Work Life & Job Hunting · SPI Prep Lab
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A car travelling at 72 km per hour keeps going for 2 hours 30 minutes. How many kilometres does it cover?
Distance = speed × time. 2 hours 30 minutes is 2.5 hours, so 72 × 2.5 = 180 km. 144 km covers only the 2 hours. 174 km comes from 72 × 2 + 30, adding the 30 minutes straight onto the distance. 216 km works it as 3 hours.
How long does it take to cover 108 km at 45 km per hour?
Time = distance ÷ speed = 108 ÷ 45 = 2.4 hours. Multiplying the decimal part 0.4 by 60 gives 24 minutes, so 2 hours 24 minutes. '2 hours 40 minutes' is the classic slip of reading the 4 of 2.4 as 40 minutes. '2 hours 4 minutes' takes the 0.4 straight off as 4 minutes.
You walked 1200 m in 15 minutes. What is this speed in km per hour?
The speed per minute is 1200 ÷ 15 = 80 m. An hour is 60 minutes, so 80 × 60 = 4800 m, that is 4.8 km per hour. '80 km per hour' reads the figure 80 m per minute straight off as an hourly speed. '48 km per hour' gets a digit wrong when converting 4800 m into kilometres.
Converted to km per hour, what is 15 m per second?
An hour is 3600 seconds and a kilometre is 1000 m, so to go from metres per second to kilometres per hour you multiply by 3.6: 15 × 3.6 = 54 km. '90 km per hour' multiplies by 6 rather than 3.6. '900 km per hour' stops at 15 × 60, which is 900 m per minute, without converting the unit. '0.9 km per hour' reads that 900 m per minute as 0.9 km and then mislabels it as an hourly speed.
A round trip was made between points A and B at 30 km per hour going out and 20 km per hour coming back. What is the average speed over the whole trip?
Suppose one way is 60 km: the outward leg takes 2 hours and the return 3 hours, making 5 hours in all for 120 km. 120 ÷ 5 = 24 km per hour. '25 km per hour' takes the plain average of 30 and 20, but an average speed is not the average of the speeds. '50 km per hour' merely adds them. '12 km per hour' comes from 30 × 20 ÷ 50, forgetting to double.
Of a 12 km route, the first 6 km was covered at 9 km per hour and the remaining 6 km at 18 km per hour. What is the average speed over the whole route?
The first 6 km takes 6 ÷ 9 = 2/3 hour and the rest takes 6 ÷ 18 = 1/3 hour, exactly 1 hour in total. 12 ÷ 1 = 12 km per hour. '13.5 km per hour' is the plain average of 9 and 18. '27 km per hour' merely adds them. '6 km per hour' comes from 9 × 18 ÷ 27, forgetting to double.
Walking from home to the station at 60 m per minute makes you 5 minutes late for a meeting, while walking at 80 m per minute gets you there 3 minutes early. How many metres is it from home to the station?
Let the distance be d metres. The gap between d ÷ 60 and d ÷ 80 is 5 + 3 = 8 minutes. d × (1/60 − 1/80) = d ÷ 240 = 8, so d = 1920 m. Checking, the two times are 32 and 24 minutes, a gap of 8 minutes. 480 m multiplies the 8 minutes by 60 m per minute, and 640 m multiplies it by 80 m per minute.
Covering a certain route by car at 40 km per hour takes 9 minutes longer than at 50 km per hour. How long is the route?
9 minutes is 0.15 hours. Let the distance be d km; then d ÷ 40 − d ÷ 50 = d ÷ 200 = 0.15, so d = 30 km. Checking, the two times are 45 and 36 minutes, a gap of 9 minutes. 6 km is 40 × 0.15 and 7.5 km is 50 × 0.15, each multiplying the time gap by one of the speeds.
Setting out at 9:40 a.m., you walked a 6 km route at 4 km per hour. What time did you arrive?
The journey takes 6 ÷ 4 = 1.5 hours, that is 1 hour 30 minutes. An hour and a half after 9:40 is 11:10. '10:55 a.m.' misreads 1.5 hours as 1 hour 15 minutes. '11:40 a.m.' works it as 2 hours and '10:40 a.m.' as 1 hour.
Over the same route, A's speed is 1.2 times B's. When A takes 30 minutes, how many minutes does B take?
