CBSE Class 8 Mathematics Direct and Inverse Proportions Assignment Set 04

Question. A does a work in 10 days and B does the same work in 15 days. In how many days they together will do the same work?
(a) 5 days
(b) 6 days
(c) 8 days
(d) 9 days
Answer: (b) 6 days

Question. A can finish a work in 18 days and B can do the same work in half the time taken by A. Then, working together, what part of the same work they can finish in a day?
(a) \( \frac{1}{6} \)
(b) \( \frac{2}{9} \)
(c) \( \frac{1}{9} \)
(d) \( \frac{2}{7} \)
Answer: (a) \( \frac{1}{6} \)

Question. A tyre has two punctures. The first puncture alone would have made the tyre flat in 9 minutes and the second alone would have done it in 6 minutes. If air leaks out at a constant rate, how long does it take both the punctures together to make if flat?
(a) \( 1\frac{1}{2} \) minutes
(b) \( 3\frac{1}{2} \) minutes
(c) \( 3\frac{3}{5} \) minutes
(d) \( 4\frac{1}{4} \) minutes
Answer: (c) \( 3\frac{3}{5} \) minutes

Question. A family has enough rice for 8 members for 30 days. Find the number of days the rice will last if the members increased to 10.
Answer: 24 days

Question. 125 people finish a piece of work in 120 days. Find the number of people required to finish the same work in 100 days.
Answer: 150 people

Question. Three taps take 6 hours, 8 hours and 12 hours respectively to fill a tank. How long will it take for all three taps together to fill the tank?
Answer: \( \frac{8}{3} \) hours

Question. 16 men can do a piece of work in 7 days. How long will it take 14 men to do the same work?
Answer: 8 days

Question. A restaurant has food stock to supply food for 900 people for 40 days. If only 600 people visited the restaurant, how long will the food last?
Answer: 60 days

Question. If apples are sold at ₹ 5 each, Sushma can buy 2 dozen apples with the money in her purse. How much will she buy if the cost is increased by ₹ 1 each?
Answer: 20 apples

Question. 8 horses consume a certain quantity of grains in 50 days. How long will the grains last if the number of horses is increased to 20?
Answer: 20 days

Question. A school has enough food for 400 children for 12 days. How long will the food last for 80 more children?
Answer: 10 days

Question. Ramesh pours \( 10^5 \) granules of sand per second in a bottle and it takes him 320 days to fill it. How many days will it take to fill that bottle if he pours \( 10^6 \) granules per second?
Answer: 32 days

Question. Fill in the tables
(a)
x (number of men): 2, 3, —, 16, 48
Y (time taken): 24, —, 8, —, —
(b)
x (men): 8, 2, 18, 9, —, —
y (days): 9, —, —, —, 12, 1
(c)
x (men): 12, —, 28, —, 6
y (time): 7, 4, —, 42, —
Answer: (a) 16, 6, 3, 1 (b) 36, 4, 8, 6, 72 (c) 21, 3, 2, 14

Question. A tank can be filled by two pipes A and B in 12 hours and 16 hours, respectively. While pipe C empties it in 8 hours, find the time taken to empty the tank if all three pipes are open (emptying implies negative).
Answer: 48 hrs

Question. 2 men working together take 6 days to finish a piece of work. If one of them takes 15 days if he works alone, find how long the second man will take to finish it, working alone?
Answer: 10 days

Question. A and B take 12 days to finish a piece of work. If B and C work together they take 15 days. While A and C together take 20 days. Find the time taken by each to work alone.
Answer: A = 30 days, B = 20 days, C = 60 days

Question. A contractor hires 120 men to complete a job in 6 months. How many men should he hire if he has to complete the job in 4 months?
Answer: 180 men

Question. A and B together can write a manuscript in 20 days. If A takes 25 days when working along, find the time taken by B working alone.
Answer: 100 days

Question. 30 men can harvest a field in 14 days. Find the time taken by 20 men for the same job.
Answer: 21 days

Question. Meeta, Veena and Kamlesh can sweep a playground in 4, 6 and 8 hours respectively. What portion of that ground can they sweep in one hour working together?
Answer: \( \frac{13}{24} \) th portion

