Chapter 3 Rule Of Three
Introduction
There are some problems in arithmetic that may be solved by more than one method. For example, the problems which can be solved by the unitary method can also be solved by applying the idea of proportion. So in this chapter, our aim is to solve those problems which we have solved earlier by using the unitary method. This new technique of solving arithmetical problems on the basis of the idea of proportion is the ‘Rule of Three’.
The Rule of Three
You know that when three terms of a proportion are known we can find the fourth term. Since the fourth term of a proportion can be identified when any three terms of the said proportion are given, this method is known as the ‘Rule of Three’. In fact, this method is nothing but finding a fourth proportional to the three given quantities.
When 5, 7, or more odd numbers of quantities in any proportion are given and we are to find from them the 6th, 8th, etc. quantity then this method is known as the multiple rules of three.
Read And Learn More WBBSE Solutions For Class 8 Maths
How to solve a problem by the Rule of Three
1. The numerical value of the required quantity is taken as x, and it is placed in the fourth position of the proportion. Let us take a problem. If 5 bottles of medicines cost 1000, then how much would 8 bottles cost? In this problem, the cost of 8 bottles is the required
quantity to be found out. Hence the numerical value of the required quantity (i.e. the cost of 8 bottles) is to be taken as x and it is to be placed in the fourth position of the proportion.
2. The given similar quantity is placed in the third position of the proportion. In the above problem, a similar quantity as x is the given cost of 5 bottles. Hence, 1000 is to be placed in the third position of the proportion.
3. We have to determine from the subject- a matter of the problem whether the required quantity will be less than or greater than the given similar quantity. These two cases are better understood by the concept of variation.
1. Direct variation: In a direct variation, the increase in one quantity (the number) causes an increase in the other quantity (the cost), and a decrease in one quantity causes a decrease in the other quantity.
Our problem with the cost of bottles of medicines is of this type since the more the number of bottles, the more the cost, and the less the number of bottles, the less the cost. The required quantity of cost of 8 bottles is more than the cost of 5 bottles.
2. Inverse variation In an inverse variation the increase in one quantity causes a decrease in the other quantity and a decrease in one quantity causes an increase in the other quantity. Let us take another example. If 8 men can do a piece of work in 12 days then how long will 16 men take to do the same work?
Let the required time be x days. In this problem, since the number of men increases, therefore, the number of days would decrease. Hence, the value of x will be less than 12 days.
4. When x is smaller than the quantity placed in the third position then between the other two quantities the greater one is placed in the first position and the smaller one is placed in the second position of the proportion.
5. When x is greater than the quantity placed in the third position then between the other two quantities the smaller one is placed in the first position and the greater one is placed in the second position of the proportion.
Some problems with the Rule of Three
Example 1
12 men can do a piece of work in 15 days. How long will 10 men take to do the same work?
Solution :
Given:
12 men can do a piece of work in 15 days.
Let, the required time be x days. Then the data can be arranged as,
Number of men Number of pens
12 8
10 12
Here, since the number of men decreases, therefore the number of days will increase. Hence, the value of x will be greater than 15
∴ 10/12 = 15/x
or, 10x = 15 x 13
or, x = 15 x 12 / 10
= 18
The required time is 18 days
18 days will 10 men take to do the same work
Example 2
A man bought 15 kg of rice for 390. How much he would require if he would have bought 17 kg of rice?
Solution:
Given:
A man bought 15 kg of rice for 390.
Let, the required price be ₹ x. Then the data can be arranged as
Quantity of rice (in kg) Expenditure (in ₹)
15 360
17 x
In this case, the quantity of rice has increased, therefore expenditure will also increase. Hence, the value of x will be greater than 390.
∴ 15/17 = 390/x
or, x = 390 x 17 / 17
= 442
The required expenditure is ₹ 442
Example 3
The price of 8 pens is 120. Find the price of 12 such pens.
Solution :
Given:
The price of 8 pens is 120.
