Direct And Inverse Proportions

Given: Cost of 5 m of cloth = Rs. 210

To Find: Cost of 2m, 4m, 10m and 13m of cloth.

Formula:

(i) Substituting x₁ = 5, y₁ = 210 and x₂ = 2 in the formula,

Gives y₂ = 84

∴The cost of 2m cloth is Rs. 84

(ii) Substituting x₁ = 5, y₁ = 210 and x₂ = 4 in the formula,

Gives y₂ = 168

∴The cost of 4m cloth is Rs. 168

(iii) Substituting x₁ = 5, y₁ = 210 and x₂ = 10 in the formula,

Gives y₂ = 420

∴The cost of 10m cloth is Rs. 420

(iv) Substituting x₁ = 5, y₁ = 210 and x₂ = 13 in the formula,

Gives y₂ = 546

∴The cost of 2m cloth is Rs. 546

Given: Cost of 1 Apple = Rs. 8

To Find: Cost of 4, 7, 12, 20 apples.

Formula:

(i) Substituting x₁ = 1 y₁ = 8 and x₂ = 4 in the formula,

Gives y₂ = 32

∴The cost of 4 apples is Rs. 32

(ii) Substituting x₁ = 1 y₁ = 8 and x₂ = 7 in the formula,

Gives y₂ = 56

∴The cost of 7 apples is Rs. 56

(iii) Substituting x₁ = 1 y₁ = 8 and x₂ = 12 in the formula,

Gives y₂ = 96

∴The cost of 12 apples is Rs. 96

(iv) Substituting x₁ = 1 y₁ = 8 and x₂ = 20 in the formula,

Gives y₂ = 160

∴The cost of 20 apples is Rs. 160

Substituting x₁ = 48, y₁ = 16800, x₂ = 36 in the formula,

Gives y₂ =

=

=

= 12600

∴The cost of 36 bags of paddy is Rs. 12,600

Given: Monthly average expenditure of 4 members = Rs. 2800

To find: Monthly average expenditure of 3 members

Monthly average expenditure of 1 member = Rs. 2800/4 = Rs. 700

Monthly average expenditure of 3 members = 3 × Rs. 700 = Rs. 2100

Given: Height of the mast in a 28m long ship = 12m

To find: Height of the mast in a 9cm long ship

Formula:

Substituting x₁ = 28, y₁ = 12, x₂ = 9 in the formula,

Gives y₂ = cm

= 3.85 cm

∴The height of the mast in a 9 cm long ship is 3.85 cm

Conclusion: Always maintain consistency in the units of x₁ and y₁, x₂ and y_2.

Given: (i) Pole Height = 5.6m, Shadow Length = 3.2m
(ii) Pole Height = 10.5 m

(iii) Shadow Length = 5m

To Find: (ii) Shadow Length

(iii) Pole Height

(i)

The length of the shadow cast by a pole 10.5m long is 6m.

(ii)

The height of the pole that casts a 5m long shadow is 8.75m.

Given: Weight of 12 sheets of paper = 40 grams

To find: Number of papers weighing

Formula:

Substituting x₁ = 12, y₁ = 40, y₂ = in the formula,

Gives x₂ =

= 5000

∴Number of papers weighing 16 kg are 5000.

Given: Speed of train = 75kmph

To Find: Distance Travelled in 20 minutes, Time required to cover 250km.

Formulation: Speed of the train is 75kmph, i.e, it requires 1 hour (60 minutes) for the train to cover 75km.

Formula:

(i) Substituting x₁ = 60 y₁ = 75 and x₂ = 20 in the formula,

Gives y₂ = 25 km

∴The train will move 25km in 20 minutes

(ii) Substituting x₁ = 60 y₁ = 75 and y₂ = 250 in the formula,

Gives x₂ = 200 minutes or 3 hours 20 minutes

∴The time required to cover a distance of 250km is 3 hours 20 minutes.

