Direct and Inverse Proportion (Term 1) — Exercise Solutions
How to Use This Page
This page provides complete, step-by-step solutions for every exercise in the TN Class 7 Maths Term 1 — Direct and Inverse Proportion chapter. Work through each problem on your own first, then check your working against the solutions below. Each solution follows the format: Problem → Working → Answer so you can identify exactly where your approach differs.
Key Concepts (Quick Recap)
Direct Proportion
- Two quantities are in direct proportion when an increase in one causes a proportional increase in the other, and a decrease in one causes a proportional decrease in the other.
- Their ratio remains constant: x₁/y₁ = x₂/y₂
- Examples: cost and quantity, distance and time (at constant speed), work and wages
- In direct proportion, if x doubles, y also doubles
Inverse Proportion
- Two quantities are in inverse proportion when an increase in one causes a proportional decrease in the other.
- Their product remains constant: x₁ × y₁ = x₂ × y₂
- Examples: workers and days to finish a job, speed and time, taps and filling time
- In inverse proportion, if x doubles, y halves
Identifying Proportion Type
| Question to ask | Answer means |
|---|
| Does more of A mean more of B? | Direct proportion |
| Does more of A mean less of B? | Inverse proportion |
Tip: Think about real-life meaning. More items bought = more cost (direct). More workers = less time (inverse).
Unitary Method
The unitary method is a two-step approach: 1. Find the value of one unit (divide) 2. Find the value of the required units (multiply)
For direct proportion: value of 1 unit = total value / number of units, then multiply.
For inverse proportion: product is constant, so use x₁ × y₁ = x₂ × y₂.
Exercise 4.1 — Direct Proportion
Fill in the Blanks
Q1. (i) Cost of 8 apples is ₹56; cost of 12 apples = ?
Solution
Cost and number of apples are in direct proportion (more apples = more cost).
Cost of 8 apples = ₹56
Cost of 1 apple = 56 / 8 = ₹7
Cost of 12 apples = 7 × 12 = ₹84
Answer₹84
Q1. (ii) Weight of one fruit box is 3½ kg; weight of 6 such boxes = ?
Solution
Weight and number of boxes are in direct proportion.
Weight of 1 box = 3½ kg = 7/2 kg
Weight of 6 boxes = (7/2) × 6 = 42/2 = 21 kg
Answer21 kg
Q1. (iii) Car travels 60 km with 3 litres of petrol; for 200 km requires = ?
Solution
Distance and petrol are in direct proportion (more distance = more petrol).
Petrol for 60 km = 3 litres
Petrol for 1 km = 3 / 60 = 1/20 litre
Petrol for 200 km = (1/20) × 200 = 10 litres
Answer10 litres
Q1. (iv) 7 m cloth costs ₹294; cost of 5 m = ?
Solution
Cost and length of cloth are in direct proportion.
Cost of 7 m = ₹294
Cost of 1 m = 294 / 7 = ₹42
Cost of 5 m = 42 × 5 = ₹210
Answer₹210
Q1. (v) Machine fills 600 bottles in 5 hrs; bottles in 3 hours = ?
Solution
Number of bottles and time are in direct proportion (more time = more bottles).
Bottles in 5 hrs = 600
Bottles in 1 hr = 600 / 5 = 120
Bottles in 3 hrs = 120 × 3 = 360
Answer360
True or False
Q2. (i) Distance travelled by a bus and time taken are in direct proportion.
Solution
At constant speed, more time means more distance. So they are directly proportional.
AnswerTrue
Q2. (ii) Expenditure of a family and number of members of the family are in direct proportion.
Solution
More family members means more expenditure. So they are directly proportional.
AnswerTrue
Q2. (iii) Number of students in a hostel and consumption of food are not in direct proportion.
Solution
More students means more food consumed. They are directly proportional. The statement says they are not, which is incorrect.
AnswerFalse (they are in direct proportion)
Q2. (iv) If Mallika walks 1 km in 20 minutes, then she can cover 3 km in 1 hour.