Let, the required expenditure be ₹ x. Then the data can be arranged as,
Number of pens Price(in ₹)
8 120
12 x
Here, since the number of pens increases, therefore the price will also increase.
Hence, the value of x will be greater than 1120
∴ 8/12 = 120/x
or, 8x = 12 x 120
or, x = 12 x 120 / 8
= 180
The required price is ₹ 180.
Example 4
A man will make 4 shirts of the same size with 20 meters of cloth. How many meters of cloth he will have to purchase for making 12 such shirts?
Solution:
Given:
A Man Will Make 4 Shirts Of The Same Size With 20 Meters Of Cloth.
Let, he will have to purchase x meters of cloth. Then the data can be arranged as,
No. of shirts Cloth (in meters)
4 20
12 X
Since the number of shirts has increased, therefore the quantity of cloth will also increase. Hence, the value of x will be greater than 20.
∴ 4/12 = 20/x
or, x = 20 x 12 / 4
= 60
He will have to purchase 60 meters of clothes.
Example 5
If the price of 7 meters of cloth is 126, then find how long cloth can be obtained by ₹ 81.
Solution:
Given:
If the price of 7 meters of cloth is 126.
Let, the required length of the cloth be x meters. Then the data can be arranged as,
Price (in ₹) Length of cloth (in meters)
126 7
81 x
Here, since the price decreases, therefore the length of the cloth will also decrease.
Hence, the value of x will be less than 7.
∴ 126/81 = 7/x
or, x x 126 = 81 x 7
or, x = 81 x 7 / 126
= 4.5
4.5 meters long cloth can be obtained.
Example 6
It took 15 days for 30 laborers to dig a pond. In how many days 25 labourers could have completed the said work?
Solution:
Given:
It took 15 days for 30 laborers to dig a pond
Let, the required time be x days. Then the data can be arranged as,
No. of laborers No. of days
30 15
25 x
Here, since the no. of laborers has decreased therefore no. of days will increase. Hence, the value of x will be greater than 15.
∴ 25/30 = 15/x
or, x = 30 x 15 / 25
= 18
The required time is 18 days.
Example 7
12 men can do a piece of work in 30 days. How many more men will be required to do it in 20 days?
Solution:
Given:
12 men can do a piece of work in 30 days.
Let, x men will do the work in 20 days.
Number of days Number of men
30 12
20 x
Here, since the number of days decreases, therefore the number of men will increase.
Hence, the value of x will be greater than 12.
∴ 20/30 = 12/x
or, 20x = 30 x 12
or, x = 30 x 12 / 20
= 18
Since 12 men were already there, si (18-12) or, 6 more men will be required.
6 more men will be required.
Example 8
A man reached a certain place in 5 hours by driving the car at a speed of 40 km/hr. How long it would have taken to reach the same place if he would drive the car at 50 km/hr?
Solution:
Let, the required time be x hours. Then the data can be arranged as,
Speed of the car (km/hr) Time (hour)
40 5
50 x
Since the speed of the car has increased, therefore, less time will be required.
Hence, the value of x will be less than 5.
∴ 50/40 = 5/x
or, x = 5 x 40 / 50
= 4
The Required time is 4 hours.
Example 9
2 men or 3 boys can finish a work in 48 days. In how many days will 4 men and 6 boys finish the work?
Solution:
Work of 2 men = Work of 3 boys
∴ Work of 1 men = work of 3/2 boys
Work of 4 men = work of 3/2 x 4 boys
= work of 6 boys
∴ Work of 4 men and 6 boys (6+6) boys = Work of 12 boys.
Now, the problem is, if 3 boys can finish a work in 48 days, then in how many days will 12 boys finish the work?
Let, the required number of days be x.
Number of boys Number of days
3 48
12 x
Here, since the number of boys increases, therefore the number of days will decrease.
Hence, the value of x will be less than 48.
∴ 12/3 = 48/x
or, 12 x x = 48 x 3
or, x = 48 x 3 /12
= 12
The required time is 12 days.
