Episode 8 — Aptitude and Reasoning / 8.11 — Speed Distance and Time
8.11.b Tips, Tricks, and Shortcuts -- Speed, Distance, and Time
Tip 1: The 5/18 and 18/5 Multiplier -- Memorize, Don't Calculate
Instead of dividing by 3.6 or multiplying by 3.6, use the fraction form:
km/h --> m/s: Multiply by 5/18
m/s --> km/h: Multiply by 18/5
Shortcut for multiples of 18:
18 km/h = 5 m/s (just divide by 18, multiply by 5)
36 km/h = 10 m/s
54 km/h = 15 m/s
72 km/h = 20 m/s
90 km/h = 25 m/s
Pattern: Every 18 km/h increment = 5 m/s increment
Quick mental math: If the speed in km/h is not a multiple of 18, break it:
75 km/h = 72 + 3 = 20 + 3x(5/18) = 20 + 5/6 = 20.83 m/s
Tip 2: Use LCM to Assume Convenient Distances
When the problem gives speeds but not distance, assume the distance to be the LCM of the speeds. This eliminates fractions.
Example: Speeds are 40 km/h and 60 km/h for equal distances.
Assume distance = LCM(40, 60) = 120 km for each leg.
Time at 40 km/h = 120/40 = 3 hours
Time at 60 km/h = 120/60 = 2 hours
Total = 5 hours for 240 km
Average speed = 240/5 = 48 km/h
Tip 3: The Percentage Change Shortcut
If speed changes by a certain fraction, time changes by the reciprocal fraction (for the same distance).
Speed increases by 1/n --> Time decreases by 1/(n+1)
Speed decreases by 1/n --> Time increases by 1/(n-1)
Examples:
Speed increases by 25% (= 1/4)
--> Time decreases by 1/5 = 20%
Speed decreases by 20% (= 1/5)
--> Time increases by 1/4 = 25%
Speed increases by 33.33% (= 1/3)
--> Time decreases by 1/4 = 25%
Ready Reckoner
Speed Change | Time Change
-----------------|------------------
+25% (+1/4) | -20% (-1/5)
+20% (+1/5) | -16.67% (-1/6)
+50% (+1/2) | -33.33% (-1/3)
+100% (doubled) | -50% (halved)
-25% (-1/4) | +33.33% (+1/3)
-20% (-1/5) | +25% (+1/4)
-50% (halved) | +100% (doubled)
Tip 4: The "Late and Early" Formula
A person travelling at speed S1 reaches 't1' minutes LATE.
Travelling at speed S2, he reaches 't2' minutes EARLY.
Distance = S1 x S2 x (t1 + t2) / (S2 - S1)
IMPORTANT: Convert t1 and t2 to hours if speeds are in km/h.
Example:
At 30 km/h, a man reaches 10 min late.
At 40 km/h, he reaches 10 min early.
t1 + t2 = 10 + 10 = 20 min = 1/3 hour
D = 30 x 40 x (1/3) / (40 - 30)
= 1200/3 / 10
= 400 / 10
= 40 km
Variation: Both Late (or Both Early)
At S1, reaches t1 late. At S2, reaches t2 late (t2 < t1).
Distance = S1 x S2 x (t1 - t2) / (S2 - S1)
Tip 5: Ratio Method for Time Difference
When two speeds are given and time difference is known:
Speeds ratio = S1 : S2 = a : b
Time ratio = b : a (inverse)
Difference in time parts = |b - a|
If actual difference = T hours:
Each part = T / |b - a|
Example:
Speeds = 40 km/h and 50 km/h
Speed ratio = 4 : 5
Time ratio = 5 : 4
Difference = 5 - 4 = 1 part
If actual time difference = 2 hours:
1 part = 2 hours
Time at 40 km/h = 5 x 2 = 10 hours
Time at 50 km/h = 4 x 2 = 8 hours
Distance = 40 x 10 = 400 km
Tip 6: Meeting Point Shortcut
Two people start simultaneously from A and B (distance D apart) and walk towards each other.
They meet at a point that divides the distance in the ratio of their speeds.
Distance from A : Distance from B = Sa : Sb
This means:
Meeting point from A = D x Sa / (Sa + Sb)
Meeting point from B = D x Sb / (Sa + Sb)
Tip 7: Catching Up / Gaining Distance
When object A is faster than object B and they move in the same direction:
Distance gained by A per hour = Sa - Sb
Think of it as:
"A gains (Sa - Sb) km on B every hour."