Given: Length of the microchip design = 18cm, Scale Ratio = 40:1

To find: Actual Length of microchip

Formula:

Substituting x₁ = 1, y₁ = 40 y₂ = 18 in the formula,

Gives x₂ =

= 0.45 cm

∴The actual length of the microchip is 0.45 cm

Given:

Average age of doctors and lawyers = 40 Average age of doctors = 35

Average age of lawyers = 50

To find: Ration of number of doctors to the number of lawyers

Let a = Number of doctors

b = Number of lawyers

x = total age of doctors

y = total age of lawyers

According to the problem,

ð x = 35a and y = 50b

Total age of doctors and lawyers = x+y

According to the problem,

ð

ð

ð

ð

ð

Therefore, the ratio of ages of doctors and lawyers is 2:1

x₁ = 50, y₁ = 5 x₁y₁ = 250

x₂ = 40, y₂ = 6 x₂y₂ = 240

x₃ = 30, y₃ = 7 x₃y₃ = 210

x₄ = 20, y₄ = 8 x₄y₄ = 160

x₁y_1≠x₂y_2≠x₃y_3≠x₄y₄

Therefore, x and y are not in inverse proportions.

x₁ = 100, y₁ = 60 x₁y₁ = 6000

x₂ = 200, y₂ = 30 x₂y₂ = 6000

x₃ = 300, y₃ = 20 x₃y₃ = 6000

x₄ = 400, y₄ = 15 x₄y₄ = 6000

x₁y_{1 =} x₂y_{2 =} x₃y_{3 =} x₄y₄

Therefore, x and y are in inverse proportions.

x₁ = 90, y₁ = 10 x₁y₁ = 900

x₂ = 60, y₂ = 15 x₂y₂ = 900

x₃ = 45, y₃ = 20 x₃y₃ = 900

x₄ = 30, y₄ = 25 x₄y₄ = 750

x₅ = 20, y₅ = 30 x₅y₅ = 600

x₆ = 5, y₆ = 25 x₆y₆ = 175

x₁y₁ =x₂y₂ =x₃y₃≠x₄y₄≠x₅y₅≠x₆y₆

(x₁,y₁), (x₂,y₂), (x₃, y₃) are in inverse proportions.

Formula: x_ny_n = 6000 where n = 1,2,3….

x₁ = 40, y₁ = 150 x₁y₁ = 6000

x₂ = 50, y₂ = ? x₂y₂ = 6000 y₂ =

x₃ = ?, y₃ = 100 x₃y₃ = 6000 x₃ =

x₄ = 75, y₄ = ? x₄y₄ = 6000 y₄ =

x₅ = ?, y₅ = 75 x₅y₅ = 6000 x₅ =

The completed table would be

(i) Is R₁ : R₂ = C₂ : C₁?

Yes

(ii) Is R₃ : R₄ = C₄ : C₃?

Yes

(iii) Is R and C inversely proportional to each other?

Yes

(iv) Do this activity with 36 squares.

Yes

Given: Price of potatoes = Rs. 8 per kg, Siri can buy = 5kg

To find: Amount of potatoes if Price is Rs. 10 per kg

Formula:

Substituting x₁ = 8, y₁ = 5 x₂ = 10 in the formula,

Gives y₂ =

∴Amount of potatoes Siri can buy is 4 kg

Given: Number of people living off the food stock for 70 days = 500

To find: Number of days the stock will last if 700 people live off.

Formula:

Substituting x₁ = 500, y₁ = 70 x₂ = 700 in the formula,

Gives y₂ =

∴If 200 more people join, the food will last for 50 days.

Given: Number of men required to do a work in 12 days = 36

To find: Number of days required for 9 men to do the same work

Formula:

Substituting x₁ = 36, y₁ = 12 x₂ = 9 in the formula,

Gives y₂ =

∴Number of days required for 9 men to do the work is 48 days.

Given: Distance covered in 2 hours = 28km

To find: Time taken to cover 56km in same speed.

Formula:

Substituting x₁ = 28, y₁ = 2 x₂ = 56 in the formula,

Gives y₂ = 4 hours

∴It takes 4 hours for the cyclist to cover 56km.

Given: Speed in 10 hours = 16 nautical miles per hour

To find: Increase in speed required if the ship has to cover the same distance in 8 hours.

Formula:

Substituting x₁ = 10, y₁ = 16 x₂ = 8 in the formula,

Gives y₂ = 20 nautical miles per hour

Change in speed = Final Speed – Initial Speed

20-16 = 4 nautical miles per hour.

∴ The speed must be increased by 4 nautical miles per hour for the ship to cover the same distance in 8 hours.