Example 10
There was a stock of food for 9 days to cater for the needs of 4000 people at a shelter camp. After 3 days 1000 people left the camp for another place. In how many days the rest of the people will consume the remaining food?
Solution:
After 3 days there was food for 6 days and the number of people reduced to 3000. Let, the required no. of days be x. Then the data can be arranged as,
No. of people No. of days 6
4000 6
3000 x
Since the number of people has reduced therefore the number of days will increase.
Hence, the value of x will be greater than 6.
∴ 3000/4000 = 6/x
or, x = 6 x 4000 / 3000
= 8
The required no .of days = is 8 days.
Example 11
In a camp of 4000 men, there was a stock of food for 190 days. After 30 days 800 men left the camp. How many days will the rest of the food last?
Solution :
After 30 days, the number of days left = (190-30)
= 160
and present number of men = (4000-800)
= 3200.
Now the problem is if there is a stock of food for 4000 men for 160 days, then how many days will it last for 3200 men.
Let the number of days be x. Then,
Number of men Number of days
4000 160
3200 x
Here, since the number of men decreases, therefore the number of days will increase.
Hence, the value of x will be greater than 160.
∴ 3200/4000 = 160/x
or, x x 3200 = 160 x 4000
or, x = 160 x 4000/3200
=200
The rest of the food will last for 200
Example 12
42 laborers of a farm can cultivate the entire land of the farm in 24 days. But suddenly 6 laborers became ill. How many days it will take to cultivate the entire land of the farm by the rest of the laborers?
Solution:
Due to the illness of 6 laborers the number of laborers became 42 – 6 = 36. Let, the required time be x days. Then the data can be arranged as,
No. of laborers No. of days
42 24
36 x
Since the number of laborers has decreased, therefore the number of days will increase. Hence, the value of x will be greater than 24.
∴ 36/42 = 24/x
or, x = 42 x 24 / 36
= 28
The required time is 28 days
Example 13
15 men can earn ₹ 1200 in 30 days. How much will 75 men earn in 5 days?
Solution:
Given:
15 men can earn ₹ 1200 in 30 days
Let the required money be ₹ x. Then the problem is
Number of men Number of days Earned money
15 30 ₹ 1200
75 5 ₹ x
Since the earned money increases with the increase in the number of men and decreases with the decrease in the number of days.
∴ \(\left.\begin{array}{l}
15: 75 \\
30: 5
\end{array}\right\}:: 1200: x\)
or, x = 75 x 5 x 1200 / 15 x 30
= 1000
They will earn ₹ 1000.
Example 14
It takes 27 days to make 1000 spare parts by 16 numbers of machines. How many days will it take to make the same number of spare parts if additional two machines are installed?
Solution:
Given:
27 days to make 1000 spare parts by 16 numbers of machines.
Since two machines are installed the number of machines becomes 18.
Let, the required time be x days. Then the data can be arranged as,
No. of machines Time (in days)
16 27
18 x
Since the number of machines has increased therefore less time will be required.
Hence the value of x will be less than 27.
∴ 18/16 = 27/x
or, x = 27 x 16 / 18
= 24
The required time is 24 days.
Example 15
A contractor undertook a 12 km long road construction job scheduled to be completed in 350 days. After employing 45 men for 200 days, he found that only 4½ km of road was completed. How many additional men must be engaged to finish the work in time?
Solution:
Given:
A contractor undertook a 12 km long road construction job scheduled to be completed in 350 days
After employing 45 men for 200 days, he found that only 4½ km of road was completed.
After 200 days, the number of days left = (350-200) = 150.
The length of the road remained unfinished = (12 – 9/2)km
= 15/2 km.
Now let us calculate, if 45 men can finish 9/2 km of road in 200 days then how many men will finish 15/2 km of road in 150
Number of days Length of road Number of men
200 9/2km 45
150 15/2km x
Since, the number of men increases with the increase in the length of the road, the decrease in the number of days and it increases in the length of the road,
\(\left.\begin{array}{c}150: 200 \\
\frac{9}{2}: \frac{15}{2}
\end{array}\right\}:: 45: x\)
or, x = 200 x 445 x 15/2 / 150 x 9/2
= 200 x 45 x 15 / 150 x 9
= 100
∴ Additional men required = (100-45)
= 55
55 more men are required.
