This module will teach you the basics of direct and indirect proportions. These concepts will further help you in time and work questions.
Important Formulas – chain rule
- Direct Proportion
Two quantities are said to be directly proportional, if on the increase or decrease of the one, the other increases or decreases the same extent.
Examples- Cost of the goods is directly proportional to the number of goods. (More goods, More cost)
- Amount of work done is directly proportional to the number of persons who did the work. (More persons, More Work)
- Indirect Proportion (inverse proportion)
Two quantities are said to be indirectly proportional (inversely proportional) if on the increase of the one, the other decreases to the same extent and vice-versa.
Examples
-
- Number of days needed to complete a work is indirectly proportional (inversely proportional) with the number of persons who does the work (More Persons, Less Days needed)
- The time taken to travel a distance is indirectly proportional (inversely proportional) with the speed in which one is travelling (More Speed, Less Time)
Solved Examples
Level 1
| 1. 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? | |
| A. Rs. (xd/y) | B. Rs. x/d |
| C. Rs. (yd/x) | D. Rs. y/d |
Answer : Option C
Explanation :
cost of x metres of wire = Rs. d
cost of 1 metre of wire = Rs.(d/x)
cost of y metre of wire = Rs.(y×d/x)=Rs. (yd/x)
| 2. 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? | |
| A. 50 | B. 30 |
| C. 40 | D. 10 |
Answer : Option B
Explanation :
Meal for 200 children = Meal for 120 men
Meal for 1 child = Meal for 120/200 men
Meal for 150 children = Meal for (120×150)/200 men=Meal for 90 men
Total mean available = Meal for 120 men
Renaming meal = Meal for 120 men – Meal for 90 men = Meal for 30 men
| 3. 36 men can complete a piece of work in 18 days. In how many days will 27 men complete the same work? | |
| A. 26 | B. 22 |
| C. 12 | D. 24 |
Answer : Option D
Explanation :
Let the required number of days be x
More men, less days (indirect proportion)
Hence we can write as
Men36:27}::x:18 ?36×18=27×x ?12×18=9×x
?12×2=x
?x=24
| 4. A wheel that has 6 cogs is meshed with a larger wheel of 14 cogs. If the smaller wheel has made 21 revolutions, what will be the number of revolutions made by the larger wheel? | |
| A. 15 | B. 12 |
| C. 21 | D. 9 |
Answer : Option D
Explanation :
Let the number of revolutions made by the larger wheel be x
More cogs, less revolutions (Indirect proportion)
Hence we can write as
Cogs 6:14}: x: 21?6×21=14×x ?6×3=2×x ?3×3=x ?x=9
| 5. 3 pumps, working 8 hours a day, can empty a tank in 2 days. How many hours a day should 4 pumps work in order to empty the tank in 1 day? | |
| A. 10 | B. 12 |
| C. 8 | D. 15 |
Answer : Option B
Explanation :
Let the required hours needed be x
More pumps, less hours (Indirect proportion)
More Days, less hours (Indirect proportion)
Hence we can write as
Pumps 3:4
::x:8
Days 2:1
?3×2×8=4×1×x
?3×2×2=x
?x=12
| 6. 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. 9 | B. 12 |
| C. 10 | D. 13 |
Answer : Option D
Explanation :
Let the required number of days be x
More persons, less days (indirect proportion)
More hours, less days (indirect proportion)
Hence we can write as
Persons 39:30
::x:12
Hours 5:6
?39×5×12=30×6×x ?39×5×2=30×x ?39=3×x ?x=13
| 7. A certain industrial loom weaves 0.128 meters of cloth every second. Approximately how many seconds will it take for the loom to weave 25 meters of cloth? | |
| A. 205 | B. 200 |
| C. 180 | D. 195 |
Answer : Option D
Explanation :
Let the required number of seconds be x
More cloth, More time, (direct proportion)
Hence we can write as
Cloth 0.128:25} :: 1:x
?0.128x=25 ?x=25/0.128 ?25000/128=3125/16?195
| 8. 21 goats eat as much as 15 cows. How many goats each as much as 35 cows? | |
| A. 49 | B. 32 |
| C. 36 | D. 41 |
Answer : Option A
Explanation :
15 cows ? 21 goats
1 cow ?21/15 goats
35 cows ? (21×35)/15 goats?(21×7)/3 goats?7×7 goats ? 49 goats
Level 2
| 1. 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. 1 | B. 40 |
| C. 20 | D. 26 |
Answer : Option B
Explanation :
Assume that in x days, one cow will eat one bag of husk.
