13 . Direct and Inverse Proportions

Chapter 13 . Direct and Inverse Proportions

13. Direct and Inverse Proportions

Exercise 13.1

1. Following are the car parking charges near a railway station upto
         4 hours            Rs 60
         8 hours             Rs 100
       12 hours             Rs 140
       24 hours             Rs 180
Check if the parking charges are in direct proportion to the parking time.

Solution: We have,

, ,

and

So ,

Therefore, the parking charges are not in direct proportion to the parking time .

2. A mixture of paint is prepared by mixing 1 part of red pigments with 8 parts of base. In the following table, find the parts of base that need to be added.

Parts of red pigment	1	4	7	12	20
Parts of base	8	...	...	...	...

Solution: let and are the parts of base respectively , then

Parts of red pigment	1	4	7	12	20
Parts of base	8

So,

We have ,

Parts of red pigment	1	4	7	12	20
Parts of base	8	32	56	96	160

3. In Question 2 above, if 1 part of a red pigment requires 75 mL of base, how much red pigment should we mix with 1800 mL of base?

Solution: let part of a red pigment requires 1800 mL of base .

A/Q,

Therefore, 24 parts of a red pigment requires 1800 mL of base .

4. A machine in a soft drink factory fills 840 bottles in six hours. How many bottles will it fill in five hours?

Solution: let be the number of bottles .

A/Q,

Therefore, the number of bottles is 700 .

5. A photograph of a bacteria enlarged 50,000 times attains a length of 5 cm as shown in the diagram. What is the actual length of the bacteria? If the photograph is enlarged 20,000 times only, what would be its enlarged length?

Solution: The actual length of the bacteria

Let be the length of a photograph of a bacteria .

A/Q ,

Therefore, the length of a photograph of the bacteria is 2 cm .

6. In a model of a ship, the mast is 9 cm high, while the mast of the actual ship is 12 m high. If the length of the ship is 28 m, how long is the model ship?

Solution: let be the length of the model ship (in metres) .

We have,

High of mast (in cm)

Length of ship (in m)

A/Q,

Therefore, the length of the model ship is 21 m

7. Suppose 2 kg of sugar contains 9 × 106 crystals. How many sugar crystals are there in (i) 5 kg of sugar? (ii) 1.2 kg of sugar?

Solution: (i) let be the number of sugar crystals .

We have ,

Weight of sugar (In kg)	No. of sugar crystals
2
5

A/Q,

Therefore, the number of sugar crystals is .

(ii) let be the number of sugar crystals .

We have ,

Weight of sugar (In kg)	No. of sugar crystals
2
1.2

A/Q,

Therefore, the number of sugar crystals is .

8. Rashmi has a road map with a scale of 1 cm representing 18 km. She drives on a road for 72 km. What would be her distance covered in the map?

Solution: let be the distance covered in the map (in cm) .

We have,

Scale (in cm)	Distance (in km)
1	18
	72

A/Q,

Therefore, the distance covered in the map is 4 cm .

9. A 5 m 60 cm high vertical pole casts a shadow 3 m 20 cm long. Find at the same time (i) the length of the shadow cast by another pole 10 m 50 cm high (ii) the height of a pole which casts a shadow 5m long.

Solution: (i) let be the length of the shadow cast by vertical pole (in cm) .

We have,

5m + 60 cm = 5×100+60 = 500+60 = 560 cm

10m + 50 cm = 10×100+50 = 1000+50 = 1050 cm

3m + 20 cm = 3×100+20 = 300+20 = 320 cm

Height of pole	Length of shadow
560	320
1050

A/Q,

Therefore, the length of the shadow cast by vertical pole is 6 m .

(ii) let be the length of the vertical pole (in metres) .

We have,

5m + 60 cm = 5.60 m = 5.6 m

3m + 20 cm = 3.20 m = 3.2 m

Height of pole	Length of shadow
5.6	3.2
	5

A/Q,

m = 875 cm = 8m 75cm

Therefore , the length of the vertical pole is 8m75 cm .

10. A loaded truck travels 14 km in 25 minutes. If the speed remains the same, how far can it travel in 5 hours? Photo

Solution: let be the distance travels by the truck in 5 hours .

We have,

Distance (in km)	Time (in minutes)
14	25
	5 × 60 = 300

A/Q,

Therefore, the distance travels by the truck is 168 km .

Exercise 13.2

1. Which of the following are in inverse proportion?
(i) The number of workers on a job and the time to complete the job.
(ii) The time taken for a journey and the distance travelled in a uniform speed.
(iii) Area of cultivated land and the crop harvested.
(iv) The time taken for a fixed journey and the speed of the vehicle.
(v) The population of a country and the area of land per person.