Over the same distance, time is inversely proportional to speed. If A's speed is 1.2 times B's, B's time is 1.2 times A's: 30 × 1.2 = 36 minutes. '25 minutes' comes from 30 ÷ 1.2, applying the ratio of speeds directly to the times. '24 minutes' comes from 30 × 0.8. Noticing that B is the slower one, so the time must exceed 30 minutes, narrows it down at once.
A round trip between towns A and B was made by bicycle at 12 km per hour going out and 18 km per hour coming back, taking 5 hours in all. How far apart are towns A and B?
Let one way be d km; then d ÷ 12 + d ÷ 18 = 5. Over a common denominator, 5d ÷ 36 = 5, so d = 36 km. Checking, the legs take 3 hours and 2 hours, making 5 hours. 75 km multiplies 5 hours by the plain average of 12 and 18, 15 km per hour. 72 km is the round-trip distance, not the one-way distance asked for.
From point A to point B, one third of the route was walked at 6 km per hour and the rest cycled at 15 km per hour, taking 1 hour 24 minutes in all. How far is it from A to B?
Let the distance be d km. The time is d/3 ÷ 6 + 2d/3 ÷ 15 = d/18 + 2d/45 = 9d/90 = d/10 hours. 1 hour 24 minutes is 1.4 hours, so d ÷ 10 = 1.4 and d = 14 km. 14.7 km multiplies 1.4 hours by the plain average of 6 and 15, 10.5 km per hour. 8.4 km is 6 × 1.4 and 21 km is 15 × 1.4, each using only one of the speeds for the whole route.
From two points 1800 m apart, A and B set off walking towards each other at the same moment. A walks at 70 m per minute and B at 80 m per minute. After how many minutes do they meet?
Since they face each other, the gap closes at the sum of the speeds: 1800 ÷ (70 + 80) = 1800 ÷ 150 = 12 minutes. '180 minutes' uses the catching-up formula, dividing by the difference of 10. '24 minutes' divides by the average speed of 75 m per minute. '22.5 minutes' divides by B's 80 m per minute alone.
A younger brother left home at 60 m per minute, and 10 minutes later the elder brother set off along the same road at 90 m per minute to catch him. How many minutes after the elder brother sets off does he catch up?
When the elder brother sets off the younger is 60 × 10 = 600 m ahead. They travel the same way, so the gap closes at the difference of the speeds: 600 ÷ (90 − 60) = 20 minutes. '4 minutes' uses the meeting formula, dividing by the sum of 150. '30 minutes' is the time from the younger brother's departure, but the question asks from the elder brother's.
Round a pond 1200 m in circumference, A walks at 90 m per minute and B at 70 m per minute, both setting off from the same point in the same direction at the same moment. After how many minutes does A first catch B?
Going the same way, A catches B once the lead reaches one full lap: 1200 ÷ (90 − 70) = 60 minutes. '7.5 minutes' uses the sum of the speeds, 160, which is the formula for walking in opposite directions. '120 minutes' works with a lead of two laps rather than one.
A train 180 m long travels at 72 km per hour and crosses a railway bridge 900 m long. How many seconds does it take from starting to cross to finishing?
72 km per hour is 72 ÷ 3.6 = 20 m per second. From the front entering to the rear leaving, the train covers 900 + 180 = 1080 m, so 1080 ÷ 20 = 54 seconds. '45 seconds' forgets to add the train's own length, 900 ÷ 20. '15 seconds' fails to convert to metres per second, 1080 ÷ 72. '9 seconds' covers only the train's length, 180 ÷ 20.
Train A, 150 m long, travels at 90 km per hour and train B, 250 m long, at 54 km per hour, running towards each other. How many seconds does it take from the moment they begin to pass to the moment they have passed?
Facing each other, the relative speed is 90 + 54 = 144 km per hour, that is 40 m per second. The distance to cover is the two lengths together, 150 + 250 = 400 m, so 400 ÷ 40 = 10 seconds. '40 seconds' uses the difference of 36 km per hour, that is 10 m per second, which is the overtaking formula. '20 seconds' takes the plain average of the speeds, 72 km per hour or 20 m per second.
A train travelling at a constant speed took 25 seconds from entering a 400 m tunnel to leaving it, and 40 seconds from starting to cross a 700 m bridge to finishing. How long is the train?