Solution
Time for 1 km = 20 minutes
Time for 3 km = 3 × 20 = 60 minutes = 1 hour
This is correct.
AnswerTrue
Q2. (v) If 12 men can dig a pond in 8 days, then 18 men can dig it in 6 days.
Solution
More men means fewer days — this is inverse proportion, not direct.
12 men × 8 days = 96 man-days
18 men → 96 / 18 = 5.33 days (not 6 days)
The statement claims 6 days, which is incorrect.
AnswerFalse
Solved Problems
Q3. A dozen bananas costs ₹20. What is the price of 48 bananas?
Solution
Given12 bananas cost ₹20. Find: cost of 48 bananas.
Number of bananas and cost are in direct proportion.
Cost of 12 bananas = ₹20
Cost of 1 banana = 20 / 12 = ₹5/3
Cost of 48 bananas = (5/3) × 48 = 240/3 = ₹80
Alternatively, using proportion: 12/20 = 48/x → x = (48 × 20)/12 = ₹80
Answer₹80
Q4. 21 students paid ₹840 as entry fee. How many students entered if total paid was ₹1680?
Solution
Given21 students paid ₹840. Find: number of students who paid ₹1680.
Number of students and total fee are in direct proportion.
Students for ₹840 = 21
Students for ₹1 = 21 / 840 = 1/40
Students for ₹1680 = (1/40) × 1680 = 42
Alternatively: 21/840 = x/1680 → x = (21 × 1680)/840 = 42
Answer42 students
Q5. 120 people take 8 trips in a lift. How many people can take 12 trips?
Solution
Given120 people take 8 trips. Find: people for 12 trips.
Number of trips and people are in direct proportion (more trips = more people).
People for 8 trips = 120
People for 1 trip = 120 / 8 = 15
People for 12 trips = 15 × 12 = 180
Answer180 people
Q6. A pole of height 8 m casts a 6 m shadow. Another pole casts a 30 m shadow; find its height.
Solution
Given8 m pole → 6 m shadow. Find: height for 30 m shadow.
Height and shadow length are in direct proportion (taller pole = longer shadow).
8 / 6 = x / 30
x = (8 × 30) / 6 = 240 / 6 = 40 m
Answer40 m
Q7. A postman sorts 738 letters in 6 hours. How many letters in 9 hours?
Solution
Given738 letters in 6 hours. Find: letters in 9 hours.
Time and letters sorted are in direct proportion.
Letters in 6 hrs = 738
Letters in 1 hr = 738 / 6 = 123
Letters in 9 hrs = 123 × 9 = 1107
Answer1107 letters
Q8. Half a meter of cloth costs ₹15. Find the cost of 8⅓ meters.
Solution
Given½ m cloth costs ₹15. Find: cost of 8⅓ m = 25/3 m.
Length and cost are in direct proportion.
Cost of 1/2 m = ₹15
Cost of 1 m = 15 / (1/2) = 15 × 2 = ₹30
Cost of 25/3 m = 30 × (25/3) = 750/3 = ₹250
Alternatively, using proportion:
(1/2) / 15 = (25/3) / x
x = (25/3 × 15) / (1/2)
x = (25 × 15/3) × 2
x = 125 × 2 = ₹250
Answer₹250
Q9. Weight of 72 books is 9 kg. Find the weight of 40 such books (unitary method).
Solution
Given72 books weigh 9 kg. Find: weight of 40 books.
Weight of 72 books = 9 kg = 9000 g
Weight of 1 book = 9000 / 72 = 125 g
Weight of 40 books = 125 × 40 = 5000 g = 5 kg
Answer5 kg
Q10. Thamarai pays ₹7500 as rent for 3 months. How much for 1 year? (unitary method)
Solution
GivenRent for 3 months = ₹7500. Find: rent for 1 year = 12 months.
Rent for 3 months = ₹7500
Rent for 1 month = 7500 / 3 = ₹2500
Rent for 12 months = 2500 × 12 = ₹30,000
Answer₹30,000
Q11. 30 men can reap a field in 15 days. In how many days can 20 men reap the same field? (unitary method)
Solution
Given30 men reap in 15 days. Find: days for 20 men.
