Chain Rule
Name: _____________________
Date: _____________________
Instructions: Answer all questions. Write your answers clearly in the space provided.
If 12 carpenters, working 6 hours a day, can make 460 chairs in 24 days, how many chairs will 18 carpenters make in 36 days, each working 8 hours a day ?
If 5 spiders can catch five files in five minutes, how many files can hundred spiders catch in 100 minutes ?
2 persons working 2 hours a day assemble 2 machine in 2 days. The number of machines assemble by 6 persons working 6 hours a day in 6 day is ?
A wall of 100 meters can be built by 7 men or 10 women in 10 days. How many days will 14 men and 20 women take to build a wall of 600 metres ?
21 binders can bind 1400 books in 15 days. How many binders will required to bind 800 books in 20 days ?
20 men complete one-third of a piece of work in 20 days. How many more men should be employed to finish the rest of the work in 25 more days ?
A rope makes 70 rounds of the circumference of a cylinder whose radius of the base is 14 cm. How many times can it go round a cylinder with radius 20 cm ?
A contract is to be complete in 46 days and 117 men were set to work, each working 8 hours a day. After 33 days, $$frac{4}{7}{ ext{ }}$$ 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 ?
12 persons can do a piece of work in 4 days. How many persons are required to complete 8 times the work in half the time ?
On a certain map of India the actual distance of 1450 km between two cities Delhi and Kolkata is shown as 5 cm. What scale is used to draw the map ?
A flagstaff 17.5 m high caste a shadow of length 40.25 m. The height of the building, which casts a shadow of length 28.75 m under similar conditions will be ?
A man completes $$frac{5}{8}$$ of a job in 10 days. At this rate, how many more days will it take him to finish the job ?
56 men can complete a piece of work in 24 days. In how many days can 42 men complete the same piece of work ?
30 men can do a piece of work in 16 days. How many men would be required to do the same work in 20 days ?
4 mat-weavers can weave 4 mats in 4 days. At the same rate, how many mats would be woven by 8 mat-weavers in 8 days ?
If 7 maids with 7 mops cleaned 7 floors in 7 hours, how long would it take 3 maids to mop 3 floors with 3 mops ?
Four gardeners with four grass mowers mow 400 sq. m of ground in 4 hours. How long would it take for eight gardeners with eight grass mowers to mow 800 sq. m of ground ?
Running at the same constant rate, 6 identical machines can produce a total of 180 bottles per hour. How many bottles could 15 such machines produce on 30 minutes ?
If 6 persons working 8 hours a day earn Rs. 8400 per week, then 9 persons working 6 hours a day will earn per week ?
If 5 workers can collect 60 kg wheat in 3 days, how many kilograms of wheat will 8 workers collect in 5 days ?
50 people consume 350 kg of rice in 30 days. In how many days will 35 people consume 50 kg of rice ?
Working 8 hours a day, 12 men can do a work in 30 days. Working 4 hours a day, 18 men can do work in ?
5 persons can prepare an admission list in 8 days working 7 hours a day. If 2 persons join them so as to complete the work in 4 days, they need to work per day for ?
4 persons can prepare an admission list in 8 days working 7 hours a day. If 3 persons join them so as to complete the work in 4 days, they need to work per day for ?
3 pumps , working 8 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work to empty the tank in 1 day ?
If 5 men or 7 women can earn Rs. 5250 per day, how much would 7 men and 13 women earn per day ?
3 men or 6 women can do a piece of work in 20 days. In how days will 12 men and 8 women do the same work ?
49 pumps can empty a reservoir in $$6frac{1}{2}$$ days, working 8 hours a day. If 196 pumps are used for 5 hours each day, then the same work will be complete in ?
30 labourers working 7 hours a day can finish a piece of work in 18 days. If the labourers work 6 hours a day, then the number of labourers to finish the same piece of work in 30 days, will be ?
3 pumps, working 8 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work to empty the tank in 1 day?
If the cost of x metres of wire is d rupees, then what is the cost of y metres of wire at the same rate?
Running at the same constant rate, 6 identical machines can produce a total of 270 bottles per minute. At this rate, how many bottles could 10 such machines produce in 4 minutes?
A fort had provision of food for 150 men for 45 days. After 10 days, 25 men left the fort. The number of days for which the remaining food will last, is:
39 persons can repair a road in 12 days, working 5 hours a day. In how many days will 30 persons, working 6 hours a day, complete the work?
A man completes $$frac{5}{8}$$ of a job in 10 days. At this rate, how many more days will it takes him to finish the job?
If a quarter kg of potato costs 60 paise, how many paise will 200 gm cost?
In a dairy farm, 40 cows eat 40 bags of husk in 40 days. In how many days one cow will eat one bag of husk?
A wheel that has 6 cogs is meshed with a larger wheel of 14 cogs. When the smaller wheel has made 21 revolutions, then the number of revolutions mad by the larger wheel is:
If 7 spiders make 7 webs in 7 days, then 1 spider will make 1 web in how many days?
