chapter 13 Direct and Indirect proportions

Explanation:

Charge per hour:

i). Rs 60 for 4 hours- 60/4 

                                = Rs. 15

ii) Rs 100 for 8 hours- 100/8

                                   = Rs. 12.50

iii) Rs 140 for 12 hours- 140/12

                                      = Rs. 11.67

iv) Rs 180 for 24 hours- 180/24

                                     = Rs 7.50

Since, the per hour charge in the above cases are not same,i.e., i≠ii≠iii≠iv 

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

Explanation:

Let the ratio of parts of red pigment to the part of base be a:b and we denote a:b as k

Case 1-

Here, 1 part of red pigment is mixed with 8 parts of base

Thus, k= 1:8 or 1 / 8; a1=1, b1= 8

Case 2, 

Here, k= 4:?; a2=4, b2= ?

K= a² / b² 

1 / 8 =4 / b2

b2= 32

Case 3,

Here k= 7:?; a3= 7, b3= ?

K= a3 / b3

1/ 8 = 7/ b3

b3= 56

Case4,

Here k= 12:?; a4= 12, b4= ?

K= a4 / b4

1/ 8 = 12/ b4

b4= 96

Case 5,

Here k= 20:?; a4= 20, b4= ?

K= a5/ b5

1/8 = 20/b5

b5=160

Combining results for all the cases, we get

Explanation:

Let part of red pigment mixed with 1800ml of base be x

75ml of base= 1 part of pigment

1800ml of base= x part of pigment

Since, it is in direct proportion

1 / 75 = x / 1800

75x= 1800

x=1800/75

x=24

Hence, 24 parts of red pigment should be mixed with base of 1800ml.

Explanation:

In 6hrs, the machine fills 840 bottles

In 1hr, the machine will fill 840/6 bottles or 140 bottles

Per hour yield of machine is 140 bottles

In 5 hours, the machine will fill 140x5 bottles or 700 bottles.

Explanation:

Let enlarged length of bacteria be x

Actual length of bacteria= 5/50,000 cm

                                          = 1/10,000 cm

                                          = 10-4 cm

Bacteria enlarged 50,000 times attains length of 5cm

Bacteria enlarged 20,000 times, attains length of x cm

Here, the length and enlarged length of bacteria is in direct proportion

5/50,000 = x/20,000

50,000x = 1,00,000

X= 2

Hence, the enlarged length of bacteria is 2cm.

Explanation:

Let the length of model ship be x cm

Length of actual ship= 28m; mast of actual ship= 12m

Length of model ship= x cm; mast of model ship= 9cm

Since, the length of the mast and the actual length of the ship are in direct proportion.

12/9 =28/x

12x= 252

x= 21

Hence, the length of model ship is 21cm.

Explanation:

i) Let crystal content in 5kg sugar be x

And 2kg sugar contains 9×106 crystal

Here, the weight of sugar and the number of crystals are in direct proportion.

2x= 9 x 5 x 106

x=  9 x 5 x 106

            2

x= 22.5 x 106

x= 2.25 x 107

Hence, the number of sugar crystals is 2.25 x 107

ii)

 Let sugar crystals in 1.2kg sugar be x.

And 2kg sugar contains 9×106 crystal

Here, the weight of sugar and the number of crystals are in direct proportion.

2x= 1.2 x 9 x 106

x= 1.2 x 9 x 106

              2

x= 5.4 x 106

Hence, the number of sugar crystals is 5.4 x 106

Explanation:

Let the distance covered in the map be x

Here, the actual distance and distance covered in the map are in direct proportion.

18/1  = 72/x

18x= 72

x= 4

Hence, the distance covered on the map is 4 cm.

Explanation:

(i) Height of pole= 5m 60cm or 560cm (Since 1m=100cm)

Length of shadow it casts= 3m 20cm or 320cm 

Height of another pole= 10m 50cm

Let the length of the shadow of another pole be x.

Here, the height of the pole and the length of the shadow are in direct proportion.

560320 = 1050x

=> 560x= 1050 x 320

=> x= 1050 x 320

                 560

x= 600 cm = 6m

Hence, the length of the shadow of another pole is 6 m.

(ii) Let the height of the pole be x.

Length of its shadow=  5m or 500cm

560320 = x500

=> x= 500 x 560

               320

= 875 cm

 = 8 m 75 cm

Hence, the height of the pole is 8 m 75 cm.