Example 16
A pump of 5 H.P. can lift 36000 liters of water in 8 hours. How long it will take to lift 63000 liters of water by a 7 H.P. pump?
Solution:
Given:
A pump of 5 H.P. can lift 36000 liters of water in 8 hours.
Let, the required time be x hours. Then the data can be arranged as,
H.P. of pump Volume of water Time
5 36000 liters 8 hours
7 63000 litres x hours
Since, if the H.P. of the pump increases, then the time decreases, and if the volume of water increases the time increases, therefore,
\(\left.\begin{array}{c}7: 5 \\
36000: 63000
\end{array}\right\}:: 8: x\)
or, x = 5 x 63000 x 8 / 7 x 36000
= 10
The required time is 10 hours.
Example 17
There are two motors of 5 H.P. and 3 H.P. in a factory. The 5 H.P. motor requires 20 units of electricity in 8 hours. How many units will be required if 3 H.P. motor works for 10 hours?
Solution :
Given:
There are two motors of 5 H.P. and 3 H.P. in a factory.
The 5 H.P. motor requires 20 units of electricity in 8 hours.
Let, x units of electricity will be required. Then the data can be arranged as,
H.P. of motor Time Electricity required
5 8 hours 20 units
3 10 hours x units
Since if the H.P. of the motor decreases less electricity is required and if time is increased more electricity is required, therefore,
\(\left.\begin{array}{l}5: 3 \\
8: 10
\end{array}\right\}:: 20: x\)
or, x = 20 x 3 x 10 / 5 x 8
= 15
15 units will be required.
Example 18
In a loom, 14 weavers can weave 210 sarees in 12 days. How many extra weavers have to be employed to weave 300 sarees in 10 days?
Solution:
Given:
In a loom, 14 weavers can weave 210 sarees in 12 days
Let x number of weavers can weave 300 sarees in 10 days. Then the data can be
No. of days No. of sarees No. of weavers
12 210 14
10 300 x
Since the quantity of work done increases if the number of people increases and the number of sarees increases if the number of weavers increases, therefore,
\(\left.\begin{array}{l}10: 12 \\
210: 300
\end{array}\right\}:: 14: x\)
or, x = 14 x 12 x 300 / 10 x 210
= 24
∴ Extra weavers have to be employed
= (24-14)
= 10
10 extra weavers have to be employed.
Example 19
A company has got the work of unloading goods from a ship in 10 days. 280 people have been employed for this purpose. After 3 days it is seen that 1/4th of the work has been completed. How many extra people are to be engaged to complete the work in time?
Solution:
Given:
A company has got the work of unloading goods from a ship in 10 days.
280 people have been employed for this purpose. After 3 days it is seen that 1/4th of the work has been completed.
Work left = (1-1/4) = 3/4th part and number of days left = (10-3) days = 7 days.
Let, x people are to be engaged to complete the work in time. Then the data can be arranged as,
Quantity of work No. of days No. of people
1/4 3 280
3/4 7 x
Since the number of days decreases if the number of weavers increases and the number of days increases if the number of people decreases,
\(\left.\begin{array}{c}\frac{1}{4}: \frac{3}{4} \\
7: 3
\end{array}\right\}:: 280: x\)
or, x = 3/4 x 3 x 280 / 1/4x 7
= 360
∴ Extra people will be required
= (360 – 280)
= 80
80 Extra people will be required.
Example 20
A power-loom is 2 1/4 times more powerful than a hand-loom. 12 hand-looms weave 1080 meters in length of cloth in 18 days. How many power looms will be required to weave 2700 meters length of cloth in 15 days?
Solution:
Given:
A power-loom is 2 1/4 times more powerful than a hand-loom. 12 hand-looms weave 1080 meters in length of cloth in 18 days.
