14. CHAIN RULE
_IMPORTANT FACTS AND FORMULAE
1. Direct Proportion: Two quantities are said to be directly proportional, if on the
increase (or decrease) of the one, the other increases (or decreases) to the
same
Ex. 1. Cost is directly proportional to the
number of articles.
(More Articles,
More Cost)
Ex. 2. Work
done is directly proportional to the number of men working on it
(More Men, More
Work)
2. Indirect Proportion: Two quantities are said to be indirectly proportional,if on the
increase of the one, the other decreases to the same extent and vice-versa.
Ex. 1. The time taken by a
car in covering a certain distance is inversely proportional to the speed of
the car.
(More speed,
Less is the time taken to cover a distance)
Ex. 2. Time
taken to finish a work is inversely proportional to the num of persons working
at it.
(More persons, Less is the time taken to finish a job)
Remark: In solving questions by chain rule, we compare every item with the
term to be found out.
SOL VED EXAMPLES
Ex. 1. If 15 toys cost Rs, 234, what do 35 toys cost?
Sol. Let the required cost be Rs. x. Then,
More
toys, More cost (Direct
Proportion)
. 15 : 35 : : 234 : x ó (15 x x) = (35 x 234) ó x=(35
X 234)/15 =546
Hence, the cost of 35 toys is Rs.
546.
Ex. 2. If
36 men can do a piece of work in 25 hours, in how many hours will 15 men
do it ?
Sol. Let
the required number of hours be x. Then,
Less men, More hours (Indirect Proportion)
15 : 36 : : 25 : x ó(15 x
x) = (36 x 25) ó(36 x 25)/15 = 60
Hence, 15 men can do it in 60 hours.
Ex. 3. If the wages of 6 men for
15 days be Rs.2100, then find the wages of
for 12 days.
Sol. Let the required
wages be Rs. x.
More
men, More wages (Direct Proportion)
Less
days, Less wages (Direct
Proportion)
Men 6: 9 : :2100:x
Days 15:12
Therefore (6 x 15 x x)=(9 x 12 x
2100) ó x=(9 x 12 x 2100)/(6 x 15)=2520
Hence the required wages are Rs. 2520.
Ex. 4. If 20 men can build a wall 66 metres long in 6
days, what length of a similar can be built by 86 men in 8 days?
Sol. Let the required
length be x metres
More men, More length built (Direct Proportion)
Less days, Less length built (Direct Proportion)
Men 20: 35
Days 6: 3 : : 56
: x
Therefore (20 x 6 x x)=(35 x 3 x 56)óx=(35 x 3 x 56)/120=49
Hence, the required length is 49 m.
Ex. 5. If 15 men, working 9 hours
a day, can reap a field in 16 days, in how many
days will 18 men reap the field, working 8 hours a
day?
Sol. Let the
required number of days be x.
More men, Less days
(indirect proportion)
Less hours per day, More days (indirect proportion)
Men 18 : 15
Hours per day 8: 9 } : :16 : x
(18 x 8 x x)=(15 x 9 x 16)ó x=(44 x 15)144 = 15
Hence,
required number of days = 15.
Ex. 6. If 9 engines consume 24 metric tonnes of coal,
when each is working 8 hours
day, bow much coal will be
required for 8 engines, each running 13hours a day, it being given that 3
engines of former type consume as much as 4 engines of latter type?
Sol. Let 3 engines of former type consume 1 unit in 1 hour.
Then,
4 engines of latter type consume 1 unit in 1 hour.
Therefore 1 engine of former type
consumes(1/3) unit in 1 hour.
1 engine of latter type consumes(1/4) unit in 1 hour.
Let the required consumption of coal be x
units.
Less engines, Less coal consumed (direct proportion)
More working hours, More coal consumed (direct
proportion)
Less rate of consumption, Less coal consumed(direct
prportion)
Number of engines
9: 8
Working hours 8 : 13 } :: 24 : x
Rate of consumption (1/3):(1/4)
[
9 x 8 x (1/3) x x) = (8 x 13 x (1/4) x
24 ) ó 24x = 624 ó x = 26.
Hence, the required consumption of coal = 26 metric
tonnes.
Ex. 7. A contract is to be completsd in 46 days sad
117 men were said to work 8 hours a day.
After 33 days, (4/7) of the work is completed. How many additional men
may be employed so that the work may be completed in time, each man now working
9 hours a day?
Sol.
Remaining work = (1-(4/7) =(3/7)
Remaining period = (46 - 33)
days = 13days
Let the total men working at it be x.
Less
work, Less men (Direct
Proportion)
Less
days, More men (Indirect
Proportion)
More
Hours per Day, Less men (Indirect
Proportion)
Work (4/7): (3/7)
Days 13:33 } : : 117: x
Hrs/day 9 : 8
Therefore (4/7) x 13 x 9 x x =(3/7) x 33 x 8 x 117 or
x=(3 x 33 x 8 x 117)/(4 x 13 x 9)=198
Additional men to be employed = (198 - 117) =
81.
Ex. 8. A garrison of 3300 men had provisions for 32
days, when given at the rate of 860 gns per head. At the end of 7 days, a
reinforcement arrives and it was for that the provisions wi1l last 17 days
more, when given at the rate of 826 gms per head, What is the strength of the
reinforcement?
Sol. The problem becomes:
3300 men taking
850 gms per head have provisions for (32 - 7) or 25 days,
How many men taking 825 gms each have provisions for 17
days?
Less
ration per head, more men (Indirect
Proportion)
Less
days, More men (Indirect
Proportion)
Ration 825 : 850
Days 17: 25 } : : 3300 : x
(825 x 17 x x) = 850 x 25 x 3300 or x = (850 x 25 x
3300)/(825 x 17)=5000
Strength of reinforcement = (5500 - 3300) =
1700.
