Two Data Plans and a Rs 2,000 Cap
Your cadet scholarship pays a monthly internet allowance capped at Rs 2,000; anything above the cap must come from your own pocket, which you can afford only in small amounts. Two providers serve the hostel, and both bill per whole gigabyte with no carry-over of unused data. Plan A charges a fixed Rs 1,200 per month, includes 10 GB, and bills Rs 60 for each additional gigabyte. Plan B charges a fixed Rs 1,800 per month, includes 15 GB, and bills Rs 30 for each additional gigabyte. Your data counter shows 19, 22, 18 and 24 GB over the last four months, and you expect the same range for the coming year. Plan B's operator also offers a 10 per cent reduction on the fixed charge if the full year is paid in advance - Rs 19,440 now - which your savings could just cover. You must choose one plan for twelve months.
1. Write a cost expression for each plan for a month of x GB (x at least 15) and find the usage at which the plans cost the same before any discount.
Read the worked answer
For x at least 15, the included data must be subtracted before the per-gigabyte charge applies:
Plan A: C(A) = 1200 + 60(x − 10) Plan B: C(B) = 1800 + 30(x − 15)
Simplify: C(A) = 600 + 60x and C(B) = 1350 + 30x.
Set equal: 600 + 60x = 1350 + 30x, so 30x = 750 and x = 25 GB.
Check: at 25 GB, Plan A = 1200 + 60(15) = 2100 and Plan B = 1800 + 30(10) = 2100. Correct.
Interpretation matters more than the algebra. Plan A has the lower fixed charge, so it wins for light use; Plan B has the lower marginal rate (Rs 30 against Rs 60 per extra gigabyte), so its cost line is flatter and it overtakes only after a large usage. Below 25 GB, A is cheaper; above 25 GB, B is cheaper. Note the domain: the expressions hold only for x at least 15. For x between 10 and 15 the correct comparison is C(A) = 600 + 60x against the flat C(B) = 1800, and for x of 10 or less it is 1200 against 1800. Ignoring those branches is the commonest error in this kind of comparison.
2. Using the observed 18–24 GB range and the Rs 2,000 allowance, which plan do you choose, and what caveat must you accept?
Read the worked answer
Compute both plans across the observed range rather than at the average, because the cap is a hard constraint.
At 18 GB: A = 600 + 1080 = 1680; B = 1350 + 540 = 1890. At 19 GB: A = 1740; B = 1920. At 22 GB: A = 1920; B = 2010. At 23 GB: A = 1980; B = 2040. At 24 GB: A = 2040; B = 2070.
Plan A is cheaper at every point in the range, by Rs 150–210, so choose A. But the cap bites: at 24 GB, Plan A costs Rs 2,040, which is Rs 40 over the allowance; Plan B is Rs 70 over. The average of the four months is 20.75 GB, where A costs about Rs 1,845 - comfortable - but an average hides the bad month.
Caveats to accept knowingly: (a) A gives less headroom, since at 23 GB it is already at Rs 1,980; (b) one heavy month costs you a small top-up from your own pocket, which is acceptable only if you can actually pay it; (c) the ranking flips above 25 GB, so if usage drifts upward (new phone, online classes) the decision must be revisited.
Decision: take Plan A, set a personal alert at 22 GB, and keep Rs 100 in reserve for the occasional 24 GB month.
3. Now apply the 10 per cent discount on Plan B's fixed charge. Recompute the indifference point, revise your advice, and state what the discount costs you beyond money.
Read the worked answer
The discount applies only to the fixed charge: 1800 × 0.9 = 1620. So C(B') = 1620 + 30(x − 15) = 1170 + 30x for x at least 15.
Equality against Plan A: 600 + 60x = 1170 + 30x, so 30x = 570 and x = 19 GB.
Check: at 19 GB, A = 1740 and B' = 1620 + 120 = 1740. Yes.
Above 19 GB, B' is cheaper. At 24 GB: B' = 1620 + 270 = 1890 against A = 2040 - Rs 150 cheaper, and inside the Rs 2,000 cap.
Revised advice: if you expect 20 GB or more in most months, take the discounted Plan B. If usage could fall to 18 GB or less, Plan A is still cheaper (A at 18 = 1680 against B' at 18 = 1710) and carries no lock-in.
What the discount costs beyond money: Rs 19,440 of prepayment is a year of cash committed upfront, which is a real risk if you leave the hostel, fall ill, or the operator's service degrades - refunds are rarely quick. It also removes your annual choice, so a better offer appearing in six months cannot be taken. A 10 per cent saving on the fixed charge, worth Rs 2,160 a year, has to be weighed against that loss of flexibility, not just against the monthly bill.