An Emptying Pipe Is the Only One You Ever Subtract
Learn to recognise inverse variation by a constant product, solve speed-time and workers-days problems, combine rates for people working together, and handle filling and emptying pipes.
Why do you add two filling pipes but subtract an emptying one?
Because what you are adding is not time but work done per hour — and an emptying pipe does negative work.
A pipe filling a tank in hours does of the job each hour. One filling it in hours does each hour. Open both and the tank gains of its capacity per hour.
An outlet emptying it in hours removes per hour, so it enters the sum with a minus sign. With all three open the tank gains per hour, and fills in hours.
Adding times would have been meaningless. This page covers the second part of the ICSE Class 8 Mathematics chapter on variation.
A pipe filling a tank in hours does of the job each hour. One filling it in hours does each hour. Open both and the tank gains of its capacity per hour.
An outlet emptying it in hours removes per hour, so it enters the sum with a minus sign. With all three open the tank gains per hour, and fills in hours.
Adding times would have been meaningless. This page covers the second part of the ICSE Class 8 Mathematics chapter on variation.
Formula
What is inverse variation and how do you test for it?
Two quantities are in inverse variation when an increase in one produces a proportionate decrease in the other, so that their product remains constant.
Written and read * varies inversely as *, it means
Compare this with direct variation, where the ratio was constant. Here it is the product, and that single change is the whole difference between the two halves of the chapter.
Worked example 1 — testing a table.
- : , , ,
- : , , ,
Taking each product:
Every product is , so the quantities do vary inversely, with and .
Worked example 2 — a table that fails.
- : , ,
- : , ,
The last product breaks the pattern, so this is not an inverse variation — even though falls as rises, and even though the first two pairs agree. Two matching products are not enough; every pair must agree.
Everyday pairs that do vary inversely:
- Speed and time for a fixed distance
- Number of workers and days to finish a fixed job
- Number of people sharing a fixed quantity and the share each gets
- Length and breadth of a rectangle of fixed area
What the constant means. In each case is the total that does not change — the distance, the whole job, the quantity to be shared, the area. Naming that total is the quickest way into any problem, and it is what makes the arithmetic obvious.
The boundary case worth stating. The relationship has **no value at , since dividing by zero is undefined. Practically that is exactly right: at zero speed the journey takes no finite time at all, and zero workers** never finish the job. Direct variation passes neatly through the origin; inverse variation never reaches either axis.
Written and read * varies inversely as *, it means
Compare this with direct variation, where the ratio was constant. Here it is the product, and that single change is the whole difference between the two halves of the chapter.
Worked example 1 — testing a table.
- : , , ,
- : , , ,
Taking each product:
Every product is , so the quantities do vary inversely, with and .
Worked example 2 — a table that fails.
- : , ,
- : , ,
The last product breaks the pattern, so this is not an inverse variation — even though falls as rises, and even though the first two pairs agree. Two matching products are not enough; every pair must agree.
Everyday pairs that do vary inversely:
- Speed and time for a fixed distance
- Number of workers and days to finish a fixed job
- Number of people sharing a fixed quantity and the share each gets
- Length and breadth of a rectangle of fixed area
What the constant means. In each case is the total that does not change — the distance, the whole job, the quantity to be shared, the area. Naming that total is the quickest way into any problem, and it is what makes the arithmetic obvious.
The boundary case worth stating. The relationship has **no value at , since dividing by zero is undefined. Practically that is exactly right: at zero speed the journey takes no finite time at all, and zero workers** never finish the job. Direct variation passes neatly through the origin; inverse variation never reaches either axis.
How do you solve speed-time and workers-days problems?
Find the constant product first — it is the fixed total the question is really about — then divide.
Worked example 1 — speed and time. A car covers a certain distance in hours at . How long will the same journey take at ?
The constant is the distance:
So hours minutes. A check on the direction: the speed went up, so the time must come down — and .
Worked example 2 — workers and days. workers can build a wall in days. How long will workers take?
The constant is the total work, measured in worker-days:
Worked example 3 — the same job, asking for workers. How many workers would finish that wall in days?