More men means fewer days — this is inverse proportion.
30 men × 15 days = 450 man-days (constant work)
20 men → 450 / 20 = 22.5 days
Answer22.5 days
Q12. Valli purchases 10 pens for ₹180; Kamala buys 8 pens for ₹96. Who bought cheaper?
Solution
Compare the price per pen for each person.
Valli: Price per pen = 180 / 10 = ₹18
Kamala: Price per pen = 96 / 8 = ₹12
Since ₹12 < ₹18, Kamala got the cheaper deal.
AnswerKamala bought the pen cheaper (₹12 per pen vs ₹18 per pen)
Q13. A motorbike requires 2 litres of petrol for 100 km. How many litres for 250 km?
Solution
Given2 litres for 100 km. Find: petrol for 250 km.
Distance and petrol are in direct proportion.
Petrol for 100 km = 2 litres
Petrol for 1 km = 2 / 100 = 0.02 litres
Petrol for 250 km = 0.02 × 250 = 5 litres
Answer5 litres
Objective Type
Q14. If the cost of 3 books is ₹90, find the cost of 12 books. (i) ₹180 (ii) ₹270 (iii) ₹360 (iv) ₹90
Solution
Cost of 3 books = ₹90
Cost of 1 book = 90 / 3 = ₹30
Cost of 12 books = 30 × 12 = ₹360
Answer(iii) ₹360
Q15. Mani buys 5 kg potatoes for ₹75. How many kg for ₹105? (i) 8 kg (ii) 7 kg (iii) 6 kg (iv) 5 kg
Solution
5 kg costs ₹75
1 kg costs ₹75/5 = ₹15
For ₹105: 105 / 15 = 7 kg
Answer(ii) 7 kg
Q16. 35 cycles produced in 5 days; how many in 21 days? (i) 120 (ii) 130 (iii) 140 (iv) 147
Solution
Cycles in 5 days = 35
Cycles in 1 day = 35 / 5 = 7
Cycles in 21 days = 7 × 21 = 147
Answer(iv) 147
Q17. An aircraft accommodates 280 people in 2 trips. How many trips for 1400 people? (i) 5 (ii) 10 (iii) 15 (iv) 20
Solution
280 people need 2 trips
1 trip holds 280 / 2 = 140 people
1400 people need 1400 / 140 = 10 trips
Answer(ii) 10
Q18. 3 kg sugar for 50 members; how much for 150 members? (i) 9 kg (ii) 6 kg (iii) 12 kg (iv) 8 kg
Solution
Sugar for 50 members = 3 kg
Sugar for 1 member = 3 / 50 = 0.06 kg
Sugar for 150 members = 0.06 × 150 = 9 kg
Answer(i) 9 kg
Exercise 4.2 — Inverse Proportion
Fill in the Blanks
Q1. (i) 16 taps can fill a petrol tank in 18 minutes. The time taken for 9 taps to fill the same tank = ?
Solution
Fewer taps means more time — inverse proportion.
Let the required time be x minutes.
16 taps × 18 min = 9 taps × x min
16 × 18 = 9x
288 = 9x
x = 288 / 9 = 32
Answer32 minutes
Q1. (ii) If 40 workers can do a project work in 8 days, then ___ workers can do it in 4 days.
Solution
Fewer days means more workers — inverse proportion.
Let the required number of workers be x.
x × 4 = 40 × 8
4x = 320
x = 320 / 4 = 80
Answer80 workers
Solved Problems
Q2. 6 pumps are required to fill a water sump in 1 hr 30 minutes. What will be the time taken to fill the sump if one pump is switched off?
Solution
Given6 pumps fill in 1 hr 30 min. Find: time for 5 pumps (one switched off).
1 hr 30 min = 90 minutes
Fewer pumps means more time — inverse proportion.