The difference in distance is 700 − 400 = 300 m and the difference in time is 40 − 25 = 15 seconds, so the speed is 300 ÷ 15 = 20 m per second. In the tunnel it covers 25 × 20 = 500 m, so the train is 500 − 400 = 100 m long. '20 m' mistakes the speed of 20 m per second for a length. '300 m' takes the difference in distance as the length.
A boat whose speed in still water is 15 km per hour goes 54 km downstream on a river flowing at 3 km per hour. How long does it take?
The downstream speed is 15 + 3 = 18 km per hour, so 54 ÷ 18 = 3 hours. '4 hours 30 minutes' uses the upstream speed 15 − 3 = 12 km per hour, giving 54 ÷ 12 = 4.5 hours. '3 hours 36 minutes' ignores the current, 54 ÷ 15 = 3.6 hours.
A boat covers the same 24 km stretch of river in 2 hours downstream and 3 hours upstream. Which combination of the boat's speed in still water and the speed of the current is correct?
Downstream is 24 ÷ 2 = 12 km per hour and upstream is 24 ÷ 3 = 8 km per hour. Still water is (12 + 8) ÷ 2 = 10 km per hour and the current is (12 − 8) ÷ 2 = 2 km per hour. '20 and 4' forgets to halve the sum and the difference. 'Still water 12' takes the downstream speed as the still-water speed.
A boat took 5 hours to go 45 km upstream and 3 hours to come back down the same stretch. What is the speed of the current?
Upstream is 45 ÷ 5 = 9 km per hour and downstream is 45 ÷ 3 = 15 km per hour. The current is (15 − 9) ÷ 2 = 3 km per hour. '6 km per hour' forgets to halve the difference. '12 km per hour' is the still-water speed, (15 + 9) ÷ 2, not the current. '24 km per hour' merely adds the two speeds.
A boat whose speed in still water is 20 km per hour goes 45 km upstream on a river flowing at 5 km per hour. How long does it take?
The upstream speed is 20 − 5 = 15 km per hour, so 45 ÷ 15 = 3 hours. '1 hour 48 minutes' uses the downstream speed 20 + 5 = 25 km per hour, giving 45 ÷ 25 = 1.8 hours. '2 hours 15 minutes' ignores the current, 45 ÷ 20 = 2.25 hours.
Round a pond 3600 m in circumference, A and B set off from the same point in opposite directions at the same moment. A walks at 100 m per minute and B at 80 m per minute. After how many minutes do they meet for the second time?
Going opposite ways, they meet once for every full lap the two cover between them. The first meeting is at 3600 ÷ (100 + 80) = 20 minutes, and the second is twice that, 40 minutes. '20 minutes' is the first meeting. '180 minutes' uses the catching-up formula with the difference of 20. '360 minutes' is when the second catch-up would happen if they went the same way.
From two points 4500 m apart, A and B set off towards each other at the same moment. A moves at 90 m per minute and B at 60 m per minute, but A took a 5-minute rest starting 10 minutes after setting off. After how many minutes from the start do they meet?
At 15 minutes after the start, A has covered 90 × 10 = 900 m and B 60 × 15 = 900 m, leaving 4500 − 1800 = 2700 m. From there the gap closes at 150 m per minute between them, so 2700 ÷ 150 = 18 minutes, giving 15 + 18 = 33 minutes. '30 minutes' is 4500 ÷ 150, the answer with no rest. '35 minutes' adds the 5-minute rest straight onto that 30, but B keeps going during the rest, so they in fact meet sooner.
A job takes A 12 days alone and B 6 days alone. How many days does it take the two of them working together?
Taking the whole job as 1, they do 1/12 + 1/6 = 1/4 a day, so 4 days. Setting the whole job at 12 gives 1 and 2 a day, that is 3 together, and 12 ÷ 3 = 4 days in whole numbers. '9 days' is the plain average of 12 and 6, and '18 days' adds the day counts. Since two are working, it must come out shorter than B's 6 days alone.
Filling a tank takes 15 minutes with pipe A alone and 10 minutes with pipe B alone. Using both at once, how many minutes does it take to fill?
Per minute they fill 1/15 + 1/10 = 2/30 + 3/30 = 5/30 = 1/6, so 6 minutes. '12.5 minutes' is the plain average of 15 and 10, '25 minutes' adds them and '5 minutes' subtracts. With both running it must come out shorter than pipe B's 10 minutes alone.