The same constant answers every version of the question, which is why finding first is worth the line it takes.
**Worked example 4 — by proportion rather than by .** The inverse-variation proportion is
So for the wall with workers:
Notice this is not the direct-variation proportion . For inverse variation the fractions are inverted on one side, so the equation becomes a product — and using the direct form here would have given , so days, an answer that is larger than when more workers must clearly take less time.
The check that catches every reversed variation. Before calculating, decide whether the answer should be bigger or smaller than the value given. More workers means fewer days; greater speed means less time; more people sharing means a smaller share. An answer that moves the wrong way has used the wrong variation, and no amount of correct arithmetic afterwards will save it.
Worked example 1 — speed and time. A car covers a certain distance in hours at . How long will the same journey take at ?
The constant is the distance:
So hours minutes. A check on the direction: the speed went up, so the time must come down — and .
Worked example 2 — workers and days. workers can build a wall in days. How long will workers take?
The constant is the total work, measured in worker-days:
Worked example 3 — the same job, asking for workers. How many workers would finish that wall in days?
The same constant answers every version of the question, which is why finding first is worth the line it takes.
**Worked example 4 — by proportion rather than by .** The inverse-variation proportion is
So for the wall with workers:
Notice this is not the direct-variation proportion . For inverse variation the fractions are inverted on one side, so the equation becomes a product — and using the direct form here would have given , so days, an answer that is larger than when more workers must clearly take less time.
The check that catches every reversed variation. Before calculating, decide whether the answer should be bigger or smaller than the value given. More workers means fewer days; greater speed means less time; more people sharing means a smaller share. An answer that moves the wrong way has used the wrong variation, and no amount of correct arithmetic afterwards will save it.
How do you solve time-and-work problems?
Convert each person's time into a rate of work per day, add the rates, then invert.
If someone finishes a job in days, they complete of it each day. That fraction is their rate, and rates are what can be added.
Worked example 1 — two people together. A can do a piece of work in days and B in days. How long will they take working together?
Inverting the combined rate gives the time:
So days. A check on the size: working together must take less than the faster person alone, so the answer has to be under days — and is.
Why you cannot average the times. Averaging and gives days, which is longer than A takes alone — plainly absurd when B is helping. Times cannot be added or averaged; only rates can.
Worked example 2 — finding one person's time from the pair. A and B together finish a job in days, and A alone takes days. How long would B alone take?
Subtract A's rate from the combined rate:
Checking: , which is days together. Correct.
Worked example 3 — three people. A, B and C can do a job in , and days respectively. Working together?
Worked example 4 — part of a job. A can finish a job in days. How much does he complete in days?
So one third remains — and that remaining fraction is what the harder versions of these questions hand to a second worker.
The one habit that makes all of these routine. Write down every rate as a fraction before doing anything else. Once the problem is a list of fractions to add or subtract, it is ordinary fraction arithmetic, and the only remaining step is remembering to invert at the end.
If someone finishes a job in days, they complete of it each day. That fraction is their rate, and rates are what can be added.
Worked example 1 — two people together. A can do a piece of work in days and B in days. How long will they take working together?
Inverting the combined rate gives the time:
So days. A check on the size: working together must take less than the faster person alone, so the answer has to be under days — and is.
Why you cannot average the times. Averaging and gives days, which is longer than A takes alone — plainly absurd when B is helping. Times cannot be added or averaged; only rates can.
Worked example 2 — finding one person's time from the pair. A and B together finish a job in days, and A alone takes days. How long would B alone take?
Subtract A's rate from the combined rate:
Checking: , which is days together. Correct.
Worked example 3 — three people. A, B and C can do a job in , and days respectively. Working together?
Worked example 4 — part of a job. A can finish a job in days. How much does he complete in days?
So one third remains — and that remaining fraction is what the harder versions of these questions hand to a second worker.
The one habit that makes all of these routine. Write down every rate as a fraction before doing anything else. Once the problem is a list of fractions to add or subtract, it is ordinary fraction arithmetic, and the only remaining step is remembering to invert at the end.