Solution: (i) The number of workers on a job and the time to complete the job.
(iv) The time taken for a fixed journey and the speed of the vehicle.
(v) The population of a country and the area of land per person.

2. In a Television game show, the prize money of ` 1,00,000 is to be divided equally amongst the winners. Complete the following table and find whether the prize money given to an individual winner is directly or inversely proportional to the number of winners?

Number of winners	1	2	4	5	8	10	20
Prize for each winner (in Rs)	1,00,000	50,000	...	...	...	....	…

Solution: Firstly ,we observe , 1×100000 = 2×50000 = 100000 . So, it is inversely proportional .

Let ,

Number of winners	1	2	4	5	8	10	20
Prize for each winner (in Rs)	1,00,000	50,000				p

Now,

Again,

Again ,

Again,

And

3. Rehman is making a wheel using spokes. He wants to fix equal spokes in such a way that the angles between any pair of consecutive spokes are equal. Help him by completing the following table.

(i) Are the number of spokes and the angles formed between the pairs of consecutive spokes in inverse proportion?
(ii) Calculate the angle between a pair of consecutive spokes on a wheel with 15 spokes.
(iii) How many spokes would be needed, if the angle between a pair of consecutive spokes is 40°?

Solution: let

Number of spokes	Angle between a pair of consecutive spokes
4	90°
6	60°
8
10
12

Now,

Again,

and

We have,

Number of spokes	Angle between a pair of consecutive spokes
4	90°
6	60°
8	45°
10	36°
12	30°

(i) Firstly , we observe 4 × 90° = 6 × 60° = 360°

So, the number of spokes and the angles formed between the pairs of consecutive spokes are in inverse proportion .

(ii) let be the angle between a pair of consecutive spokes on a wheel .

A/Q,

Therefore, the angle between a pair of consecutive spokes on a wheel is 24° .

(iii) let be the number of spokes .

A/Q,

Therefore, the number of spokes is 9 .

4. If a box of sweets is divided among 24 children, they will get 5 sweets each. How many would each get, if the number of the children is reduced by 4?

Solution: let be the number of sweets box .

No. of children	No. of sweets box
24	5
24 – 4 = 20

A/Q,

Therefore, the number of sweets box is 6 .

5. A farmer has enough food to feed 20 animals in his cattle for 6 days. How long would the food last if there were 10 more animals in his cattle?

Solution: let be the number of days .

We have ,

No. of cattle	No. of days
20	6
20+10 = 30

A/Q,

Therefore, the number of days is 4 .

6. A contractor estimates that 3 persons could rewire Jasminder’s house in 4 days. If, he uses 4 persons instead of three, how long should they take to complete the job?

Solution: let be the days take to complete the job .

We have,

No. of persons	No. of days
3	4
4

A/Q,

Therefore, the days take to complete the job is 3 .

7. A batch of bottles were packed in 25 boxes with 12 bottles in each box. If the same batch is packed using 20 bottles in each box, how many boxes would be filled?

Solution: let be the number of boxes .

We have,

No. of boxes	No. of bottles
25	12
	20

A/Q,

Therefore, the number of boxes is 15

8. A factory requires 42 machines to produce a given number of articles in 63 days. How many machines would be required to produce the same number of articles in 54 days?

Solution: let be the number of machines .

We have,

No. of machine	No. of days
42	63
	54

A/Q,

Therefore, the number of machines is 49 .

9. A car takes 2 hours to reach a destination by travelling at the speed of 60 km/h. How long will it take when the car travels at the speed of 80 km/h?

Solution: let be the time taken travel by the car .

We have ,

Speed (in km)	Times (in hours)
60	2
80

A/Q,

Therefore ,the time taken travel by the car is hours .

10. Two persons could fit new windows in a house in 3 days.
(i) One of the persons fell ill before the work started. How long would the job take now?
(ii) How many persons would be needed to fit the windows in one day?

Solution: (i) let be the number of days .

We have ,

No. of persons	No. of days
2	3
(2 – 1 = ) 1

A/Q,

Therefore, the number of days is 6 .

(ii) Let be the number of persons .

We have,

No. of persons	No. of days
2	3
	1

A/Q,

Therefore, the number of persons is 6 .

11. A school has 8 periods a day each of 45 minutes duration. How long would each period be, if the school has 9 periods a day, assuming the number of school hours to be the same?

Solution: let be the number of period duration (in minute) .

We have ,

Periods	Duration (minutes)
8	45
9

A/Q,

Therefore , the number of period duration is 40 minute .