More Workers Means Fewer Days, Not More
Learn to spot direct from inverse variation, solve both by the unitary method, handle men-and-days and provisions problems, and work out how long two taps take together.
If 12 workers take 15 days, how long do 18 workers take?
10 days — fewer, not more. More workers share the same job, so the time falls, and multiplying by instead of is the mistake to avoid.
Deciding which way the numbers go is the whole skill here. This page covers everything in the ICSE Class 7 Mathematics chapter on the unitary method: telling direct from inverse variation, solving each by the unitary method, and time-and-work problems.
Deciding which way the numbers go is the whole skill here. This page covers everything in the ICSE Class 7 Mathematics chapter on the unitary method: telling direct from inverse variation, solving each by the unitary method, and time-and-work problems.
How do you tell direct variation from inverse variation?
Ask what happens to the second quantity when the first increases.
In direct variation, both change the same way — more of one means more of the other, and their ratio stays constant. More pens cost more money, more petrol takes you further, more workers produce more goods.
In inverse variation, they change opposite ways — more of one means less of the other, and their product stays constant. More workers take fewer days, a higher speed takes less time, more people make provisions last fewer days.
Test it on numbers. If 5 pens cost ₹75, then 10 pens cost ₹150 — doubled and doubled, so direct, and the ratio is constant. If 12 workers take 15 days, 24 workers take about 7.5 days — doubled and halved, so inverse, and the product is constant.
So the reliable check is arithmetic rather than instinct: if the ratio stays the same it is direct, and if the product stays the same it is inverse.
One pair catches almost everyone. Speed and time for a fixed distance are inversely related, while distance and time at a fixed speed are directly related — so the same three quantities behave both ways depending on which one is held fixed.
In direct variation, both change the same way — more of one means more of the other, and their ratio stays constant. More pens cost more money, more petrol takes you further, more workers produce more goods.
In inverse variation, they change opposite ways — more of one means less of the other, and their product stays constant. More workers take fewer days, a higher speed takes less time, more people make provisions last fewer days.
Test it on numbers. If 5 pens cost ₹75, then 10 pens cost ₹150 — doubled and doubled, so direct, and the ratio is constant. If 12 workers take 15 days, 24 workers take about 7.5 days — doubled and halved, so inverse, and the product is constant.
So the reliable check is arithmetic rather than instinct: if the ratio stays the same it is direct, and if the product stays the same it is inverse.
One pair catches almost everyone. Speed and time for a fixed distance are inversely related, while distance and time at a fixed speed are directly related — so the same three quantities behave both ways depending on which one is held fixed.
How does the unitary method solve a direct variation problem?
Find the value for one unit first, then multiply up to the number you need.
Worked example. If 5 pens cost ₹75, what do 8 pens cost?
Cost of 1 pen
Cost of 8 pens
Worked example. A car travels km on litres of petrol. How far on litres?
Distance on 1 litre km
Distance on 20 litres km
Worked example, running backwards. If m of cloth costs ₹294, how much cloth for ₹504?
Cost of 1 m , so cloth for ₹504 m.
Market shopping is unitary method in daily use — knowing the price per kilo lets you price any quantity at once.
A size check protects the answer. In direct variation, more units must cost more, so 8 pens at ₹120 against 5 at ₹75 is believable, while an answer under ₹75 would signal that the multiplication went the wrong way.
Worked example. If 5 pens cost ₹75, what do 8 pens cost?
Cost of 1 pen
Cost of 8 pens
Worked example. A car travels km on litres of petrol. How far on litres?
Distance on 1 litre km
Distance on 20 litres km
Worked example, running backwards. If m of cloth costs ₹294, how much cloth for ₹504?
Cost of 1 m , so cloth for ₹504 m.
Market shopping is unitary method in daily use — knowing the price per kilo lets you price any quantity at once.
A size check protects the answer. In direct variation, more units must cost more, so 8 pens at ₹120 against 5 at ₹75 is believable, while an answer under ₹75 would signal that the multiplication went the wrong way.
How do you solve inverse variation problems?
Find the total work as a product first — the constant — then divide by the new quantity.
Men and days. If 12 workers build a wall in 15 days, how long do 18 workers take?
Total work worker-days
Time for 18 workers days
The traditional phrasing is *1 worker takes days*, which is the same calculation.
Provisions. A hostel has food for 40 students for 30 days. If 10 students leave, how long will it last?
Total food student-days
For 30 students days
Speed, distance and time. A bus covers a route in 4 hours at km/h. How long at km/h?
Distance km
Time at 80 km/h hours
The direction of the change is the built-in check. Fewer students means the food lasts longer (40 days beats 30), and a higher speed means less time (3 hours beats 4). If your answer moves the other way, the multiplication and division have been swapped.
Men and days. If 12 workers build a wall in 15 days, how long do 18 workers take?
Total work worker-days
Time for 18 workers days
The traditional phrasing is *1 worker takes days*, which is the same calculation.