How do you solve pipes and cisterns problems?
Exactly like time-and-work, with one addition: a pipe that empties the tank is subtracted.
Worked example 1 — two filling pipes. Pipe A can fill a tank in hours and pipe B in hours. How long will both together take?
That is hours and minutes, since of an hour is minutes. Converting the decimal part of an hour into minutes is expected in the answer.
Worked example 2 — with an outlet. Pipe A fills a tank in hours, pipe B fills it in hours, and pipe C empties it in hours. If all three are opened together, how long will the tank take to fill?
The outlet has slowed the filling from hours to hours, which is the direction it must go.
The sign rule, stated plainly. A filling pipe contributes a positive rate and an emptying pipe a negative one. Nothing else about the method changes — and forgetting the minus sign is the single mistake this question type exists to catch.
Worked example 3 — when the outlet wins. Pipe A fills a tank in hours and pipe C empties it in hours. What happens if both are opened?
The combined rate is negative, so the tank does not fill at all — it empties, at per hour. A full tank would take hours to drain.
A negative answer here is not an error but a meaningful result, and it is worth interpreting in words rather than reporting as a negative time.
Worked example 4 — a tank already part full. Pipes A and B together fill a tank in hours, as found above. If the tank is already full, how long to fill the rest?
The remaining work is of a tank at a rate of per hour:
That is hour minutes. Time equals work divided by rate — and since the full tank at that rate takes hours, two-thirds of it taking hours is exactly proportional, as it should be.
Worked example 1 — two filling pipes. Pipe A can fill a tank in hours and pipe B in hours. How long will both together take?
That is hours and minutes, since of an hour is minutes. Converting the decimal part of an hour into minutes is expected in the answer.
Worked example 2 — with an outlet. Pipe A fills a tank in hours, pipe B fills it in hours, and pipe C empties it in hours. If all three are opened together, how long will the tank take to fill?
The outlet has slowed the filling from hours to hours, which is the direction it must go.
The sign rule, stated plainly. A filling pipe contributes a positive rate and an emptying pipe a negative one. Nothing else about the method changes — and forgetting the minus sign is the single mistake this question type exists to catch.
Worked example 3 — when the outlet wins. Pipe A fills a tank in hours and pipe C empties it in hours. What happens if both are opened?
The combined rate is negative, so the tank does not fill at all — it empties, at per hour. A full tank would take hours to drain.
A negative answer here is not an error but a meaningful result, and it is worth interpreting in words rather than reporting as a negative time.
Worked example 4 — a tank already part full. Pipes A and B together fill a tank in hours, as found above. If the tank is already full, how long to fill the rest?
The remaining work is of a tank at a rate of per hour:
That is hour minutes. Time equals work divided by rate — and since the full tank at that rate takes hours, two-thirds of it taking hours is exactly proportional, as it should be.
Exam tip
Exam tip: add the rates, never the times
In every work or pipe problem, turn each time into a rate — a person finishing in days works at per day — then add the rates and invert at the end.
Never add or average the times. Averaging and days gives , which is longer than one worker takes alone and is plainly wrong.
An emptying pipe is subtracted. If the combined rate comes out negative, the tank empties rather than fills, and the answer should say so in words.
For inverse variation, use the product , not the direct-variation ratio. Test a table by checking every product, and one unequal product disproves it.
Predict the direction before calculating: more workers means fewer days, greater speed means less time, more sharers means a smaller share. An answer moving the wrong way means the wrong variation was used.
Name the constant and its unit — of distance, worker-days of work. The same constant answers every version of the question.
Convert a decimal part of an hour into minutes: hours is hours minutes.
And for a partly finished job, use .
Never add or average the times. Averaging and days gives , which is longer than one worker takes alone and is plainly wrong.
An emptying pipe is subtracted. If the combined rate comes out negative, the tank empties rather than fills, and the answer should say so in words.
For inverse variation, use the product , not the direct-variation ratio. Test a table by checking every product, and one unequal product disproves it.