A flagstaff 17.5 m high casts a shadow of length 40.25 m. The height of the building, which casts a shadow of length 28.75 m under similar conditions will be:
In a camp, there is a meal for 120 men or 200 children. If 150 children have taken the meal, how many men will be catered to with remaining meal?
An industrial loom weaves 0.128 metres of cloth every second. Approximately, how many seconds will it take for the loom to weave 25 metres of cloth?
36 men can complete a piece of work in 18 days. In how many days will 27 men complete the same work?
4 mat-weavers can weave 4 mats in 4 days. At the same rate, how many mats would be woven by 8 mat-weavers in 8 days?
The price of 357 mangoes is Rs. 1517.25. What will be the approximate price of 49 dozens of such mangoes ?
An industrial loom weaves 0.128 metres of cloth every second. Approximately, how many seconds will it take for the loom to weave 25 metres of cloth ?
A snapshot $${ ext{1}}frac{7}{8} imes 2frac{1}{2}$$ xa0 is to be enlarged so that the longer dimension is 4. What will be the dimension of the shorter side ?
A canteen requires 651 bananas for a week. Totally, how many bananas will it require for the months of April, May and June ?
If $$frac{4}{9}$$ th of a bucket is filled in 1 minute, the rest of it will be filled in ?
If 5 engines consume 6 metric tonnes of coal when each is running 9 hours a day, how many metric tonnes of coal will be needed for 8 engines, each running 10 hours a day, it begin given that 3 engines of the former type consume as much as 4 engines of the latter type ?
If 9 men working $${ ext{7}}frac{1}{2}$$ hours a day can finish a piece of work in 20 days, then how many days will be taken by 12 men, working 6 hours a day to finish the work ? (It is being given that 2 men of latter type work as much as 3 men of the former type.)
15 men take 21 days of 8 hours each to do a piece of work. How many days of 6 hours each would 21 women take, 3 women do as much work as 2 men ?
In a barrack of soldiers there was stock of food for 190 days for 4000 soldiers. After 30 days 800 soldiers left the barrack. For how many days shall the left over food last for the remaining soldiers ?
A garrison of 500 men had provisions for 27 days. After 3 days a reinforcement of 300 men arrived. For how many more days will the remaining food last now ?
A garrison had provision for a certain number of days. After 10 days, $$frac{1}{5}$$ of the men desert and it is found that the provisions will now last just as long as before. How long was that ?
A fort has provisions for 50 days. If after 10 days they are strengthened by 500 men and the food lasts for 35 days longer, the number of men originally in the fort were ?
A team of workers was employed by a contractor who undertook to finish 360 pieces of an article in a certain number of days. Making four more pieces per day than was planned, they could complete the job a day ahead of schedule. How many days did they take to complete the job ?
The work done by a women in 8 hours is equal to the work done by a man in 6 hours and by a boy in 12 hours. If working 6 hours per day 9 men can complete a work in 6 days, then in how many days can 12 men, 12 women and 12 boys together finish the same work, working 8 hours per day ?
12 men and 18 boys, working $$7frac{1}{2}$$ hours a day, can do a piece of work in 60 days. If a man works equal to 2 boys, then how many boys will be required to help 21 men to do twice the work in 50 days, working 9 hours a day ?
If 2 m, 60 cm cloths is required for one shirt, then the cloth required for 7 shirts is ?
The cost of 4 dozen papers is Rs. 24. The cost of 1 score of papers (in rupees) is ?
The cost of 8 fans and 14 ovens is Rs. 36520. What is the cost of 12 fans and 21 ovens ?
The cost of 5 kgs of apples is Rs. 450. The cost of 12 dozen mangoes is Rs. 4320 and the cost of 4 kgs of oranges is Rs. 240. What is the total cost of 8 kg of apples, 8 dozen of mangoes and 8 kg of oranges ?
The cost of 21 pencils and 9 clippers is Rs. 819. The cost price of 7 pencils and 3 clippers is = ?
Large, medium and small ships are used to bring water. 4 large ships carry as much water as 7 small ships, 3 medium ships carry the same amount of water as 2 large ships and 1 small ship. 15 large, 7 medium and 14 small ships, each made 36 journeys and brought a certain quantity of water. In how many journeys would 12 large, 14 medium and 21 small ships bring the same quantity ?
If 17 labourers can dig a ditch 26 m long in 18 days, working 8 hours a day
how many more labourers should be engaged to dig a similar ditch 39 m long in 6 days, each labourer working 9 hours a day ?