Explanation:

Let the distance covered in 5 hours be x km.

1 hour = 60 minutes

Therefore, 5 hours = 5×60 = 300 minutes

Here, the distance covered and time are in direct proportion.

1425 = x300

25x = 300 x 24

x = 168

Therefore, the truck can travel 168 km in 5 hours.

Explanation:

i) The number of workers on a job and time to complete the job are in inverse proportion because if more are the number of workers less time will be taken to complete the job and vice versa.

ii) Time taken for a journey and distance travelled is directly proportional because with more time more distance can be covered.

iii) It is direct proportion because more area of cultivated land will yield more harvest.

iv) Speed and time are in inverse proportion because with more speed less time will be taken.

v) It is inverse proportion because with an increase in population the area of land per person decreases.

Explanation:

Here, the number of winners and prize money are in inverse proportion because prize money is decreasing with increase in number of winners.

When there are 4 winners, each winner will get =100000/4 = Rs. 25,000

When there are 5 winners, each winner will get =100000/5 = Rs. 20,000

When there are 8 winners, each winner will get =100000/8 = Rs. 12,500

When there are 10 winners, each winner will get = 100000/10 = Rs. 10,000

When there are 20 winner, each winner will get = 100000/20 = Rs. 5,000

Explanation:

a) When there are 4 spokes, then the angle between a pair of consecutive spokes = 360/4 = 90 degree

b) When there are 8 spokes, then the angle between a pair of consecutive spokes = 360/8= 45 degree.

C) When there are12 spokes, then the angle between a pair of consecutive spokes = 360/12 = 30 degree.

(i) Yes, the number of spokes and the angles formed between a pair of consecutive spokes is in inverse proportion as it is clear above that as the number of spokes is increasing, the angle between a pair of consecutive spokes is decreasing.

(ii) When the number of spokes is 15, then the angle between a pair of consecutive spokes = 360/15= 24 degree.

(iii) If the angle between a pair of consecutive spokes is 40 degree, then number of spokes required will be 360/40= 9 spokes

Explanation:

When each child gets 5 sweets, 24 children will get 24 x 5 sweets, i.e., 120 sweets

Therefore, total number of sweets= 120

When the number of students is reduced by 4, then the children left is ( 24-4) or 20

Noe, each child will get 120/20 sweets, i.e., 6 sweets.

Explanation:

20 animals can be fed for 6 days. 

When total number of animals is increased by 10, let us assume that they can be fed for x days.

Here, the number of animals and the number of days are in inverse proportion.

2030 = x6

30x = 120

x = 4

Hence, the food will last for 4 days.

Explanation:

Let the time taken to complete the job by 4 person be x.

Here, the number of persons and the number of days are in inverse proportion.

3/4= x/4

3×4 = 4x

x = 3

Hence, it will take 3 days to complete the job by 4 person.

Explanation:

Let the number of boxes filled with 20 bottles in each box be x.

Here, the number of bottles and the number of boxes are in inverse proportion.

12/20 = x/25

20x=12×25 

x = 12×25/20 = 15

Hence, 15 boxes would be filled.

Explanation:

Let the number of machines required be x.

Here, the number of machines and the number of days are in inverse proportion.

63/54 = x/42

63×42 = 54x

x = 63×42/54

x= 49

Hence, 49 machines would be required.

Explanation:

Let the number of hours be x when speed is 80km/hr

Here, the speed of the car and time are in inverse proportion.

60/80 = x/2

60×2 = 80x

x = 60×2/80

x= 3/2

x= 1.5

Hence the car will take 1.5 hours to reach its destination.

Explanation:

(i) Let the number of days be x.

Here, the number of persons and the number of days are in inverse proportion.

2/1 = x/3

x = 6 

Hence, when one person fell ill the same job will take 6 days now.

(ii) Let the number of persons be x.

Here, the number of persons and the number of days are in inverse proportion.

2/x = 1/3

x = 6 

Hence, to fit the window in one day 6 person would be needed.

Explanation:

Let the duration of each period be x when school has 9 periods a day.

Here, the number of periods and the duration of periods are in inverse proportion.

8/9 = x/45

9x= 8 x 45

x = 40

Hence, the duration of each period would be 40 minutes.