Predict the direction before calculating: more workers means fewer days, greater speed means less time, more sharers means a smaller share. An answer moving the wrong way means the wrong variation was used.
Name the constant and its unit — of distance, worker-days of work. The same constant answers every version of the question.
Convert a decimal part of an hour into minutes: hours is hours minutes.
And for a partly finished job, use .
Did you know
Why can nine women not dig a well in one day?
Inverse variation says that if one worker takes nine days, then nine workers take one day — and by the same arithmetic, nine hundred workers would take about fourteen minutes.
The arithmetic is correct and the conclusion is nonsense, which tells you something important about the model rather than about the arithmetic.
Inverse variation assumes every worker is equally productive and that they do not interfere with one another. Both assumptions hold reasonably well for a few workers sharing a large job. Neither survives nine hundred people around one well — most of them cannot reach it, they obstruct each other, and the tools run out long before the space does.
So an examination question saying assuming all workers work at the same rate is not padding. It is the condition that makes the inverse-variation model apply at all, and every such question quietly relies on it.
It is worth knowing where a model stops working. The mathematics of this chapter is exact; what it describes is an idealisation, and the idealisation breaks down long before the formula does.
The arithmetic is correct and the conclusion is nonsense, which tells you something important about the model rather than about the arithmetic.
Inverse variation assumes every worker is equally productive and that they do not interfere with one another. Both assumptions hold reasonably well for a few workers sharing a large job. Neither survives nine hundred people around one well — most of them cannot reach it, they obstruct each other, and the tools run out long before the space does.
So an examination question saying assuming all workers work at the same rate is not padding. It is the condition that makes the inverse-variation model apply at all, and every such question quietly relies on it.
It is worth knowing where a model stops working. The mathematics of this chapter is exact; what it describes is an idealisation, and the idealisation breaks down long before the formula does.
Key takeaways
Inverse variation, work and pipes: quick revision
- Inverse variation means the product is constant: , so or . Compare direct variation, where the ratio is constant.
- Test a table by checking every product. with all give , so it varies inversely with . But with gives , , — not inverse variation.
- Examples: speed and time, workers and days, sharers and share, length and breadth at fixed area. The constant is the fixed total.
- has **no value at — zero speed never completes the journey.
- Speed and time**: , so at the time is hours.
- Workers and days: worker-days, so workers take days and days needs workers.
- The inverse proportion is , not .
- Time and work: a job done in days means a rate of per day. Add the rates and invert.
- A in days and B in gives , so together days. Never average the times.
- Together in days with A alone in : B's rate is , so B alone takes days.
- Three workers at , and days give per day, so days. And A working of his days completes .
- Pipes: filling pipes are added, an emptying pipe is subtracted. A in h and B in h give , so hours — h min.
- With C emptying in h: , so hours.
- A negative combined rate means the tank empties: , draining in hours.
- For a partly filled tank, : two-thirds at per hour takes hours.
Work a set of pipe problems including one with an outlet that wins — interpreting a negative rate in words is the step that shows you understood the method rather than copied it.
- Test a table by checking every product. with all give , so it varies inversely with . But with gives , , — not inverse variation.
- Examples: speed and time, workers and days, sharers and share, length and breadth at fixed area. The constant is the fixed total.
- has **no value at — zero speed never completes the journey.
- Speed and time**: , so at the time is hours.
- Workers and days: worker-days, so workers take days and days needs workers.
- The inverse proportion is , not .
- Time and work: a job done in days means a rate of per day. Add the rates and invert.
- A in days and B in gives , so together days. Never average the times.
- Together in days with A alone in : B's rate is , so B alone takes days.
- Three workers at , and days give per day, so days. And A working of his days completes .
- Pipes: filling pipes are added, an emptying pipe is subtracted. A in h and B in h give , so hours — h min.
- With C emptying in h: , so hours.
- A negative combined rate means the tank empties: , draining in hours.
- For a partly filled tank, : two-thirds at per hour takes hours.
Work a set of pipe problems including one with an outlet that wins — interpreting a negative rate in words is the step that shows you understood the method rather than copied it.