Answer Key
,,,,{ ext{Carpenters 12}}:18 hfill \
{ ext{Hours per day 6}}:8 hfill \
,,,,,,,,,,,,,,,,,{ ext{Days 24}}:{ ext{36}} hfill \
end{gathered}
ight}::460:x] $$eqalign{
& herefore { ext{ }}12 imes 6 imes 24 imes x = 18 imes 8 imes 36 imes 460 cr
& Leftrightarrow x = frac{{left( {18 imes 8 imes 36 imes 460}
ight)}}{{left( {12 imes 6 imes 24}
ight)}} cr
& Leftrightarrow x = 1380 cr} $$ ∴ Required number of chairs = 1380
,,,{ ext{Spiders 5}}:100 hfill \
{ ext{Minutes 5}}:100 hfill \
end{gathered}
ight}::5:x] $$eqalign{
& herefore { ext{ }}5 imes 5 imes x = 100 imes 100 imes 5 cr
& Leftrightarrow x = frac{{left( {100 imes 100 imes 5}
ight)}}{{left( {5 imes 5}
ight)}} cr
& Leftrightarrow x = 2000 cr} $$
,,,,,,,,,,,,,,,,,,,{ ext{Persons 2}}:6 hfill \
,,,, { ext{Working hours 2}}:6 hfill \
,,,,,,,,,,,,,,,,,,,,,,,,,{ ext{Days 2}}:{ ext{6}} hfill \
end{gathered}
ight}::2:x] $$eqalign{
& herefore { ext{ }}2 imes 2 imes 2 imes x = 6 imes 6 imes 6 imes 2 cr
& Leftrightarrow x = frac{{left( {6 imes 6 imes 6 imes 2}
ight)}}{{left( {2 imes 2 imes 2}
ight)}} cr
& Leftrightarrow x = 54 cr} $$
{ ext{Length 100}}:600 hfill \
,,{ ext{Women 40}}:10 hfill \
end{gathered}
ight}::10:x] $$eqalign{
& herefore { ext{ }}100 imes 40 imes x = 600 imes 10 imes 10 cr
& Leftrightarrow x = frac{{left( {600 imes 10 imes 10}
ight)}}{{left( {100 imes 40}
ight)}} cr
& Leftrightarrow x = 15 cr} $$
{ ext{Books 1400}}:800 hfill \
,,,,,,,,,{ ext{Days 20}}:15 hfill \
end{gathered}
ight}::21:x] $$eqalign{
& herefore { ext{ }}1400 imes 20 imes x = 800 imes 15 imes 21 cr
& Leftrightarrow x = frac{{left( {800 imes 15 imes 21}
ight)}}{{left( {1400 imes 20}
ight)}} cr
& Leftrightarrow x = 9 cr} $$
ight) = frac{2}{3}$$ More work, More men (Direct proportion) More days, Less men (Indirect proportion) [left. x08egin{gathered}
,{ ext{Work }}frac{1}{3}:frac{2}{3} hfill \
{ ext{Days 25}}:20 hfill \
end{gathered}
ight}::20:x] $$eqalign{
& herefore { ext{ }}left( {frac{1}{3} imes 25 imes x}
ight) = left( {frac{2}{3} imes 20 imes 20}
ight) cr
& Leftrightarrow x = frac{{800}}{{25}} cr
& Leftrightarrow x = 32 cr} $$ ∴ More men to be employed = (32 - 20) = 12
& herefore { ext{ }}20:14::70:x cr
& Leftrightarrow left( {20 imes x}
ight) = left( {14 imes 70}
ight) cr
& Leftrightarrow x = frac{{left( {14 imes 70}
ight)}}{{20}} cr
& Leftrightarrow x = 49 cr} $$
ight){ ext{ = }}frac{3}{7}$$ Remaining period = (46 - 33) = 13 days Let the total men working at it be x Less work, Less men (Direct proportion) Less days, More men (Indirect proportion) More hour/day, Less men (Indirect proportion) [left. x08egin{gathered}
,,,,,{ ext{Work }}frac{4}{7}:frac{3}{7} hfill \
,,,,,,,{ ext{Men 13}}:33 hfill \
{ ext{Hour/day }}9:8 hfill \
end{gathered}
ight}::117:x] $$eqalign{
& herefore { ext{ }}frac{4}{7} imes 13 imes 9 imes x = frac{3}{7} imes 33 imes 8 imes 117 cr
& Leftrightarrow x = frac{{left( {3 imes 33 imes 8 imes 117}
ight)}}{{left( {4 imes 13 imes 9}
ight)}} cr
& Leftrightarrow x = 198 cr} $$ ∴ Addition men to be employed = (198 - 117) = 81
,{ ext{Days 2}}:4 hfill \
{ ext{Work 1}}:8 hfill \
end{gathered}
ight}::12:x] $$eqalign{
& herefore ,2 imes 1 imes x = 4 imes 8 imes 12 cr
