Unit I Aptitude: Time and Work Problems, Speed, Distance, and Related Quantitative Aptitude Topics
1. Time and Work Problems
Time and Work is a fundamental concept in quantitative aptitude. It deals with the amount of work that can be completed by individuals or groups within a certain period, considering their individual rates of work.
1.1 Basic Concepts
The core idea is that the amount of work done is directly proportional to the rate of work and the time spent working. Conversely, the time taken is inversely proportional to the rate of work for a fixed amount of work.
Work Done = Rate of Work × Time Taken
Rate of Work = Work Done / Time Taken
Time Taken = Work Done / Rate of Work
In most Time and Work problems, the "Work" is considered as one complete unit of task. Therefore, the formulas simplify to:
Work Done = Rate × Time
Rate = Work / Time
Time = Work / Rate
When the work is considered as 1 unit:
Rate = 1 / Time
Time = 1 / Rate
1.2 Rates of Work
If a person A can complete a work in 'x' days, then A's rate of work is 1/x of the work per day.
If a person B can complete the same work in 'y' days, then B's rate of work is 1/y of the work per day.
1.3 Working Together
If A and B work together, their combined rate of work is the sum of their individual rates.
Combined Rate = Rate of A + Rate of B = (1/x) + (1/y)
The time taken by A and B to complete the work together is the reciprocal of their combined rate.
Time Taken Together = 1 / (Combined Rate) = 1 / ((1/x) + (1/y)) = xy / (x + y) days.
1.4 Working with a Fixed Amount of Work
Let's say A can complete a work in x days. In 1 day, A completes 1/x of the work. In 'd' days, A completes d/x of the work.
1.5 Example:
A can complete a piece of work in 10 days. B can complete the same work in 15 days. In how many days can they complete the work together?
Rate of A = 1/10 work per day.
Rate of B = 1/15 work per day.
Combined Rate = (1/10) + (1/15)
To add these fractions, find a common denominator, which is 30.
Combined Rate = (3/30) + (2/30) = 5/30 = 1/6 work per day.
Time Taken Together = 1 / (Combined Rate) = 1 / (1/6) = 6 days.
1.6 Men, Days, and Work
This type of problem involves the number of men, the number of days they work, and the amount of work they complete. The relationship is often expressed as:
(M1 * D1) / W1 = (M2 * D2) / W2
Where:
- M1 = Number of men in the first case
- D1 = Number of days in the first case
- W1 = Amount of work done in the first case
- M2 = Number of men in the second case
- D2 = Number of days in the second case
- W2 = Amount of work done in the second case
1.7 Example:
10 men can complete a work in 20 days. How many men are required to complete the same work in 5 days?
Here, M1 = 10, D1 = 20, W1 = 1 (assuming the same work).
We need to find M2 when D2 = 5, W2 = 1.
(10 * 20) / 1 = (M2 * 5) / 1
200 = 5 * M2
M2 = 200 / 5 = 40 men.
1.8 Efficiency
Sometimes, the problem involves people with different efficiencies or rates of working.
If A is twice as efficient as B, it means A can do twice the work B can do in the same amount of time. Or, A can do the same work in half the time B takes.
If B takes 'x' days, A takes 'x/2' days.
The ratio of their efficiencies is the inverse ratio of the time taken.
Efficiency Ratio (A:B) = Time Ratio (B:A)
1.9 Example:
A is twice as efficient as B and can complete a work in 30 days less than B. Find the time taken by A and B to complete the work together.
Let time taken by B be 'x' days. Then time taken by A is 'x - 30' days.
Since A is twice as efficient as B, A takes half the time B takes for the same work.
So, x - 30 = x / 2
Multiply by 2: 2(x - 30) = x
2x - 60 = x
x = 60 days (Time taken by B)
Time taken by A = x - 30 = 60 - 30 = 30 days.
Now, find the time taken together:
Time Together = (Time of A * Time of B) / (Time of A + Time of B)
Time Together = (30 * 60) / (30 + 60) = 1800 / 90 = 20 days.
1.10 Pipes and Cisterns
This is a variation of Time and Work problems. Here, pipes fill the cistern (positive work) and leaks empty the cistern (negative work).
If a pipe can fill a cistern in 'x' hours, its rate is 1/x (filling rate).
If a pipe can empty a cistern in 'y' hours, its rate is 1/y (emptying rate).
