
(x - 3)(x - 4) = 0
x = 3 or x = 4
So, A's speed can be 4 mph or 3 mph.
If A's speed is 4 mph, then B's speed
7 - x = 7 - 4
7 - x = 3 mph
If A's speed is 3 mph, then B's speed
7 - x = 7 - 3
7 - x = 4 mph
Given : A is faster than B.
So, the speed of A is 4 miles per hour and B is 3 miles per hour.
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