& Leftrightarrow x = frac{{left( {4 imes 8 imes 12}
ight)}}{{left( 2
ight)}} cr
& Leftrightarrow x = 192 cr} $$
ight)$$xa0 km = 290 km Hence, the scale is 1 cm : 290 km i.e., $$1:29 imes {10^6}{ ext{ }}$$ $$left[ {x08ecause 290{ ext{km}} = left( {29 imes {{10}^6}}
ight){ ext{cm}}}
ight]$$
& herefore 40.25:28.75::17.5:x cr
& Leftrightarrow 40.25 imes x = 28.75 imes 17.5 cr
& Leftrightarrow x = frac{{left( {28.75 imes 17.5}
ight)}}{{40.25}} cr
& Leftrightarrow x = 12.5 cr} $$
ight)$$ xa0 $$ = frac{3}{8}$$ Less work, Less days ( Direct proportion) Let the required number of days be x Then, $$eqalign{
& Leftrightarrow frac{5}{8}:frac{3}{8}::10:x cr
& Leftrightarrow frac{5}{8} imes x = frac{3}{8} imes 10 cr
& Leftrightarrow x = left( {frac{3}{8} imes 10 imes frac{8}{5}}
ight) cr
& Leftrightarrow x = 6 cr} $$
& herefore 42:56::24:x cr
& Leftrightarrow 42 imes x = 56 imes 24 cr
& Leftrightarrow x = left( {frac{{56 imes 24}}{{42}}}
ight) cr
& Leftrightarrow x = 32 cr} $$
& herefore { ext{ }}20:16::30:x cr
& Leftrightarrow 20x = 16 imes 30 cr
& Leftrightarrow x = left( {frac{{16 imes 30}}{{20}}}
ight) cr
& Leftrightarrow x = 24 cr} $$
{ ext{Weavers }}4:8 hfill \
,,,,,,,,{ ext{Days }}4:8 hfill \
end{gathered}
ight}::4:x] $$eqalign{
& herefore 4 imes 4 imes x = 8 imes 8 imes 4 cr
& Leftrightarrow x = frac{{left( {8 imes 8 imes 4}
ight)}}{{left( {4 imes 4}
ight)}} cr
& Leftrightarrow x = 16 cr} $$
{ ext{Maids & Mops 3}}:7 hfill \
,,,,,,,,,,,,,,,,,,,,,{ ext{Floors 7}}:3 hfill \
end{gathered}
ight}::7:x] $$eqalign{
& herefore { ext{ }}3 imes 7 imes x = 7 imes 3 imes 7 cr
& Leftrightarrow x = frac{{left( {7 imes 3 imes 7}
ight)}}{{left( {3 imes 7}
ight)}} cr
& Leftrightarrow x = 7 cr} $$
{ ext{Gardeners & grass mowers 8}}:4 hfill \
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,{ ext{Area 400}}:800 hfill \
end{gathered}
ight}::4:x] $$eqalign{
& herefore { ext{ }}8 imes 400 imes x = 4 imes 800 imes 4 cr
& Leftrightarrow x = frac{{left( {4 imes 800 imes 4}
ight)}}{{left( {8 imes 400}
ight)}} cr
& Leftrightarrow x = 4 cr} $$
{ ext{Machines 6}}:15 hfill \
,,,,,,{ ext{Times 60}}:30 hfill \
end{gathered}
ight}::180:x] $$eqalign{
& herefore { ext{ }}6 imes 60 imes x = 15 imes 30 imes 180 cr
& Leftrightarrow x = frac{{left( {15 imes 30 imes 180}
ight)}}{{left( {6 imes 60}
ight)}} cr
& Leftrightarrow x = 225 cr} $$
,,,,,,,,,,,,,,,,{ ext{Persons 6}}:9 hfill \
{ ext{Working hours 8}}:6 hfill \
end{gathered}
ight}::8400:x] $$eqalign{
& herefore { ext{ }}6 imes 8 imes x = 9 imes 6 imes 8400 cr
& Leftrightarrow x = frac{{left( {9 imes 6 imes 8400}
ight)}}{{left( {6 imes 8}
ight)}} cr
& Leftrightarrow x = 9450 cr} $$
{ ext{Workers 5}}:8 hfill \
,,,,,,,,{ ext{Days 3}}:5 hfill \
end{gathered}
ight}::60:x] $$eqalign{
& herefore { ext{ }}5 imes 3 imes x = 8 imes 5 imes 60 cr
& Leftrightarrow x = frac{{left( {8 imes 5 imes 60}
ight)}}{{left( {5 imes 3}
ight)}} cr
& Leftrightarrow x = 160 cr} $$
,,,,,,,{ ext{People 35}}:50 hfill \
{ ext{Quantity 350}}:50 hfill \
end{gathered}
ight}::30:x] $$eqalign{
& herefore { ext{ }}35 imes 350 imes x = 50 imes 50 imes 30 cr
& Leftrightarrow x = frac{{left( {50 imes 50 imes 30}
ight)}}{{left( {35 imes 350}
ight)}} cr
& Leftrightarrow x = frac{{300}}{{49}} cr