A job takes A 20 days alone and B 30 days alone. The two worked together for 6 days and A then finished the rest alone. How many days did A work alone?
Together they do 1/20 + 1/30 = 1/12 a day, so in 6 days 6/12 = 1/2 is done. Doing the remaining 1/2 alone, A needs 1/2 ÷ 1/20 = 10 days. '16 days' is A's total working days, 6 + 10, not the days worked alone. '12 days' is how long it would take with both working right to the end.
A job takes A and B together 12 days and B alone 20 days. How many days does it take A alone?
A does 1/12 − 1/20 = 5/60 − 3/60 = 2/60 = 1/30 a day, so 30 days. '8 days' subtracts the day counts, 20 − 12, and '32 days' adds them. Checking, 1/30 + 1/20 = 2/60 + 3/60 = 1/12, which matches the condition.
A piece of construction work takes 12 people 20 days. How many people are needed to finish the same work in 15 days?
The total labour is fixed at 12 × 20 = 240 person-days, so 240 ÷ 15 = 16 people. '9 people' comes from 12 × 15 ÷ 20, making people proportional to days, which contradicts itself: you cannot cut the days by cutting the workforce. '4 people' is the extra needed, 16 − 12, not the total required.
A job takes A 10 days alone and B 15 days alone. They take turns a day at a time, A on day one, B on day two, and so on. On which day is the job finished?
Setting the whole job at 30, A does 3 a day and B does 2, so 5 gets done every 2 days. Six such pairs, that is 12 days, come to exactly 5 × 6 = 30, so it ends on day 12. 'Day 6' is how long it would take with both working at once; taking turns takes nearly twice as long. 'Day 25' adds 10 and 15.
A tank fills in 20 minutes with pipe A alone and in 30 minutes with pipe B alone. Pipe C alone drains it in 15 minutes. Starting from empty with all three open, how many minutes does it take to fill?
Per minute the level changes by 1/20 + 1/30 − 1/15 = (3 + 2 − 4) ÷ 60 = 1/60, so 60 minutes. '12 minutes' ignores the drain, 1/20 + 1/30 = 1/12. '35 minutes' works with the raw times, 20 + 30 − 15, instead of fractions. Since the inflow exceeds the outflow, the tank does fill in the end, however long it takes.
A 480 L tank has one pipe delivering 20 L a minute and another delivering 12 L a minute. Using both at once, how many minutes does it take to fill from empty?
Together they deliver 20 + 12 = 32 L a minute, so 480 ÷ 32 = 15 minutes. '24 minutes' uses only the 20 L pipe, 480 ÷ 20, and '40 minutes' only the 12 L pipe, 480 ÷ 12. With both running it must come out shorter than the faster pipe's 24 minutes.
A 600 L tank was filled first by pipe A alone (25 L a minute) for 8 minutes, and pipe B (15 L a minute) was then added until it was full. How many minutes did it take from adding pipe B until the tank was full?
In 8 minutes 25 × 8 = 200 L goes in, leaving 400 L. The two pipes together deliver 25 + 15 = 40 L a minute, so 400 ÷ 40 = 10 minutes. '15 minutes' ignores the first 8 minutes, 600 ÷ 40. '16 minutes' fills the remaining 400 L with pipe A alone, 400 ÷ 25. '24 minutes' is the time to fill all 600 L with pipe A alone.
A job takes A and B together 8 days, B and C together 12 days, and A and C together 24 days. How many days does it take all three together?
Adding the daily rates of the three pairs gives 1/8 + 1/12 + 1/24 = 3/24 + 2/24 + 1/24 = 1/4, which is exactly twice the rate of the three of them. The three together do 1/4 ÷ 2 = 1/8 a day, so 8 days. '4 days' takes the 1/4 as the three-person rate and forgets to halve it. '44 days' adds 8, 12 and 24.
A job takes A 18 days alone and B 9 days alone. A worked alone for the first 6 days, after which B joined and the two finished it. How many days did it take from start to finish?
A's 6 days complete 6 × 1/18 = 1/3, leaving 2/3. Together they do 1/18 + 1/9 = 3/18 = 1/6 a day, so the rest takes 2/3 ÷ 1/6 = 4 days. The total is 6 + 4 = 10 days. '4 days' gives only the days worked together. '12 days' is 6 + 6, the case where B finishes the rest alone. '27 days' adds 18 and 9.
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