Which of the following vary directly and which vary inversely with each other and which are neither of the two?
(i) Number of students in a hostel and consumption of food.
(ii) Area of the walls of a room and the cost of white washing the walls.
(iii) The number of people working and the quantity of work.
(iv) Simple interest on a given sum and the rate of interest.
(v) Compound interest on a given sum and the sum invested.

Solution:

(i) The number of students in a hostel and the food consumption are directly related.

The more the number of students in the hostel the more food will be consumed.

(ii) The area of the walls of a room and the cost of white washing them vary directly with each other. In other words the larger the area of the walls the higher will be the cost of whitewashing

(iii) The number of people working and the quantity of work will vary directly. The more the number of people working the more the quantity of work will be performed.

(iv) Simple Interest on a given sum and the rate of interest are directly proportional or related to each other. If the rate of interest is higher the simple interest on a given sum will be higher. The formula for simple Interest(SI) is given as:

SI = (P × R × T)/100

Where P = sum invested; R = Rate of Interest; T = Time for which invested

So it is evident from the formula that when R increases the simple interest also Increases.

(v) Compound interest(CI) on the sum given and the sum invested are not related to each other either directly or inversely because there are other variables like rate of interest and time which finally decide the amount of compound interest.

The formula with the help of which Compound Interest is calculated is given as: A = P(1 + r/100)ⁿ. CI = A - P. So as principal increases the Amount increases and hence CI (= A - P) also increases.But the factors rate of interest and time also decide the amount of compound interest.

✦ Try This: Which of the following vary directly and which vary inversely with each other and which are neither of the two?

(i) The amount of rice procured per month for the kitchen of the hostel having 100 students is 2 quintals. If the number of students increases by 25 then how much rice will be required?

Number of students Quantity of Rice(Quintals)

100 2

125 x = ?

Rice Required for 125 students = 2 × (125/100) = 2 × (5/4) = 2.5 quintals

Hence the more students, the more rice will be required.

(ii) The cost of white washing the walls of the house is Rs. 150 per square feet. If the area of the walls is 1600 square feet what will be the cost of white washing the walls? If the area of the walls is 2000 sq. ft. then what will be the cost of white washing?

Cost of white washing the walls(Area 1600 sq. ft.) = Rs.150 × 1600 = Rs.2,40,000

Cost of white washing the walls(Area 2000 sq ft.) = Rs. 150 × 2000 = Rs.3,00,000

We thus see that greater is the area of the walls the higher is the cost of white washing them.

(iii) If the 40 workers finish a work in 10 days then 60 workers will finish the work in how many days? Also how much work does 60 workers finish in one day when compared to 40 workers?

No. of Workers No. of Days Taken to finish a job

40 10

60 x = ?

No.of days taken by 60 workers to finish the work = 10 × (40/60) = 20/3 or 6 2/3 days

40 workers 1/10th of the work in one day while 60 workers finish 3/20th of the work in one day. We know that 3/20 > 1/10 and this means that more workers do more work.

(iv) Calculate the Simple Interest(SI) on a sum of Rs. 2000 for two years if the rate of interest is a) 6% per annum and ii) 7% per annum

(a) At 6% rate of Interest per annum

SI = (2000× 6 × 2)/100 = Rs. 240

(b) At 7% rate of interest per annum

SI = (2000 × 7 × 2)/100 = Rs. Rs. 280

Hence we conclude that the SI is directly related to the rate of interest if the sum is fixed.

(v) Calculate the Compound Interest for 2 years on (a) Rs. 1500 and (b) Rs. 2000 if the sums are invested at the rate of 6% and 3% per annum compounded yearly

(a) The Amount is calculated as:

A = P(1 + r/100)ⁿ

P = 1500; r = 6% per annum compounded annually; n = 2 years

A = 1500(1 + 6/100)²

A = 1500(1.06)²

A = 1500 × 1.1236

A = Rs. 1685.40

CI = Rs. 1685.40 - Rs. 1500 = Rs. 185.40

(b) The Amount is calculated as:

A = P(1 + r/100)ⁿ

P = 2000; r = 3% per annum compounded annually; n = 2 years

A = 2000(1 + 3/100)²

A = 2000(1.03)²

A = 2000 × 1.0609

A = Rs. 2121.80

CI = Rs. 2121.80 - Rs. 2000 = Rs. 121.80

We therefore see that even as the principal increases the compound interest decreases because the interest rate decreases. The compound interest might decrease if the time period changes. Hence compound interest is directly related to the principal invested only if the time period of investment and rate of interest is constant.

☛ Also Check: NCERT Solutions for Class 8 Maths Chapter 13

NCERT Exemplar Class 8 Maths Chapter 10 Problem 61

Which of the following vary directly and which vary inversely with each other and which are neither of the two? (i) Number of students in a hostel and consumption of food.(ii) Area of the walls of a room and the cost of white washing the walls.(iii) The number of people working and the quantity of work. (iv) Simple interest on a given sum and the rate of interest.(v) Compound interest on a given sum and the sum invested.

Summary:

i) The number of students in a hostel and the food consumption are directly related.
ii) The area of the walls of a room and the cost of white washing them vary directly with each other.
iii) The number of people working and the quantity of work will vary directly
iv) Simple Interest on a given sum and the rate of interest are directly proportional or related to each other.
v) Compound interest(CI) on the sum given and the sum invested are neither directly or inversely related to each other

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