More cows, less days (Indirect proportion)
More bags, more days (direct proportion)
Hence we can write as
Cows 40:1 ::x:40
Bags 1:40
?40×1×40=1×40×x ?x=40
| 2. If a quarter kg of potato costs 60 paise, how many paise does 200 gm cost? | |
| A. 65 paise | B. 70 paise |
| C. 52 paise | D. 48 paise |
Answer : Option D
Explanation :
Let 200 gm potato costs x paise
Cost of ¼ Kg potato = 60 Paise
=> Cost of 250 gm potato = 60 Paise (? 1 Kg = 1000 gm => ¼ Kg = 1000/4 gm = 250 gm)
More quantity, More Paise (direct proportion)
Hence we can write as
Quantity 200:250} :: x:60
?200×60=250×x ?4×60=5×x ?4×12=x ?x=48
| 3. A contract is to be completed in 56 days if 104 persons work, each working at 8 hours a day. After 30 days, 2/5 of the work is completed. How many additional persons should be deployed so that the work will be completed in the scheduled time, each person’s now working 9 hours a day. | |
| A. 160 | B. 150 |
| C. 24 | D. 56 |
Answer : Option D
Explanation :
Persons worked = 104
Number of hours each person worked per day = 8
Number of days they worked = 30
Work completed = 2/5
Remaining days = 56 – 30 = 26
Remaining Work to be completed = 1 – 2/5 = 3/5
Let the total number of persons who do the remaining work = x
Number of hours each person needs to be work per day = 9
More days, less persons(indirect proportion) More hours, less persons(indirect proportion)
More work, more persons(direct proportion)
Hence we can write as
Days 30:26
Hours 8:9 ::x:104
Work 35:25
?30×8×3/5×104=26×9×2/5×x
?x=(30×8×3/5×104)/(26×9×2/5)=(30×8×3×104)/(26×9×2)
=(30×8×104)/(26×3×2)=(30×8×4)/(3×2)=5×8×4=160
Number of additional persons required = 160 – 104 = 56
| 4. x men working x hours per day can do x units of a work in x days. How much work can be completed by y men working y hours per day in y days? | |
| A. x2/y2 units | B. y3/x2 units |
| C. x3/y2 units | D. y2/x2 units |
Answer : Option B
Explanation :
Let amount of work completed by y men working y hours per in y days = w units
More men, more work(direct proportion)
More hours, more work(direct proportion)
More days, more work(direct proportion)
Hence we can write as
Men x:y
Hours x:y ::x:w
Days x:y
?x3w=y3x ?w=y3x/x3=y3/x2
| 5. A flagstaff 17.5 m high casts a shadow of length 40.25 m. What will be the height of a building, which casts a shadow of length 28.75 m under similar conditions? | |
| A. 12.5 m | B. 10.5 m |
| C. 14 | D. 12 |
Answer : Option A
Explanation :
Let the required height of the building be x meter
More shadow length, More height (direct proportion)
Hence we can write as
Shadow length 40.25:28.75}:: 17.5:x
?40.25×x=28.75×17.5 ?x=(28.75×17.5)/40.25=(2875×175)/40250
= (2875×7)/1610=2875/230=575/46=12.5
| 6. If the price of 357 apples is Rs.1517.25, what will be the approximate price of 49 dozens of such apples? | |
| A. Rs. 2500 | B. Rs. 2300 |
| C. Rs. 2200 | D. Rs. 1400 |
Answer : Option A
Explanation :
Let the required price be x
More apples, More price (direct proportion)
Hence we can write as
Apples 357:(49×12)} :: 1517.25:x
?357x = (49×12)×1517.25?x = (49×12×1517.25)/357=(7×12×1517.25)/51
= (7×4×1517.25)/17
=7×4×89.25?2500
| 7. 9 engines consume 24 metric tonnes of coal, when each is working 8 hours day. How much coal is required for 8 engines, each running 13 hours a day, if 3 engines of former type consume as much as 4 engines of latter type? | |
| A. 20 metric tonnes | B. 22 metric tonnes |
| C. 24 metric tonnes | D. 26 metric tonnes |
Answer : Option D
Explanation :
Let required amount of coal be x metric tonnes
More engines, more amount of coal (direct proportion)
If 3 engines of first type consume 1 unit, then 1 engine will consume 1/3 unit which is its the rate of consumption.
If 4 engines of second type consume
More rate of consumption, more amount of coal (direct proportion)
More hours, more amount of coal(direct proportion)
Hence we can write as
Engines 9:8
rate of consumption 13:14 ::24:x
hours 8:13
?9×1/3×8×x=8×1/4×13×24 ?3×8×x=8×6×13 ?3xX=6×13?x=2×13=26
| 8. in a camp, food was was sufficient for 2000 people for 54 days. After 15 days, more people came and the food last only for 20 more days. How many people came? | |
| A. 1900 | B. 1800 |
| C. 1940 | D. 2000 |
Answer : Option A
Explanation :
Given that food was sufficient for 2000 people for 54 days
Hence, after 15 days, the remaining food was sufficient for 2000 people for 39 days (? 54 – 15 =39)
Let x number of people came after 15 days.
Then, total number of people after 15 days = (2000 + x)
Then, the remaining food was sufficient for (2000 + x) people for 20 days
More men, Less days (Indirect Proportion)?Men 2000:(2000+x)} :: 20:39
?2000×39=(2000+x)20?100×39=(2000+x)?3900=2000+x?x=3900?2000=1900