6 pumps × 90 min = 5 pumps × x min
540 = 5x
x = 540 / 5 = 108 minutes = 1 hr 48 min
Answer108 minutes (1 hr 48 min)
Q3. A farmer has enough food for 144 ducks for 28 days. If he sells 32 ducks, how long will the food last?
Solution
GivenFood for 144 ducks for 28 days. After selling 32: 144 − 32 = 112 ducks.
Fewer ducks means food lasts longer — inverse proportion.
144 ducks × 28 days = 112 ducks × x days
4032 = 112x
x = 4032 / 112 = 36
Answer36 days
Q4. It takes 60 days for 10 machines to dig a hole. How long will it take 30 machines to dig the same hole?
Solution
Given10 machines take 60 days. Find: days for 30 machines.
More machines means fewer days — inverse proportion.
10 machines × 60 days = 30 machines × x days
600 = 30x
x = 600 / 30 = 20
Answer20 days
Q5. Forty students stay in a hostel. They had food stock for 30 days. If the students are doubled, then for how many days will the stock last?
Solution
Given40 students, food for 30 days. Find: days if students double to 80.
More students means food lasts fewer days — inverse proportion.
40 students × 30 days = 80 students × x days
1200 = 80x
x = 1200 / 80 = 15
Answer15 days
Q6. Meena had enough money to send 8 parcels each weighing 500 grams through a courier service. What would be the weight of each parcel if she has to send 40 parcels for the same money?
Solution
Given8 parcels at 500 g each. Find: weight per parcel if 40 parcels.
More parcels means each is lighter — inverse proportion.
8 parcels × 500 g = 40 parcels × x g
4000 = 40x
x = 4000 / 40 = 100
Answer100 grams
Q7. It takes 120 minutes to weed a garden with 6 gardeners. If the same work is to be done in 30 minutes, how many more gardeners are needed?
Solution
Given6 gardeners take 120 min. Find: total gardeners for 30 min, then find additional.
Less time means more gardeners — inverse proportion.
6 gardeners × 120 min = x gardeners × 30 min
720 = 30x
x = 720 / 30 = 24 gardeners (total)
Additional gardeners = 24 − 6 = 18
Answer18 more gardeners needed
Q8. Neelaveni goes by bicycle to her school every day. Her average speed is 12 km/hr and she reaches school in 20 minutes. What is the increase in speed if she reaches the school in 15 minutes?
Solution
GivenSpeed = 12 km/hr, time = 20 min. Find: new speed for 15 min, then increase.
Higher speed means less time — inverse proportion.
12 km/hr × 20 min = x km/hr × 15 min
240 = 15x
x = 240 / 15 = 16 km/hr
Increase in speed = 16 − 12 = 4 km/hr
Answer4 km/hr increase
Q9. A toy company requires 36 machines to produce car toys in 54 days. How many machines would be required to produce the same number of car toys in 81 days?
Solution
Given36 machines in 54 days. Find: machines for 81 days.
More days means fewer machines needed — inverse proportion.
36 machines × 54 days = x machines × 81 days
1944 = 81x
x = 1944 / 81 = 24
Answer24 machines
Objective Type
Q10. 12 cows can graze a field for 10 days. 20 cows can graze the same field for ___ days. (i) 15 (ii) 18 (iii) 6 (iv) 8
Solution
More cows means fewer days — inverse proportion.
12 × 10 = 20 × x
120 = 20x
x = 120 / 20 = 6
Answer(iii) 6 days
Q11. 4 typists are employed to complete a work in 12 days. If two more typists are added, they will finish the same work in ___ days. (i) 7 (ii) 8 (iii) 9 (iv) 10
Solution
Two more typists = 6 typists total. More typists means fewer days — inverse proportion.
4 typists × 12 days = 6 typists × x days
48 = 6x
x = 48 / 6 = 8
Answer(ii) 8 days
Exercise 4.3 — Miscellaneous and Challenge Problems
Miscellaneous Practice Problems
Q1. If the cost of 7 kg of onions is ₹84, find: (i) Weight of the onions bought for ₹180 (ii) The cost of 3 kg of onions
Solution
(i) Weight for ₹180
Cost and weight are in direct proportion.