& Leftrightarrow x = 6frac{6}{{49}} cr} $$
{ ext{Working hours 4}}:8 hfill \
,,,,,,,,,,,,,,,,,,,,{ ext{Men 18}}:12 hfill \
end{gathered}
ight}::30:x] $$eqalign{
& herefore { ext{ }}4 imes 18 imes x = 8 imes 12 imes 30 cr
& Leftrightarrow x = frac{{left( {8 imes 12 imes 30}
ight)}}{{left( {4 imes 18}
ight)}} cr
& Leftrightarrow x = 40 cr} $$
{ ext{Persons 7}}:5 hfill \
{ ext{Quantity 4}}:8 hfill \
end{gathered}
ight}::7:x] $$eqalign{
& herefore { ext{ }}7 imes 4 imes x = 5 imes 8 imes 7 cr
& Leftrightarrow x = frac{{left( {5 imes 8 imes 7}
ight)}}{{left( {7 imes 4}
ight)}} cr
& Leftrightarrow x = 10 cr} $$
{ ext{Persons 7}}:4 hfill \
{ ext{Quantity 4}}:8 hfill \
end{gathered}
ight}::7:x] $$eqalign{
& herefore { ext{ }}7 imes 4 imes x = 4 imes 8 imes 7 cr
& Leftrightarrow x = frac{{left( {4 imes 8 imes 7}
ight)}}{{left( {7 imes 4}
ight)}} cr
& Leftrightarrow x = 8 cr} $$
{ ext{Pumps 4}}:3 hfill \
,,,,,,{ ext{Days 1}}:2 hfill \
end{gathered}
ight}::8:x] $$eqalign{
& herefore { ext{ }}4 imes 1 imes x = 3 imes 2 imes 8 cr
& Leftrightarrow x = frac{{left( {3 imes 2 imes 8}
ight)}}{{left( 4
ight)}} cr
& Leftrightarrow x = 12 cr} $$
ight) $$ xa0 women = $$frac{49}{5}$$ women ∴ (7 men and 13 women) $$eqalign{
& = left( {frac{{49}}{5} + 13}
ight){ ext{women}} cr
& = frac{{114}}{5}{ ext{ women}} cr} $$ Now, More women, More earning (Direct proportion) $$eqalign{
& herefore { ext{ }}7 : frac{{114}}{5}::{ ext{ 5}}250 : x cr
& Leftrightarrow 7x = left( {frac{{114}}{5} imes 5250}
ight) cr
& Leftrightarrow 7x = 119700 cr
& Leftrightarrow x = 17100 cr} $$
& herefore { ext{ }}32:6::20:x cr
& Leftrightarrow x = left( {frac{{6 imes 20}}{{32}}}
ight) cr
& Leftrightarrow x = frac{{15}}{4} cr
& Leftrightarrow x = 3frac{3}{4} cr} $$
,,,,,,,,,,,,,,,,,,,,,,{ ext{Pumps 196}}:49 hfill \
{ ext{Working hour/day 5}}:8 hfill \
end{gathered}
ight}::frac{{13}}{2}:x] $$eqalign{
& herefore { ext{ }}196 imes 5 imes x = 49 imes 8 imes frac{{13}}{2} cr
& Leftrightarrow x = left( {49 imes 8 imes frac{{13}}{2} imes frac{1}{{196 imes 5}}}
ight) cr
& Leftrightarrow x = frac{{13}}{5} cr
& Leftrightarrow x = 2frac{3}{5} cr} $$
{ ext{Working hours/day 6}}:7 hfill \
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,{ ext{Days 30}}:18 hfill \
end{gathered}
ight}::30:x] $$eqalign{
& herefore 6 imes 30 imes x = 7 imes 18 imes 30 cr
& Leftrightarrow 6x = 126 cr
& Leftrightarrow x = 21 cr} $$
{ ext{Pumps 4}}:3 hfill \
{ ext{Days}},,,,,,,,{ ext{1}}:2 hfill \
end{gathered}
ight}::8:x] $$eqalign{
& herefore 4 imes 1 imes x = 3 imes 2 imes 8 cr
& Rightarrow x = frac{{ {3 imes 2 imes 8} }}{{ 4 }} cr
& Rightarrow x = 12 cr} $$
& { ext{Cost of }}x{ ext{ metres}} cr
& = { ext{Rs}}{ ext{. }}d cr
& { ext{Coast of 1 metre}} cr
& = { ext{Rs}}{ ext{.}}left( {frac{d}{x}}
ight) cr
& { ext{Cost of }}y{ ext{ metres}} cr
& = { ext{Rs}}{ ext{.}}left( {frac{d}{x} imes y}
ight) cr
& = { ext{Rs}}{ ext{.}} {frac{{yd}}{x}} cr} $$
{ ext{Machines}},,,,,,,,,,,,,,,,{ ext{6}}:10 hfill \
{ ext{Time(in min}}{ ext{.)}},,{ ext{1}}:4 hfill \
end{gathered}
ight}::270:x] $$eqalign{
& herefore 6 imes 1 imes x = 10 imes 4 imes 270 cr
& Rightarrow x = frac{{ {10 imes 4 imes 270} }}{{ 6 }} cr