Question. If 20 men working 12 hours a day can do binding of 2500 books in a day, how many books can 24 men bind in a day if each of them work 8 hours a day?
Answer: 2000 books

Question. 4 men and 7 boys can dig a tank in 15 days. How long would 2 men and 4 boys working together take to dig the tank.
Answer: 52.5 days

Question. A can do a piece of work in 25 days, B can finish it in 20 days. They work together for 5 days and then A goes away. In how many days will B finish the remaining work?
Answer: 11 days

Question. 6 men can complete the electric fitting of a building in 7 days. How many days will it take if 21 men do the job?
Answer: 2 days

Question. A cistern can be filled by one tap in 8 hours and by another in 4 hours. How long will it take to fill the cistern if both taps are opened together?
Answer: \( \frac{8}{3} \) hrs

Question. If 12 boys earn ₹ 840 in 7 days, what will 15 boys earn in 6 days?
Answer: ₹ 900

Question. If 25 men earn ₹ 1000 in 10 days, how much will 15 men earn in 15 days?
Answer: ₹ 900

Question. Find the time taken by a train of length 200 m at 60 kmph to cross a lamp post.
Answer: 12 sec

Question. A car moving from city A to city B takes \( 8\frac{1}{2} \) hours. If the distance between the cities is 425 km, find the speed of the car.
Answer: 50 km/h

Question. What is the distance covered by a car at 73 kmph in \( 5\frac{1}{2} \) hours?
Answer: 401.5 km

Question. Suresh cycles at 30 kmph to reach a destination 285 km away. What is the time taken by him?
Answer: 9.5 h

Question. Two cars start from the same point at speeds 60 km/h and 50 km/h. The slower car starts half an hour earlier. When will the other car catch up with the slower car?
Answer: \( 2\frac{1}{2} \) hr

Question. Two trains start from the same station in the same direction but with a gap of one hour. The earlier train travels at 80 km/h while the second train travels at 95 km/h. When will the second train overtake the first?
Answer: \( \frac{16}{3} \) hrs

Question. How long will a train 600 m long take to cross a pole, if it travels at 45 kmph?
Answer: 48 sec

Question. How long will a train 650 m long take to cross a bridge 350 m long, if it travels at a speed of 54 kmph?
Answer: 1 min 7 sec (approx)

Question. A train 450 m long crosses a bridge in 72 seconds travelling at 60 kmph. Find the length of the bridge.
Answer: 750 m

Question. A 300 m long train is travelling at 85 km/h. A man is running in the same direction at 5 km/h. How long will it take the train to cross the man?
Answer: \( \frac{27}{2} \) sec

Question. A train 750 m long travelling at 45 km/h takes 72 seconds to cross another train travelling in the opposite direction at 52 km/h. Find the length of the second train.
Answer: 1190 m

Question. Two trains of length 220 m and 100 m are travelling on parallel lines in the same direction with speeds 46 km/h and 36 km/h. How long will it take the faster train to overtake the slower train? How long will it take if the trains were moving in the opposite directions?
Answer: 115.2 sec, 14.048 sec

Question. A car takes 9 hours to cover a distance from X to Y travelling at 40 km per hour. Find the time taken if its speed is increased by 20 km per hour.
Answer: 6 hr

Question. A plane takes \( 1\frac{2}{3} \) hours to cover a distance flying at 340 km per hour. Find the time taken if the speed is increased to 850 km per hour.
Answer: \( \frac{2}{3} \) hr

Assignment on Direct Proportion
 

Question. Find \(x_1\) and \(y_1\) from the following table if \(x \propto y\).

Answer: \(x_1 = 20, y_1 = 70\)
Since \(x \propto y\), we have \(\frac{x}{y} = k\) (a constant value).
From the complete columns in the table:
\(\frac{30}{60} = \frac{14}{28} = \frac{2}{4} = \frac{104}{208} = \frac{1}{2}\)
Using this constant ratio:
1. \(\frac{x_1}{40} = \frac{1}{2} \implies x_1 = 20\)
2. \(\frac{35}{y_1} = \frac{1}{2} \implies y_1 = 70\)

Question. Fill the gap if \(x \propto y\)