For ₹84, weight = 7 kg
For ₹1, weight = 7 / 84 = 1/12 kg
For ₹180, weight = (1/12) × 180 = 180/12 = 15 kg
(ii) Cost of 3 kg
Cost of 7 kg = ₹84
Cost of 1 kg = 84 / 7 = ₹12
Cost of 3 kg = 12 × 3 = ₹36
Answer(i) 15 kg (ii) ₹36
Q2. If C = kd, (i) what is the relation between C and d? (ii) Find k when C = 30 and d = 6 (iii) Find C when d = 10
Solution
(i) C = kd means C is directly proportional to d. As d increases, C increases.
AnswerDirect proportion
(ii) Find k:
C = kd
30 = k × 6
k = 30 / 6 = 5
Answerk = 5
(iii) Find C when d = 10:
C = kd = 5 × 10 = 50
AnswerC = 50
Q3. Every 3 months Tamilselvan deposits ₹5000 as savings. In how many years can he save ₹1,50,000?
Solution
Given₹5000 per 3 months. Find: time to save ₹1,50,000.
Time and savings are in direct proportion.
₹5000 saved in 3 months
₹1,50,000 saved in (3/5000) × 1,50,000 months
= 3 × 30 = 90 months
= 90 / 12 = 7.5 years = 7½ years
Answer7½ years
Q4. A printer prints a book of 300 pages at 30 pages per minute. How long will it take to print the same book if the speed is 25 pages per minute?
Solution
Given300 pages at 30 pages/min. Find: time at 25 pages/min.
Higher speed means less time — inverse proportion.
Time at 30 pages/min = 300 / 30 = 10 min
30 × 10 = 25 × x
300 = 25x
x = 300 / 25 = 12
Answer12 minutes
Q5. If the cost of 6 cans of juice is ₹210, what will be the cost of 4 cans of juice?
Solution
Given6 cans cost ₹210. Find: cost of 4 cans.
Cans and cost are in direct proportion.
Cost of 6 cans = ₹210
Cost of 1 can = 210 / 6 = ₹35
Cost of 4 cans = 35 × 4 = ₹140
Answer₹140
Q6. x varies inversely as twice of y. When y = 6, x = 4. Find x when y = 8.
Solution
x varies inversely as 2y: x = k/(2y)
When y = 6, x = 4:
4 = k / (2 × 6)
4 = k / 12
k = 4 × 12 = 48
When y = 8:
x = 48 / (2 × 8) = 48 / 16 = 3
Answerx = 3
Q7. A truck requires 108 litres of diesel for 594 km. How much diesel is needed for 1650 km?
Solution
Given108 litres for 594 km. Find: diesel for 1650 km.
Distance and fuel are in direct proportion.
Diesel for 594 km = 108 litres
Diesel for 1 km = 108 / 594 litres
Diesel for 1650 km = (108 / 594) × 1650
= (108 × 1650) / 594
= 178200 / 594
= 300 litres
Answer300 litres
Challenge Problems
Q8. If the cost of a dozen soaps is ₹396, what will be the cost of 35 such soaps?
Solution
Given1 dozen (12) soaps cost ₹396. Find: cost of 35 soaps.
Cost of 12 soaps = ₹396
Cost of 1 soap = 396 / 12 = ₹33
Cost of 35 soaps = 33 × 35 = ₹1155
Verification: 33 × 35 = 33 × 30 + 33 × 5 = 990 + 165 = ₹1155
Answer₹1155
Q9. A school has 7 periods a day each of 45 minutes duration. How long is each period if there are 9 periods a day (same total hours)?
Solution
Given7 periods of 45 min each. Find: duration of each period if 9 periods.
Total time is constant. More periods means shorter duration — inverse proportion.
7 × 45 = 9 × x
315 = 9x
x = 315 / 9 = 35
Answer35 minutes
Q10. Cost of 105 notebooks is ₹2415. How many notebooks can be bought for ₹1863?