Given: Number of pumps required to fill a tank in 1.5 hours = 5

To find: Number of pumps require to fill the tank in 0.5 hours

Formula:

Substituting x₁ = 1.5, y₁ = 5 x₂ = 0.5 in the formula,

Gives y₂ = 15 pumps

∴ 15 pumps are required to fill the tank in 0.5 hours.

Given: Number of workers that can build a wall in 48 hours = 15

To find: Number of workers that can build a wall in 30 hours

Formula:

Substituting x₁ = 48, y₁ = 15 x₂ = 30 in the formula,

Gives y₂ = 24

∴ 24 workers can build the wall in 30 hours.

Given: Number of periods per day = 8

Duration of each period = 45 minutes

To find: Duration of each period if there are 6 periods per day

Formula:

Substituting x₁ = 8, y₁ = 45 x₂ = 6 in the formula,

Gives y₂ = 60

∴ Duration of each period will be 60 minutes if there are 6 periods.

Given: z is directly proportional to x, z x

z is inversely proportional to y, z

Therefore z

To find: Percentage increase in z if there is 12% increase in x and 20% decrease in y

z

(Assuming the proportionality constant to be 1)

Change in x 100x+12x

Change in y 100y-20y

The new z value becomes

Change in z value = New z value – Initial z value

Percentage increase in z =

40%

∴ Percentage increase in z is 40%

Given: Number of days required for x+1 men to do the work = x+1

To find: Number of days required for x+2 men to do the work

Formula:

Substituting x₁ = x+1, y₁ = x+1 x₂ = x+2 in the formula,

) (

------------

0 +1

Gives y₂ = days

∴ Number of days required for x+2 men to complete the work is days

Given: Perimeter of the rectangle P = 2(l+b) = 24m

To find: Variation of width and area with length if perimeter is fixed.

Area = lxb

Add length and breadth and multiplying by 2 should give 24.

(Note: All lengths are in cm, all areas are in cm²)

l = 3, P = 24 2(l+b) = 24 = 2(3+b) b = 9, A = 27

l = 4, P = 24 2(l+b) = 24 = 2(4+b) b = 8, A = 32

l = 5, P = 24 2(l+b) = 24 = 2(5+b) b = 7, A = 35

l = 6, P = 24 2(l+b) = 24 = 2(6+b) b = 6, A = 36

l = 7, P = 24 2(l+b) = 24 = 2(7+b) b = 5, A = 35

l = 8, P = 24 2(l+b) = 24 = 2(8+b) b = 4, A = 32

l = 9, P = 24 2(l+b) = 24 = 2(9+b) b = 3, A = 27

Observation: Length and breadth are inversely proportional to each other if perimeter is kept constant. Area first increases with length then decreases.

Given: Cost of Rice for 8 members for 20 days = Rs. 480

To find: Cost of Rice for 12 members for 15 days.

Cost of Rice for 8 members for 1 day = Rs. = Rs.24

Cost of Rice for 1 member for 1 day = Rs. = Rs. 3

Cost of Rice for 12 members for 1 day = Rs. 3x12 = Rs. 36

Cost of Rice for 12 members for 15 days = Rs. 36x15 = Rs.540

∴ Cost of Rice for 12 members for 15 days is Rs. 540

Given: Number of men working 8 hours a day for 15 days = 24

24 men working at 8 hours per day can do the work in 15 days

Amount of work = 24 x 8 x 15 = 2880 man-hour


Find the number of days needed for 20 men working 9 hours a day:

Time number of man-hour = 20 x 9 = 180

Number of days needed = 2880 ÷ 180 = 16 days

Hence, 20 men working on 9 hours can complete the work in 16 days.

Given: Number of days required for 175 men to dig a canal 3150m long = 36

To find: Number of men required to dig a canal of 3900m in 24 days.

Formula: = Constant where W = Work, WF = Workforce, T = Time

Gives WF₂ = 325

∴ Number of men required to dig a canal of 3900m in 24 days is 325

Given: Number of days required for 14 typists to type the manuscript 6 hours a day = 12 days

To find: Number of days required for 4 typists to type working 7 hours a day.

14 typists work 72 hours to complete the manuscript of a work.

14 typists complete of the manuscript in 1 hour.

1 typist completes of the manuscript in 1 hour.

4 typists complete 4× of the work in 1 hour.

4 typists complete 6× of the work in 6 hours per day

It will take = 42 days for 4 typists to complete the whole manuscript.

∴ Number of days required for 4 typists to type working 7 hours a day is 42 days.