When pipes work together, their rates are added. When a filling pipe and an emptying pipe work together, the emptying rate is subtracted from the filling rate.
1.11 Example:
A tap can fill a cistern in 10 hours. Another tap can empty it in 15 hours. If both taps are opened simultaneously, in how many hours will the cistern be filled?
Rate of filling by first tap = 1/10 (positive)
Rate of emptying by second tap = 1/15 (negative)
Net rate when both are open = (1/10) - (1/15)
Net rate = (3/30) - (2/30) = 1/30 (positive, so it fills)
Time to fill = 1 / (Net rate) = 1 / (1/30) = 30 hours.
2. Speed, Distance, and Time
This topic deals with the relationship between speed, distance, and time. It's a crucial part of quantitative aptitude and is often tested in exams.
2.1 Basic Formula
The fundamental relationship is:
Distance = Speed × Time
From this, we can derive:
Speed = Distance / Time
Time = Distance / Speed
2.2 Units of Measurement
It is essential to maintain consistent units. Common units include:
- Distance: Kilometers (km), Meters (m)
- Speed: Kilometers per hour (km/h), Meters per second (m/s)
- Time: Hours (h), Minutes (min), Seconds (s)
2.3 Conversion of Units
The most common conversion required is between km/h and m/s.
- To convert km/h to m/s: Multiply by 5/18.
- To convert m/s to km/h: Multiply by 18/5.
1 km/h = 1000 m / 3600 s = 5/18 m/s
1 m/s = 18/5 km/h
2.4 Average Speed
Average speed is not simply the average of the speeds. It is calculated as:
Average Speed = Total Distance / Total Time
2.5 Cases for Average Speed
Case 1: Object travels equal distances at different speeds.
If an object travels a distance 'd' at speed S1 and another distance 'd' at speed S2, then:
Total Distance = d + d = 2d
Time taken for first part = d / S1
Time taken for second part = d / S2
Total Time = (d / S1) + (d / S2) = d * (1/S1 + 1/S2) = d * (S1 + S2) / (S1 * S2)
Average Speed = (2d) / [d * (S1 + S2) / (S1 * S2)] = 2 * (S1 * S2) / (S1 + S2)
This is the Harmonic Mean of the speeds.
Case 2: Object travels for equal time intervals at different speeds.
If an object travels for time 't' at speed S1 and then for time 't' at speed S2, then:
Distance covered in first part = S1 * t
Distance covered in second part = S2 * t
Total Distance = S1*t + S2*t = t * (S1 + S2)
Total Time = t + t = 2t
Average Speed = [t * (S1 + S2)] / (2t) = (S1 + S2) / 2
This is the Arithmetic Mean of the speeds.
2.6 Example:
A car travels from point A to point B at a speed of 40 km/h and returns from point B to point A at a speed of 60 km/h. What is the average speed for the entire journey?
Here, the distance from A to B is the same as from B to A. So, we use the formula for equal distances.
Average Speed = 2 * (40 * 60) / (40 + 60)
Average Speed = 2 * 2400 / 100
Average Speed = 4800 / 100 = 48 km/h.
Another example: A train travels for 2 hours at 50 km/h and then for 3 hours at 60 km/h. What is its average speed?
Here, the time intervals are different, but the question implies distinct segments of travel. We calculate total distance and total time.
Distance 1 = 50 km/h * 2 h = 100 km
Distance 2 = 60 km/h * 3 h = 180 km
Total Distance = 100 km + 180 km = 280 km
Total Time = 2 h + 3 h = 5 h
Average Speed = Total Distance / Total Time = 280 km / 5 h = 56 km/h.
2.7 Relative Speed
Relative speed is used when two objects are moving, and we want to find their speed with respect to each other.
Case 1: Objects moving in the same direction.
If two objects move in the same direction with speeds S1 and S2 (S1 > S2), their relative speed is (S1 - S2).
Case 2: Objects moving in opposite directions.
If two objects move in opposite directions, their relative speed is (S1 + S2).
2.8 Applications of Relative Speed
1. Trains crossing each other:
- If two trains move in the same direction, their relative speed is the difference of their speeds. The time taken to cross each other is (Sum of lengths) / (Relative Speed).
- If two trains move in opposite directions, their relative speed is the sum of their speeds. The time taken to cross each other is (Sum of lengths) / (Relative Speed).
