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Linear growth and decay · 4 marks

A telecom company charges ₹600 for a certain recharge scheme. This prepaid balance is reduced by ₹15 each day after the recharge.

  1. (i) Write an equation that models the remaining balance b(x) after using the scheme for x days. Explain why it represents linear decay.
  2. (ii) After how many days will the balance run out?
  3. (iii) Make a table of values for x varying from 1 to 10 days and show how the balance b(x), reduces with time.
Answer: (i) b(x) = 600 − 15x, linear decay (falls by ₹15 each day) (ii) 40 days (iii) ₹585, 570, 555, …, 450 for x = 1 to 10.

Step-by-step solution

Idea: Start at ₹600 and subtract a fixed ₹15 per day. After x days, 15x rupees have been used.

(i) Write an equation that models the remaining balance b(x) after using the scheme for x days. Explain why it represents linear decay.

  1. Amount used in x days = 15x, so b(x) = 600 − 15x.1 mark
  2. b(x) is linear in x with a negative coefficient (−15): the balance decreases by a constant ₹15 in each equal interval of one day. So it represents linear decay.½ mark
b(x) = 600 − 15x; linear decay.

(ii) After how many days will the balance run out?

  1. Balance runs out when b(x) = 0: 600 − 15x = 0.½ mark
  2. 15x = 600 ⇒ x = 40 days.½ mark
40 days

(iii) Make a table of values for x varying from 1 to 10 days and show how the balance b(x), reduces with time.

  1. Day, xBalance, b(x) (₹)
    1585
    2570
    3555
    4540
    5525
    6510
    7495
    8480
    9465
    10450
    1 mark
  2. The balance reduces by the same ₹15 every day.½ mark
585, 570, 555, 540, 525, 510, 495, 480, 465, 450
(i) b(x) = 600 − 15x, linear decay since it drops by a constant ₹15 each day (ii) 40 days (iii) 585, 570, 555, 540, 525, 510, 495, 480, 465, 450 rupees.

Check: b(40) = 600 − 600 = 0 ✓; b(10) = 600 − 150 = 450 ✓.

Answer to write in the exam

(i)

b(x) = 600 − 15x

b(x) decreases by a constant ₹15 each day (equal intervals)

∴ It represents linear decay

(ii)

600 − 15x = 0

15x = 600 ⇒ x = 40

∴ The balance runs out after 40 days

(iii)

x: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

b(x) (₹): 585, 570, 555, 540, 525, 510, 495, 480, 465, 450

∴ The balance reduces by ₹15 every day

Common mistakes that cost marks

  • Writing b(x) = 15x − 600 (the balance would be negative at the start).
  • Answering (ii) with 600 − 15 = 585, the balance after one day, instead of solving b(x) = 0.
  • Starting the table at x = 0. The question asks for x from 1 to 10.

How this can come in the exam

MCQ (1 mark)

A metro card has ₹450 and ₹18 is deducted for each trip. The number of trips before the balance runs out is

  1. 18
  2. 25
  3. 432
  4. 24
Show answer

(B) 25
450 − 18n = 0 ⇒ n = 25.

Assertion–Reason (1 mark)

Assertion (A): The balance b(x) = 600 − 15x is ₹300 after 20 days.
Reason (R): The balance halves every 20 days.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
Show answer

(C) A is true but R is false.
A is true: 600 − 300 = 300. R is false: after 40 days the balance is 0, not 150; it falls by equal amounts, it does not halve.

Try one yourself

A recharge of ₹399 reduces by ₹13.30 per day. Write b(x) and find when it runs out.

Show answer

b(x) = 399 − 13.30x; 13.30x = 399 ⇒ x = 30 days.

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