Provisions. A hostel has food for 40 students for 30 days. If 10 students leave, how long will it last?
Total food student-days
For 30 students days
Speed, distance and time. A bus covers a route in 4 hours at km/h. How long at km/h?
Distance km
Time at 80 km/h hours
The direction of the change is the built-in check. Fewer students means the food lasts longer (40 days beats 30), and a higher speed means less time (3 hours beats 4). If your answer moves the other way, the multiplication and division have been swapped.
How long do two people or two taps take working together?
Find the one-day work of each, add them, then turn the total upside down.
Worked example. A can finish a job in 10 days and B in 15 days. How long together?
A's one-day work , B's one-day work
Together in one day
So they finish the whole job in 6 days.
Taps. One tap fills a tank in 12 hours, another in 6 hours. Together in one hour they fill
so the tank fills in 4 hours.
A tap and an outlet. If an inlet fills a tank in 8 hours and a leak empties it in 24 hours, the leak subtracts:
so the tank fills in 12 hours — slower than 8, as it must be.
The habitual error is adding the days instead of the one-day work. Ten days and fifteen days do not give 25 days; working together must be faster than either alone, so 6 days is the only kind of answer that can be right.
Worked example. A can finish a job in 10 days and B in 15 days. How long together?
A's one-day work , B's one-day work
Together in one day
So they finish the whole job in 6 days.
Taps. One tap fills a tank in 12 hours, another in 6 hours. Together in one hour they fill
so the tank fills in 4 hours.
A tap and an outlet. If an inlet fills a tank in 8 hours and a leak empties it in 24 hours, the leak subtracts:
so the tank fills in 12 hours — slower than 8, as it must be.
The habitual error is adding the days instead of the one-day work. Ten days and fifteen days do not give 25 days; working together must be faster than either alone, so 6 days is the only kind of answer that can be right.
Exam tip
Exam tip: naming the variation before you calculate
Every question in this chapter is decided before the arithmetic, so state the type first.
Write one line naming it: this is inverse variation, since more workers means fewer days. That sentence often carries a mark and fixes the method in your own mind.
For direct variation, find the value of one unit and multiply. For inverse, find the product and divide. Showing the one-unit or the product step earns method marks.
For time-and-work, always use one-day work as a fraction and add the fractions — never add the days. Include a leak as a subtraction.
Finish with a size check: more units cost more, more workers take fewer days, working together is faster than either alone.
And carry the unit through your working — worker-days, student-days — so the final division is obviously the right one.
Write one line naming it: this is inverse variation, since more workers means fewer days. That sentence often carries a mark and fixes the method in your own mind.
For direct variation, find the value of one unit and multiply. For inverse, find the product and divide. Showing the one-unit or the product step earns method marks.
For time-and-work, always use one-day work as a fraction and add the fractions — never add the days. Include a leak as a subtraction.
Finish with a size check: more units cost more, more workers take fewer days, working together is faster than either alone.
And carry the unit through your working — worker-days, student-days — so the final division is obviously the right one.
Did you know
Why is working together always faster than the quicker person alone?
Because the help can only ever add to the work done, never take away from it.
A finishes of the job each day on his own. With B alongside, that becomes , which is strictly more. More work per day must mean fewer days.
So the combined time has a guaranteed range: it is always less than the faster person's time, and always more than half of it. For A at 10 days and B at 15, the answer had to fall between 5 and 10 days — and 6 does.
A finishes of the job each day on his own. With B alongside, that becomes , which is strictly more. More work per day must mean fewer days.
So the combined time has a guaranteed range: it is always less than the faster person's time, and always more than half of it. For A at 10 days and B at 15, the answer had to fall between 5 and 10 days — and 6 does.
Key takeaways
Unitary method and variation: quick revision
- Direct variation: both quantities move the same way and their ratio is constant. Inverse variation: they move opposite ways and their product is constant.
- Speed and time are inversely related for a fixed distance; distance and time are directly related at a fixed speed.
- Direct problems: find the value for one unit, then multiply — 5 pens at ₹75 gives ₹15 each and ₹120 for 8.
- Inverse problems: find the product and divide — worker-days, so 18 workers take 10 days.
- Provisions for 40 students for 30 days is 1200 student-days, so 30 students get 40 days.
- For work together, add the one-day work as fractions and invert the total: , so 6 days — and a leak is subtracted.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Speed and time are inversely related for a fixed distance; distance and time are directly related at a fixed speed.
- Direct problems: find the value for one unit, then multiply — 5 pens at ₹75 gives ₹15 each and ₹120 for 8.
- Inverse problems: find the product and divide — worker-days, so 18 workers take 10 days.
- Provisions for 40 students for 30 days is 1200 student-days, so 30 students get 40 days.
- For work together, add the one-day work as fractions and invert the total: , so 6 days — and a leak is subtracted.
You will remember all of this far better after answering five questions on it than after reading it twice.