A person is choosing between two mobile plans.
Plan A: ₹50 monthly fee + ₹0.20 per minute of call time.
Plan B: ₹30 monthly fee + ₹0.30 per minute of call time.
For how many minutes of calling per month is Plan A cheaper than Plan B? For how many minutes is Plan B cheaper? Also find the number of minutes at which both plans cost the same.
Step-by-step solution
Idea: Write each monthly cost as a linear expression in the minutes x: yA = 50 + 0.2x, yB = 30 + 0.3x. Find where they are equal; on either side, the plan with the smaller per-minute rate wins for large x.
- Let x = minutes per month. Cost of Plan A: y = 50 + 0.2x. Cost of Plan B: y = 30 + 0.3x.1 mark
- Equal cost: 50 + 0.2x = 30 + 0.3x ⇒ 20 = 0.1x ⇒ x = 200 minutes (each plan costs 50 + 40 = ₹90).1 mark
- Plan A is cheaper when 50 + 0.2x < 30 + 0.3x ⇒ 20 < 0.1x ⇒ x > 200. E.g. 300 min: A = ₹110, B = ₹120.1 mark
- Plan B is cheaper when x < 200. E.g. 100 min: A = ₹70, B = ₹60. (On a graph, the two lines cross at (200, 90).)1 mark
Check: At 200 minutes: A = 50 + 40 = 90, B = 30 + 60 = 90 ✓.
Answer to write in the exam
Cost A = 50 + 0.2x; Cost B = 30 + 0.3x (x = minutes)
Equal: 50 + 0.2x = 30 + 0.3x ⇒ 0.1x = 20 ⇒ x = 200
A cheaper: 50 + 0.2x < 30 + 0.3x ⇒ x > 200
B cheaper: x < 200
∴ Same cost (₹90) at 200 min; A cheaper above 200 min, B cheaper below 200 min
Common mistakes that cost marks
- Deciding by the monthly fee alone (B looks cheaper because ₹30 < ₹50).
- Writing 0.1x = 20 ⇒ x = 2 or 20 instead of 200.
How this can come in the exam
Gym X charges ₹500 joining + ₹300 per month; Gym Y charges ₹200 joining + ₹350 per month.
(i) Write the cost of each for n months. (ii) After how many months is the total cost equal? (iii) Which is cheaper for one year?
Show answer
(i) X: 500 + 300n; Y: 200 + 350n (1 mark). (ii) 500 + 300n = 200 + 350n ⇒ n = 6 (2 marks). (iii) 12 months: X = 4100, Y = 4400 ⇒ X cheaper (1 mark).Try one yourself
Taxi P: ₹40 + ₹12 per km; Taxi Q: ₹60 + ₹10 per km. For what distance do they cost the same?
Show answer
40 + 12d = 60 + 10d ⇒ d = 10 km (₹160 each).
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