Solution
Given105 notebooks cost ₹2415. Find: notebooks for ₹1863.
Cost and number of notebooks are in direct proportion.
105 / 2415 = x / 1863
x = (105 × 1863) / 2415
x = 195615 / 2415
x = 81
Answer81 notebooks
Q11. 10 farmers can plough a field in 21 days. Find the number of days reduced if 14 farmers ploughed the same field.
Solution
Given10 farmers in 21 days. Find: days for 14 farmers, then reduction.
More farmers means fewer days — inverse proportion.
10 × 21 = 14 × x
210 = 14x
x = 210 / 14 = 15 days
Days reduced = 21 − 15 = 6
Answer6 days reduced
Q12. A flood relief camp has food for 80 people for 60 days. After 10 days, 20 more people joined. Calculate the number of days of food shortage due to the addition.
Solution
Given80 people, 60 days of food. After 10 days, 20 more join (now 100 people).
Step 1: Food consumed in first 10 days
80 people × 10 days = 800 man-days consumed
Step 2: Remaining food capacity
Total food = 80 × 60 = 4800 man-days
Remaining = 4800 − 800 = 4000 man-days
Step 3: How long food lasts for 100 people
100 people × x days = 4000
x = 4000 / 100 = 40 days
Step 4: Without additional people, food would have lasted
80 people for remaining 50 days (60 − 10 = 50 days)
Step 5: Shortage
50 − 40 = 10 days
Answer10 days shortage
Q13. Six men can complete a work in 12 days. Two days later, 6 more men joined them. How many days will they take to complete the remaining work?
Solution
Given6 men complete in 12 days. After 2 days, 6 more men join (total 12 men).
Step 1: Total work
6 men × 12 days = 72 man-days
Step 2: Work done in first 2 days
6 men × 2 days = 12 man-days
Step 3: Remaining work
72 − 12 = 60 man-days
Step 4: Days for 12 men to finish remaining work
60 / 12 = 5 days
Answer5 days to complete the remaining work
Common Mistakes
1. Confusing direct and inverse proportion — Always check: does more of one quantity mean more or less of the other? More items → more cost (direct). More workers → less time (inverse).
2. Forgetting to convert units — When a problem mixes units (e.g., hours and minutes, kg and grams), convert everything to the same unit before solving. 1 hr 30 min = 90 min, not 1.30 or 130.
3. Applying direct proportion formula to inverse problems — In inverse proportion, the product is constant (x₁ × y₁ = x₂ × y₂), not the ratio. Using x₁/y₁ = x₂/y₂ for an inverse problem gives the wrong answer.
4. Skipping the unitary step — Even when using the proportion formula, showing "value of 1 unit" makes it clear whether you are dividing or multiplying and helps catch errors.
5. Mixing up "how many more" with "total" — Questions like "how many more gardeners" need you to subtract the original number from the total. Read the question carefully: "total" vs "additional."
6. Not identifying inverse proportion in work problems — Problems involving workers and days, speed and time, or taps and filling time are almost always inverse proportion. The more workers/taps, the fewer days/minutes needed.
7. Calculation errors with mixed fractions — Convert mixed fractions (like 3½ or 8⅓) to improper fractions before calculating: 3½ = 7/2, 8⅓ = 25/3.
Exam Checklist
- [ ] Can I correctly identify whether a problem is direct or inverse proportion?
- [ ] Do I know the formulas: x₁/y₁ = x₂/y₂ (direct) and x₁ × y₁ = x₂ × y₂ (inverse)?
- [ ] Can I use the unitary method to find the value of one unit first?
- [ ] Do I convert all units to be consistent (minutes vs hours, grams vs kg)?
- [ ] Do I read carefully for "how many more" vs "how many total"?
- [ ] Can I handle mixed fractions (3½, 8⅓) by converting to improper fractions?
- [ ] Do I verify my answer by checking if it makes sense (e.g., more workers should mean fewer days)?
- [ ] Can I solve multi-step problems where conditions change partway through (e.g., new workers join, food shortage)?