Let, the number of handlooms is x.
Then the data can be arranged as,
Length of cloth No.of days No.of hand-looms
1080 meters 18 12
2700 meters 15 x
Since, the length of cloth increases if the number of looms increases and the number of days decreases if the number of looms increases, therefore,
\(\begin{aligned}& 1080: 2700 \\
& 15: 18 \quad\}:: 12: x
\end{aligned}\)
or, x = 2700 x 18 x 12 / 1080 x 15
= 36
9/4 hand-looms = 1 power-loom
1 hand-loom = 4/9 power-loom
36 hand-looms = 4/9 x 36 power-looms
= 16 power-looms
Required no of power looms is 16.
Example 21
2400 bighas of land of a cooperative society can be cultivated by 25 farmers in 36 days. It was seen that half of the land of the society could be cultivated in 30 days after purchasing a tractor. Find the power of the tractor equivalent to the number of farmers.
Solution:
Given:
2400 bighas of land of a cooperative society can be cultivated by 25 farmers in 36 days
It was seen that half of the land of the society could be cultivated in 30 days after purchasing a tractor
Let, the power of the tractor is equivalent to x number of farmers. The data can be arranged as,
Quality of land No.of days No.of farmers
2400 bighas 36 25
1200 bighas 30 x
Since, if the quantity of land increases the number of farmers increases and if the number of days decreases, then the number of farmers increases, therefore,
\(\left.\begin{array}{l}2400: 1200 \\
30: 36
\end{array}\right\}:: 25: x,\)
or, x = 1200 x 36 x 25 / 2400 x 30
= 15
1 tractor is equivalent to 15 farmers.
Example 22
A ship takes 25 days to sail from Kolkata to Cochin. The ship started with 36 sailors for each of which 850 gm of food was allotted per day. But the ship rescued 15 sailors from another sinking ship after 13 days of journey and the ship reached Cochin in 10 days increasing the speed. What will be the quantity of food each sailor required to reach Cochin safely and the entire storage of food would be consumed by this time?
Solution:
Given:
If the 15 sailors are rescued then the number of sailors becomes (36 +15)= 51.
If 13 days go out of 25 days the number of days remaining = (25-13) = 12.
Now, the food which lasts 12 days for 36
sailors will have to last 10 days for 51 sailors.
The data can be arranged as,
No. of sailors No. of days Food allotted per head per day
36 12 850 gm
51 10 x gm(say)
If the number of sailors increases the food allotted per head per day decreases and if the number of days decreases the food allotted per head per day increases, therefore,
\(\left.\begin{array}{l}51: 36 \\
10: 12
\end{array}\right\}:: 850: x\)
or, x = 850 × 36 × 12 / 51 x 10
= 720
The required quantity of food is 720 gm.
Example 23
36 people of a certain village can construct 120 meters of road in 8 days by working 6 hours daily. Another 6 people were involved in this work and the duration of work each day was increased by 2 hours. Find the length of the road to be constructed now in 9 days.
Solution:
Given:
36 people of a certain village can construct 120 meters of road in 8 days by working 6 hours daily.
Another 6 people were involved in this work and the duration of work each day was increased by 2 hours.
Let x meters of the road may be constructed.
The data may be arranged as,
No.of people Daily time No.of days Length of road
36 6 hr 8 120 meters
42 8hr 9 x meters
If the number of people increases, daily time increases, and if the number of days increases, then the length of the road increases, therefore,
\(\left.\begin{array}{rl}36 & : 42 \\
6 & : 8 \\
8 & : 9
\end{array}\right\}:: 120: x\)
or, x = 42 x 8 x 9 x 120 / 36 x 6 x 8
= 120
The length of the road is 210 meters.
Example 24
250 people can excavate a pond of size 50 meters long, 35 meters wide, and 5.2 meters deep in 18 days by working 10 hours daily. How many days it will take for 300 people to excavate a pond of size 65 meters long, 40 meters wide, and 5.6 meters deep by working 8 hours daily?