To find when A catches B who has a head start of G km:
Time = G / (Sa - Sb)
Tip 8: Average Speed for Round Trip
This comes up so frequently that you should memorize the formula:
+-----------------------------------------------+
| Round trip average speed = 2ab / (a + b) |
| where a and b are the speeds for each leg |
+-----------------------------------------------+
Quick check: The average speed for a round trip is ALWAYS less than the arithmetic mean of the two speeds.
If speeds are 40 and 60:
Arithmetic mean = 50
Actual average = 48 (which is < 50, correct!)
Special case: If one speed is double the other (ratio 1:2):
Avg speed = 2(S)(2S) / (S + 2S) = 4S/3
Tip 9: Stoppage Time Shortcut
Stoppage per hour = (Difference in speeds) / (Speed without stoppage)
In minutes = [(S - S') / S] x 60
Example:
Without stoppages: 80 km/h
With stoppages: 60 km/h
Stoppage = (80-60)/80 x 60 = 20/80 x 60 = 15 minutes per hour
Tip 10: Clock-Based Speed Problems
A useful trick for problems involving specific departure/arrival times:
Step 1: Find the time difference between the two scenarios.
Step 2: Use the late/early formula or ratio method.
Pattern recognition:
"Leaves 30 min late, increases speed by 25%, reaches on time"
Speed increase = 25% = 1/4
Time saved = 1/5 of original time (from Tip 3)
Time saved = 30 minutes
Original time = 30 x 5 = 150 minutes = 2.5 hours
Tip 11: Assume Distance = Product of Speeds
For problems where you need to find time or ratio:
Assume D = S1 x S2
Time at S1 = S1 x S2 / S1 = S2
Time at S2 = S1 x S2 / S2 = S1
This gives clean numbers.
Tip 12: The 1/3rd - 2/3rd Split Trick
When a journey has the first 1/3 at one speed and rest at another:
Total time = D/3S1 + 2D/3S2
Average speed = D / (D/3S1 + 2D/3S2)
= 3.S1.S2 / (2.S1 + S2)
Similarly for 1/4, 3/4 split:
Average speed = 4.S1.S2 / (3.S1 + S2)
Tip 13: The Escalator / Moving Walkway Analogy
Walking WITH the escalator: Effective speed = Own speed + Escalator speed
Walking AGAINST the escalator: Effective speed = Own speed - Escalator speed
This is the same principle as Boats and Streams (Section 8.13).
Tip 14: "Starting at the Same Time" vs "Starting at Different Times"
Same time, same direction: relative speed = |S1 - S2|
Same time, opposite: relative speed = S1 + S2
Different time: account for head start FIRST, then use relative speed
Template for different start times:
Step 1: Calculate distance covered during head start.
Step 2: Treat this as the initial gap.
Step 3: Time to close gap = Gap / Relative speed.
Tip 15: Quick Approximation with Fractions
For exams with multiple-choice answers, you can often eliminate options:
Average speed of a round trip at speeds a and b:
- Always LESS than (a+b)/2
- Always MORE than the smaller speed
- Always LESS than the larger speed
- Closer to the smaller speed (because more time is spent at it)
Example:
Speeds: 20 km/h and 30 km/h
Average MUST be between 20 and 25 (closer to 20 than 30)
Actual: 2x20x30/50 = 1200/50 = 24 km/h (confirmed!)
Tip 16: Converting "X minutes per km" to km/h
If a person takes X minutes per km:
Speed = 60/X km/h
Example: 12 minutes per km = 60/12 = 5 km/h
Tip 17: Shortcut Table for Common Ratios
Speed Ratio | Time Ratio | Avg Speed (round trip)
------------|------------|----------------------
1 : 1 | 1 : 1 | S (same)
1 : 2 | 2 : 1 | 4S/3
1 : 3 | 3 : 1 | 3S/2
2 : 3 | 3 : 2 | 12S/5 (where S is unit)
3 : 4 | 4 : 3 | 24S/7
3 : 5 | 5 : 3 | 15S/4
Summary of Top Shortcuts
1. Multiply by 5/18 or 18/5 for unit conversion
2. Assume LCM as distance to avoid fractions
3. Speed +1/n --> Time -1/(n+1)
4. Late/Early formula: D = S1.S2.(t1+t2)/(S2-S1)
5. Meeting divides distance in ratio of speeds
6. Round trip avg = 2ab/(a+b), always < arithmetic mean
7. Stoppage/hr = (S-S')/S in hours
8. Ratio method: inverse speeds ratio = time ratio
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