2. Boats and Streams:
- Speed of boat in still water = 'b'
- Speed of stream = 's'
- Speed downstream (boat and stream in same direction) = b + s
- Speed upstream (boat against stream) = b - s
- To find 'b' and 's':
- b = (Speed Downstream + Speed Upstream) / 2
- s = (Speed Downstream - Speed Upstream) / 2
3. Cyclists, Runners, etc.:
The same principles of relative speed apply.
2.9 Example:
A train 150 meters long is running at 60 km/h. In what time will it cross a man standing on the platform?
Here, the man is stationary. The train needs to cover its own length.
Speed = 60 km/h. Convert to m/s: 60 * (5/18) = 300/18 = 50/3 m/s.
Distance = Length of train = 150 m.
Time = Distance / Speed = 150 m / (50/3 m/s) = 150 * (3/50) = 3 * 3 = 9 seconds.
Another example: Two trains, each 100 meters long, are running in the same direction with speeds 60 km/h and 40 km/h respectively. In how much time will the faster train cross the slower train?
Length of train 1 = 100 m, Speed 1 = 60 km/h.
Length of train 2 = 100 m, Speed 2 = 40 km/h.
Since they are moving in the same direction, relative speed = 60 - 40 = 20 km/h.
Convert relative speed to m/s: 20 * (5/18) = 100/18 = 50/9 m/s.
Total length to be covered = Sum of lengths = 100 m + 100 m = 200 m.
Time = Total Length / Relative Speed = 200 m / (50/9 m/s) = 200 * (9/50) = 4 * 9 = 36 seconds.
3. Quantitative Aptitude Topics - Related Concepts
Time and Work, and Speed, Distance, Time are foundational. Many other quantitative aptitude topics build upon these principles or share similar logical structures.
3.1 Problems on Trains
These are a direct application of Speed, Distance, Time and Relative Speed. Key scenarios include:
- Train crossing a stationary object (pole, man): Distance = Length of train.
- Train crossing an object of length (platform, bridge, another train): Distance = Length of train + Length of object.
- Trains moving in the same direction: Relative Speed = Difference in speeds.
- Trains moving in opposite directions: Relative Speed = Sum of speeds.
3.2 Boats and Streams
As discussed earlier, this topic relies heavily on understanding upstream and downstream speeds, which are applications of relative speed (boat speed vs. stream speed).
3.3 Percentages
Percentage is a way of expressing a number as a fraction of 100. It's fundamental for profit/loss, simple/compound interest, and ratio/proportion problems.
Percentage = (Part / Whole) × 100
Increase/Decrease by %: Original Value × (1 ± %/100)
3.4 Profit and Loss
Key terms:
- Cost Price (CP): The price at which an item is bought.
- Selling Price (SP): The price at which an item is sold.
- Profit: SP - CP (if SP > CP)
- Loss: CP - SP (if CP > SP)
- Profit % = (Profit / CP) × 100
- Loss % = (Loss / CP) × 100
Formulae:
SP = CP * (1 + Profit%/100)
SP = CP * (1 - Loss%/100)
CP = SP / (1 + Profit%/100)
CP = SP / (1 - Loss%/100)
3.5 Simple Interest (SI) and Compound Interest (CI)
Simple Interest: Interest is calculated only on the principal amount.
SI = (P × R × T) / 100
Amount (A) = P + SI = P + (PRT/100) = P(1 + RT/100)
Where P = Principal, R = Rate of interest per annum, T = Time in years.
Compound Interest: Interest is calculated on the principal plus accumulated interest.
Amount (A) = P (1 + R/100)^T
Compound Interest (CI) = A - P
3.6 Ratio and Proportion
Ratio: Comparison of two quantities by division (a:b).
Proportion: Equality of two ratios (a:b :: c:d).
Product of means = Product of extremes: b*c = a*d.
These concepts are used extensively in partnership problems, mixture problems, and as a tool to simplify calculations in other areas.
3.7 Averages
Average = Sum of items / Number of items.
This is similar to the Average Speed formula (Total Distance / Total Time).
3.8 Mensuration (Basic Formulas)
While not directly related to Time/Work or Speed/Distance, basic mensuration formulas for areas and volumes are often required in problems that might involve rates of filling or covering areas.
- Area of rectangle = l × b
- Area of square = side²
- Area of circle = πr²
- Circumference of circle = 2πr
- Volume of cuboid = l × b × h
- Volume of cube = side³
- Volume of cylinder = πr²h
- Surface area of sphere = 4πr²
- Volume of sphere = (4/3)πr³