Solution:
Given:
250 people can excavate a pond of size 50 meters long, 35 meters wide, and 5.2 meters deep in 18 days by working 10 hours daily.
Volume of the first pond = 50 x 35 x 5.2 cu m
= 9100 cu m
Volume of the second pond = 65 × 40 × 5.6 cu m
= 14560 cu m
Let, the number of days is = x. Then the data can be arranged as,
No.of people Volume of pond Daily working time No.of days
250 9100 cu m 20 hours 18
300 14560 cu m 8 hours x
250 Since, the number of people increases if the number of days decreases and the volume of the pond increases if the number of days increases and if daily working time increases the number of days decreases, therefore,
\(\left.\begin{array}{rl}300 & : 250 \\
9100 & : 14560 \\
8 & : 10
\end{array}\right\}:: 18: x\)
or, x = 18 x 250 x 14560 x 10 / 300 x 9100 x 8
= 30
The required no. of days is 30 days.
Example 25
24 men take 6 days more to finish a piece of work than 33 men take to finish it. How long will 44 men take to finish the work?
Solution:
Given:
24 men take 6 days more to finish a piece of work than 33 men take to finish it.
Let, 33 men take x days to finish the work.
Then 24 men take (x+6) days to finish it.
No. of men No. of days
33 x
24 x+6
Since, the number of days increases with the decrease in the number of men, therefore,
24/33 = x/x+6
or, 33x = 24x + 144
or, 33x – 24x = 144
or, 9x = 144
or, x = 144/9
= 16
∴ 33 men do the work in 16 days
Let, 44 men do the work in y days
No. of men No. of days
33 16
44 y
∴ 44/33 = 16/y
or, 4/3 = 16/y
or, 4y= 48
or, y = 12
They will finish the work in 12 days.
Example 26
In a camp, there is a stock of food for 20 days. After one week 100 more people took shelter in the camp and the food lasted for 11 days. In the beginning how many people were there?
Solution:
Given:
In a camp, there is a stock of food for 20 days.
After one week 100 more people took shelter in the camp and the food lasted for 11 days.
Let, in the beginning, x people were there.
After 1 week (207)
or, 13 days remain.
No.of people No.of days
x 13
(100 + x) 11
Since the number of days decreases as the number of men increases, therefore,
100 + x / x = 13/11
or, 13x = 11x + 1100
or, 13x – 11x = 1100
or, 2x = 1100
or, x = 550
In the beginning, there were 550 people.
Example 27
24 men take 12 days to prepare a pond. How many more men will be required to prepare a pond in 8 days?
Solution:
Given:
24 men take 12 days to prepare a pond.
Let, x men prepare the pond in 8 days.
No.of days No.of men
12 24
8 x
Here, since, the number of days has decreased, therefore, more men will be required. Therefore, the value of x will be greater than 24.
∴ 8/12 = 24/x
or, 8x = 24 x 12
or, x = 24 x 12 / 8
= 36
Since there were already 24 men, therefore (3624) or 12 more men will be required.
12 more men will be required.
Example 28
In a book written by hand, there are 105 pages and 25 lines on each page, and 8 words in each line. If that book is so printed that there will be 30 lines on each page and 10 words on each line, then what will be the number of pages of the book?
Solution:
Given:
In a book written by hand, there are 105 pages 25 lines on each page, and 8 words in each line.
If that book is so printed that there will be 30 lines on each page and 10 words on each line.
Let the number of pages of the book be x.
No.of lines on each page No.of words in each line No.of pages
25 8 105
30 10 x
Since, as the number of lines on each page will increase, the number of pages will decrease and as the number of words in each line will increase, the number of pages will decrease.
Therefore,
\(\left.\begin{array}{l}30: 25 \\
10: 8
\end{array}\right\}:: 105: x\)
or, \(x=\frac{25 \times 8 \times 105}{30 \times 10}=70\)
The number of pages of the book will be 70.