& Rightarrow x = 1800 cr} $$
& herefore 125:150::35:x cr
& Rightarrow 125 imes x = 150 imes 35 cr
& Rightarrow x = frac{{150 imes 35}}{{125}} cr
& Rightarrow x = 42 cr} $$
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,{ ext{Persons }},30:39 hfill \
{ ext{Working hour/day }},{ ext{6}}:{ ext{5}} hfill \
end{gathered}
ight}::12:x] $$eqalign{
& herefore 30 imes 6 imes x = 39 imes 5 imes 12 cr
& Rightarrow x = frac{{ {39 imes 5 imes 12} }}{{ {30 imes 6} }} cr
& Rightarrow x = 13 cr} $$
& { ext{Work}},{ ext{done}} = frac{5}{8} cr
& { ext{Balance}},{ ext{work}} = {1 - frac{5}{8}} = frac{3}{8} cr
& { ext{Let}},{ ext{the}},{ ext{required}},{ ext{number}},{ ext{of}},{ ext{days}},{ ext{be}},x cr
& { ext{Then}}, cr
&frac{5}{8}:frac{3}{8} :: 10:x cr
& Rightarrow frac{5}{8} imes x = frac{3}{8} imes 10 cr
& Rightarrow x = {frac{3}{8} imes 10 imes frac{8}{5}} cr
& Rightarrow x = 6 cr} $$
& { ext{Let}},{ ext{the}},{ ext{required}},{ ext{weight}},{ ext{be}},x,{ ext{kg}}. cr
& { ext{Less}},{ ext{weight,}},{ ext{less}},{ ext{cost}},left( {{ ext{Direct}},{ ext{Proportion}}}
ight) cr
& herefore 250:200::60:x cr
& Rightarrow 250 imes x = {200 imes 60} cr
& Rightarrow x = frac{{ {200 imes 60} }}{{250}} cr
& Rightarrow x = 48 cr} $$
{ ext{Cows}},,,,,1:40 hfill \
{ ext{Bags}},,,,,40:1 hfill \
end{gathered}
ight}::40:x] $$eqalign{
& herefore 1 imes 40 imes x = 40 imes 1 imes 40 cr
& Rightarrow x = 40 cr} $$
& herefore 14:6::21:x cr
& Rightarrow 14 imes x = 6 imes 21 cr
& Rightarrow x = frac{{6 imes 21}}{{14}} cr
& Rightarrow x = 9 cr} $$
{ ext{Spiders}},,,,,1:7 hfill \
,,,{ ext{Webs}},,,,,7:1 hfill \
end{gathered}
ight}::7:x] $$eqalign{
& herefore 1 imes 7 imes x = 7 imes 1 imes 7 cr
& Rightarrow x = 7 cr} $$
& herefore 40.25:28.75::17.5:x cr
& Rightarrow 40.25 imes x = 28.75 imes 17.5 cr
& Rightarrow x = frac{{28.75 imes 17.5}}{{40.25}} cr
& Rightarrow x = 12.5 cr} $$
& = left( {frac{{120}}{{200}} imes 50}
ight){ ext{men}} cr
& = { ext{30 men}} cr} $$
& herefore 0.128:25::1:x cr
& Rightarrow 0.128x = 25 imes 1 cr
& x = frac{{25}}{{0.128}} = frac{{25 imes 1000}}{{128}} cr
& Rightarrow x = 195.31 cr} $$ ∴ Required time = 195 sec (approximately)
& herefore 27:36::18:x cr
& Rightarrow 27 imes x = 36 imes 18 cr
& Rightarrow x = frac{{36 imes 18}}{{27}} cr
& Rightarrow x = 24 cr} $$
{ ext{Wavers}},,,,4:8 hfill \
,,,,,,{ ext{Days}},,,,4:8 hfill \
end{gathered}
ight}::4:x] $$eqalign{
& herefore 4 imes 4 imes x = 8 imes 8 imes 4 cr
& Rightarrow x = frac{{ {8 imes 8 imes 4} }}{{4 imes 4}} cr
& Rightarrow x = 16 cr} $$
& herefore 357:left( {49 imes 12}
ight)::1517.25:x cr
& Leftrightarrow 357x = left( {49 imes 12 imes 1517.25}
ight) cr
& Leftrightarrow x = frac{{left( {49 imes 12 imes 1517.25}
ight)}}{{357}} cr
& Leftrightarrow x = 2499 cr} $$ Hence, the approximate price is Rs. 2500
& herefore 0.128:25::1:x cr
& Leftrightarrow x = frac{{25}}{{0.128}} cr
& Leftrightarrow x = frac{{25 imes 1000}}{{128}} cr
& Leftrightarrow x = 195.31 cr} $$` ∴ Required time =195 seconds (approx)
& herefore 2frac{1}{2}:4::1frac{7}{8}:x cr
& Leftrightarrow frac{5}{2}x = 4 imes frac{{15}}{8} cr
& Leftrightarrow x = left( {frac{{15}}{2} imes frac{2}{5}}
ight) cr
& Leftrightarrow x = 3 cr} $$
