Find the Value of One Thing and Every Sum Gets Easier
Learn the unitary method for direct and inverse variation, the speed-distance-time relations, and how to convert between km/h and m/s without guessing.
What is the unitary method?
The unitary method solves a comparison problem in two steps: first find the value of one unit, then multiply to find the value of however many units you need. It turns almost every "if this much costs that much" question into two easy lines.
This page covers everything in the ICSE Class 6 Mathematics chapter on the unitary method and speed: direct and inverse variation, the three speed relations, converting between km/h and m/s, and average speed.
This page covers everything in the ICSE Class 6 Mathematics chapter on the unitary method and speed: direct and inverse variation, the three speed relations, converting between km/h and m/s, and average speed.
How do you solve a direct variation problem?
In direct variation, both quantities grow together — more items cost more, more petrol takes you further. Divide to reach one unit, then multiply.
If 8 notebooks cost Rs 240, find the cost of 13.
A second example: if a scooter covers 135 km on 3 litres of petrol, then one litre covers km, so 7 litres cover km.
Check the direction of your answer before writing it down. You asked about more notebooks than 8, so the answer must be more than Rs 240. If it came out smaller, you multiplied where you should have divided.
If 8 notebooks cost Rs 240, find the cost of 13.
A second example: if a scooter covers 135 km on 3 litres of petrol, then one litre covers km, so 7 litres cover km.
Check the direction of your answer before writing it down. You asked about more notebooks than 8, so the answer must be more than Rs 240. If it came out smaller, you multiplied where you should have divided.
How is inverse variation different?
In inverse variation, one quantity rises as the other falls — more workers finish a job in fewer days, a faster speed needs less time. The unitary step changes: you multiply to find the value for one, then divide.
If 6 workers build a wall in 20 days, how long will 8 workers take?
First find the work for one worker: worker-days. Then
More workers, fewer days — and 15 is indeed less than 20, so the direction is right.
In real life, filling a water tank behaves the same way: two taps fill it in half the time one tap needs.
The boundary case is worth stating: not every pair of quantities varies inversely. The number of workers and the wages paid in total rise together, so that is direct variation even though workers appear in both problems. Decide the type from the situation, never from the words alone.
If 6 workers build a wall in 20 days, how long will 8 workers take?
First find the work for one worker: worker-days. Then
More workers, fewer days — and 15 is indeed less than 20, so the direction is right.
In real life, filling a water tank behaves the same way: two taps fill it in half the time one tap needs.
The boundary case is worth stating: not every pair of quantities varies inversely. The number of workers and the wages paid in total rise together, so that is direct variation even though workers appear in both problems. Decide the type from the situation, never from the words alone.
Formula
What are the speed, distance and time relations?
Speed is the distance covered in one unit of time — itself a unitary idea. The three forms are:
Here distance is measured in kilometres or metres, time in hours or seconds, and speed in km/h or m/s.
Worked example: a bus covers 180 km in 4 hours.
Worked the other way: a cyclist rides at 12 km/h for 3 hours, so
And for time: to cover 150 km at 50 km/h,
The units must agree. Distance in km with time in hours gives km/h; distance in metres with time in seconds gives m/s. Mixing them — km with seconds — produces a number that means nothing.
Here distance is measured in kilometres or metres, time in hours or seconds, and speed in km/h or m/s.
Worked example: a bus covers 180 km in 4 hours.
Worked the other way: a cyclist rides at 12 km/h for 3 hours, so
And for time: to cover 150 km at 50 km/h,
The units must agree. Distance in km with time in hours gives km/h; distance in metres with time in seconds gives m/s. Mixing them — km with seconds — produces a number that means nothing.
Formula
How do you convert km/h to m/s and find average speed?
Since 1 km is 1000 m and 1 hour is 3600 s, one km/h equals m/s. That gives the two conversions:
Convert 72 km/h into m/s:
Convert 15 m/s into km/h:
Average speed is the total distance divided by the total time — never the average of the two speeds:
If a car covers 60 km in 2 hours and then 90 km in 3 hours, the total is 150 km in 5 hours, so the average speed is km/h.
Convert 72 km/h into m/s:
Convert 15 m/s into km/h:
Average speed is the total distance divided by the total time — never the average of the two speeds:
If a car covers 60 km in 2 hours and then 90 km in 3 hours, the total is 150 km in 5 hours, so the average speed is km/h.
Exam tip
The mistake most students make with average speed
Faced with a journey at 40 km/h and then at 60 km/h, most students answer 50 km/h by averaging the two numbers. That is wrong unless the times happen to be equal.
Average speed is always total distance divided by total time. Take 120 km at 40 km/h, then 120 km at 60 km/h. The times are h and h, so 240 km in 5 hours:
The method is fixed: work out each time separately, add the distances, add the times, then divide once at the end.
Average speed is always total distance divided by total time. Take 120 km at 40 km/h, then 120 km at 60 km/h. The times are h and h, so 240 km in 5 hours:
The method is fixed: work out each time separately, add the distances, add the times, then divide once at the end.
Did you know
Why is the km/h to m/s factor exactly 5 by 18?
It comes straight from the units. One kilometre per hour is 1000 metres spread over 3600 seconds, and cancels to .
That is why a speed limit of 36 km/h is exactly 10 m/s — a neat pair worth remembering, since it lets you sanity-check any conversion in your head.
That is why a speed limit of 36 km/h is exactly 10 m/s — a neat pair worth remembering, since it lets you sanity-check any conversion in your head.
Key takeaways
Unitary method and speed in 30 seconds
- The unitary method finds the value of one unit first, then scales up to the number required.
- In direct variation, divide then multiply; in inverse variation, multiply then divide — and check the answer moves in the sensible direction.
- speed = distance / time, distance = speed × time, time = distance / speed, with units kept consistent throughout.
- Multiply by 5/18 to go from km/h to m/s, and by 18/5 to go back.
- Average speed is total distance divided by total time, never the average of two speeds.
You will remember all of this far better after answering five questions on it than after reading it twice.
- In direct variation, divide then multiply; in inverse variation, multiply then divide — and check the answer moves in the sensible direction.
- speed = distance / time, distance = speed × time, time = distance / speed, with units kept consistent throughout.
- Multiply by 5/18 to go from km/h to m/s, and by 18/5 to go back.
- Average speed is total distance divided by total time, never the average of two speeds.
You will remember all of this far better after answering five questions on it than after reading it twice.