Answer: \(a = 18, b = 4, c = 27, d = 11\)
Since \(x\) is directly proportional to \(y\), the ratio \(\frac{x}{y}\) is constant.
From the given complete columns: \(\frac{15}{5} = \frac{120}{40} = \frac{9}{3} = 3\).
Thus, \(\frac{x}{y} = 3\).
1. For \(a\): \(\frac{a}{6} = 3 \implies a = 18\)
2. For \(b\): \(\frac{12}{b} = 3 \implies b = 4\)
3. For \(c\): \(\frac{c}{9} = 3 \implies c = 27\)
4. For \(d\): \(\frac{33}{d} = 3 \implies d = 11\)

Question. The cost of 8 pencils Rs 16. Find the cost of 5 pencils.
Answer: Rs 10
Let the cost of 5 pencils be \(x\). Since cost is directly proportional to the number of pencils:
\(\frac{8}{16} = \frac{5}{x}\)
\(\frac{1}{2} = \frac{5}{x} \implies x = 10\)
Thus, the cost of 5 pencils is Rs 10.

Question. A car travels 14 km in 25 minutes. If the speed remains the same, how much distance it will cover in 5 hours?
Answer: 168 km
Time in minutes = \(5 \text{ hours} \times 60 \text{ minutes/hour} = 300 \text{ minutes}\).
Since distance covered is directly proportional to time when speed is constant:
\(\frac{14}{25} = \frac{\text{Distance}}{300}\)
\(\text{Distance} = \frac{14 \times 300}{25} = 14 \times 12 = 168\text{ km}\).

Question. A 5 m 60 cm high tower casts a shadow of 3 m 20 cm. Find the height of the tower which casts a shadow of 5 m.
Answer: 8.75 m
Converting measurements to meters:
Height of first tower = \(5.6\text{ m}\)
Shadow of first tower = \(3.2\text{ m}\)
Shadow of second tower = \(5\text{ m}\)
Let the height of the second tower be \(h\). Since height and shadow length are in direct proportion:
\(\frac{5.6}{3.2} = \frac{h}{5}\)
\(\frac{7}{4} = \frac{h}{5} \implies h = \frac{35}{4} = 8.75\text{ m}\).

Question. The daily consumption of wheat in a hostel is 12 kg for 240 students. Find the consumption of wheat for 170 students.
Answer: 8.5 kg
Let the wheat consumption for 170 students be \(x\text{ kg}\). Since consumption is directly proportional to the number of students:
\(\frac{12}{240} = \frac{x}{170}\)
\(\frac{1}{20} = \frac{x}{170} \implies x = \frac{170}{20} = 8.5\text{ kg}\).

Question. 12 persons can dig 324 cubic metres earth in a day. How many persons will be required to dig 270 cubic metres earth.
Answer: 10 persons
Let the required number of persons be \(x\). Since the volume of earth dug is directly proportional to the number of persons:
\(\frac{12}{324} = \frac{x}{270}\)
\(\frac{1}{27} = \frac{x}{270} \implies x = \frac{270}{27} = 10\text{ persons}\).

Question. An aeroplane flies 4235 km in 5 hrs 30 minutes. How much distance it will cover in 3 hrs?
Answer: 2310 km
Time 1 = \(5 \text{ hours } 30 \text{ minutes} = 5.5 \text{ hours}\).
Let the distance covered in 3 hours be \(d\). Since distance is directly proportional to time:
\(\frac{4235}{5.5} = \frac{d}{3}\)
\(770 = \frac{d}{3} \implies d = 770 \times 3 = 2310\text{ km}\).

Question. The cost of 12 packets of salt is Rs 21. The weight of each packet is 900g. How many such packets of salt can be purchased in Rs 26.25?
Answer: 15
Since the weight of each packet is the same, the number of packets purchased is directly proportional to the total cost.
Let the number of packets be \(x\).
\(\frac{12}{21} = \frac{x}{26.25}\)
\(x = \frac{12 \times 26.25}{21} = \frac{315}{21} = 15\).

Question. If 120 dollars can be exchanged to Rs 5754. Find the amount obtained in 55 dollars?
Answer: Rs 2637.25
Let the amount obtained for 55 dollars be \(x\). Since the exchange value is in direct proportion:
\(\frac{5754}{120} = \frac{x}{55}\)
\(x = \frac{5754 \times 55}{120} = 47.95 \times 55 = 2637.25\text{ Rs}\).