& herefore 7:91::651:x cr
& Rightarrow 7x = 91 imes 651 cr
& Rightarrow x = left( {frac{{91 imes 651}}{7}}
ight) cr
& Rightarrow x = 8463 cr} $$
& = left( {1 - frac{4}{9}}
ight) cr
& = frac{5}{9}{ ext{ }} cr} $$ Let the required time be x minutes More volume to be filled , More time taken (Direct proportion) $$eqalign{
& herefore { ext{ }}frac{4}{9}:frac{5}{9}::1:x cr
& Rightarrow frac{4}{9}x = frac{5}{9} cr
& Rightarrow x = left( {frac{5}{9} imes frac{9}{4}}
ight) cr
& Rightarrow x = frac{5}{4}{ ext{ }} cr} $$
,,,,,,,,,,,,,,{ ext{Engines 5}}:8 hfill \
{ ext{Hours per day 9}}:10 hfill \
,,,,,,,,,,,,,,,,,,,,{ ext{Rate }}frac{1}{3}:frac{1}{4} hfill \
end{gathered}
ight}::6:x] $$eqalign{
& herefore ,,left( {5 imes 9 imes frac{1}{3} imes x}
ight) = left( {8 imes 10 imes frac{1}{4} imes 6}
ight) cr
& Leftrightarrow 15x = 120 cr
& Leftrightarrow x = 8 cr} $$
ight)$$ = 18 men of former type More men, Less days (Indirect proportion) Less working hours, More days (Indirect proportion) [left. x08egin{gathered}
,,,,,,,,,,,,,,,,{ ext{Men 18}}:9 hfill \
{ ext{Working hrs 6}}:frac{{15}}{2} hfill \
end{gathered}
ight}::20:x] $$eqalign{
& herefore ,18 imes 6 imes x = 9 imes frac{{15}}{2} imes 20 cr
& Leftrightarrow 108x = 1350 cr
& Leftrightarrow x = frac{{25}}{2} cr
& Leftrightarrow x = 12frac{1}{2} cr} $$
,,,,,,,,,,,,,,,,{ ext{Men 14}}:15 hfill \
{ ext{Hours per day 6}}:8 hfill \
end{gathered}
ight}::21:x] $$eqalign{
& herefore ,left( {14 imes 6 imes x}
ight) = left( {15 imes 8 imes 21}
ight) cr
& Leftrightarrow x = frac{{left( {15 imes 8 imes 21}
ight)}}{{left( {14 imes 6}
ight)}} cr
& Leftrightarrow x = 30 cr} $$ ∴ Required number of days = 30
& herefore ,3200:4000::160:x cr
& Leftrightarrow 3200x = 4000 imes 160 cr
& Leftrightarrow x = frac{{left( {4000 imes 160}
ight)}}{{3200}} cr
& Leftrightarrow x = 200 cr} $$
& herefore ,800:500::24:x cr
& Leftrightarrow left( {800 imes x}
ight) = left( {500 imes 24}
ight) cr
& Leftrightarrow x = frac{{left( {500 imes 24}
ight)}}{{800}} cr
& Leftrightarrow x = 15 cr} $$
ight)$$ xa0 men had food for y days $$eqalign{
& herefore ,xleft( {y - 10}
ight) = frac{{4x}}{5} imes y cr
& Leftrightarrow 5xy - 50x = 4xy cr
& Leftrightarrow xy - 50x = 0 cr
& Leftrightarrow xleft( {y - 50}
ight) = 0 cr
& Leftrightarrow y - 50 = 0 cr
& Leftrightarrow y = 50 cr} $$
& herefore ,left( {x + 500}
ight):x::40:35 cr
& Leftrightarrow 35 imes left( {x + 500}
ight) = 40x cr
& Leftrightarrow 5x = 35 imes 500 cr
& Leftrightarrow x = left( {frac{{35 imes 500}}{5}}
ight) cr
& Leftrightarrow x = 3500 cr} $$
$$frac{{360}}{x}$$ More number of pieces per day, Less days (Indirect proportion) $$eqalign{
& herefore ,left( {frac{{360}}{x} + 4}
ight):frac{{360}}{x}::x:left( {x - 1}
ight) cr
& Leftrightarrow left( {frac{{360}}{x} + 4}
ight) left( {x - 1}
ight) = frac{{360}}{x} imes x cr
& Leftrightarrow 360 - frac{{360}}{x} + 4x - 4 = 360 cr
& Leftrightarrow 4x - frac{{360}}{x} - 4 = 0 cr
& Leftrightarrow x - frac{{90}}{x} - 1 = 0 cr
& Leftrightarrow {x^2} - x - 90 = 0 cr
& Leftrightarrow left( {x - 10}
ight)left( {x + 9}
ight) = 0 cr
& Leftrightarrow x = 10 cr} $$
& = 8:6:12 cr
& = 4:3:6 cr} $$ So, 4 women ≡ 3 men ≡ 6 boy (12 mens + 12 womens + 12 boys) $$eqalign{
& = left[ {12 + left( {frac{3}{4} imes 12}
ight) + left( {frac{3}{6} imes 12}
ight)}
ight]{ ext{men}} cr
& { ext{ = }}left( {12 + 9 + 6}
ight){ ext{men}} cr
& = 27{ ext{ men}} cr} $$ Let the required number of days be x More men, Less days (Indirect proportion) More working hours, Less days (Indirect proportion) [left. x08egin{gathered}
{ ext{Working hours 8}}:6 hfill \
,,,,,,,,,,,,,,,,,,,,,{ ext{Men 27}}:9 hfill \
end{gathered}
ight}::6:x] $$eqalign{
& herefore ,27 imes 8 imes x = 9 imes 6 imes 6 cr
& Leftrightarrow x = frac{{left( {9 imes 6 imes 6}
ight)}}{{left( {27 imes 8}
ight)}} cr
& Leftrightarrow x = frac{3}{2} cr
& Leftrightarrow x = 1frac{1}{2} cr} $$
,,,,,,,,,,,,,,,,{ ext{Days 50}}:60 hfill \
{ ext{Hours per day 9}}:frac{{15}}{2} hfill \
,,,,,,,,,,,,,,,,,,,,,{ ext{Work }}1:2 hfill \
end{gathered}
ight}::42:left( {42 + x}
ight)] $$ herefore left[ {50 imes 9 imes 1 imes left( {42 + x}
ight)}
ight] = $$ xa0 xa0 $$left( {60 imes frac{{15}}{2} imes 2 imes 42}
ight)$$ $$eqalign{
& Leftrightarrow left( {42 + x}
ight) = frac{{37800}}{{450}} cr
& Leftrightarrow 42 + x = 84 cr
& Leftrightarrow x = 42 cr} $$
& = frac{{24}}{{4 imes 12}} imes 20 cr
& = { ext{Rs}}{ ext{.10}} cr} $$
& { ext{15L}} = left( {frac{7}{4} imes 15}
ight){ ext{S}} = frac{{105}}{4}{ ext{S}} cr
& { ext{Also, 2L}} = left( {frac{7}{4} imes 2}
ight){ ext{S}} = frac{7}{2}{ ext{S}} cr
& { ext{3M}} equiv 2{ ext{L}} + { ext{1S }} equiv left( {frac{7}{2} + 1}
ight){ ext{S}} = frac{9}{2}{ ext{S}} cr
& Leftrightarrow { ext{ 7M}} = left( {frac{9}{2} imes frac{1}{3} imes 7}
ight),,{ ext{S}} = frac{{21}}{2}{ ext{S}} cr
& herefore ,left( {{ ext{15L}} + { ext{7M}} + { ext{14S}}}
ight){ ext{ ships}} cr
& equiv left( {frac{{105}}{4} + frac{{21}}{2} + 14}
ight){ ext{S}} cr
& = frac{{203}}{4}{ ext{S}} cr
& left( {{ ext{12 large}} + 14{ ext{ medium}} + 2{ ext{1 small}}}
ight){ ext{ ships}} cr
& equiv left[ {left( {frac{7}{4} imes 12}
ight) + left( {frac{{21}}{2} imes 2}
ight) + 21}
ight]{ ext{S}} cr
& = left( {21 + 21 + 21}
ight){ ext{S}} cr
& = 63,{ ext{S}} cr} $$ Let the required number of journeys be x More ships, Less journeys (Indirect proportion) $$eqalign{
& herefore ,63:frac{{203}}{4}::36:x cr
& Leftrightarrow 63x = frac{{203}}{4} imes 36 = 1827 cr
& Leftrightarrow x = frac{{1827}}{{63}} cr
& Leftrightarrow x = 29 cr} $$ Alternate: Now from the question, $$eqalign{
& 4L = 7S cr
& Rightarrow frac{L}{S} = frac{7}{4} cr
& Rightarrow L = 7x
S = 4x cr
& 3M = 2L + S cr
& Rightarrow M = frac{{2 imes 7x + 4x}}{3} cr
& Rightarrow M = 6x cr} $$ Thus, ratio of large, medium and small = 7 : 6 : 4 So, numbers of journeys: $$eqalign{
& = frac{{(15 imes 7 + 7 imes 6 + 14 imes 4)36}}{{12 imes 7 + 14 imes 6 + 21 imes 4}} cr
& = frac{{7308}}{{252}} cr
& = 29 cr} $$
,,,,,,,,,,,,{ ext{Lentgh 26}}:39 hfill \
,,,,,,,,,,,,,,,,,,,{ ext{Days 6}}:18 hfill \
{ ext{Hours per day }}9:8 hfill \
end{gathered}
ight}::17:x] $$eqalign{
& herefore { ext{ }}26 imes 6 imes 9 imes x = 39 imes 18 imes 8 imes 17 cr
& Leftrightarrow x = frac{{left( {39 imes 18 imes 8 imes 17}
ight)}}{{left( {26 imes 6 imes 9}
ight)}} cr
& Leftrightarrow x = 68 cr} $$ ∴ Number of more labourers = (68 -17) = 51