Photocopy Plans: When Two Lines Meet and When They Never Do
A stationery shop beside a junior college advertises three monthly photocopy plans for teachers. Plan P: ₹120 fixed charge per month plus ₹1.50 per copy. Plan Q: no fixed charge, ₹2.50 per copy. Plan R: ₹120 fixed charge per month plus ₹2.50 per copy. Ms. Rao, a lecturer, expects to make about 90 copies a month but is not sure of the exact number. The shopkeeper tells her: 'Plan P and Plan Q cost exactly the same when you make 100 copies; Plan Q and Plan R can never cost the same, whatever the number of copies.' He also claims that Plan R is simply Plan Q with a fixed charge added, so the two graphs must be parallel. Ms. Rao wants a recommendation that stays sensible even if her monthly copying rises to about 200 pages during examination weeks, and she asks you to settle the shopkeeper's statements first.
1. Write the cost equations for Plans P and Q, find the number of copies at which they cost the same, and test the shopkeeper's 100-copy claim.
Read the worked answer
Let x = number of copies per month and y = monthly cost in ₹.
P: y = 120 + 1.5x
Q: y = 2.5x
Equal cost: 120 + 1.5x = 2.5x → 120 = x → x = 120 copies, and y = 2.5 × 120 = ₹300.
Interpretation: below 120 copies Q is cheaper (the fixed charge is not yet recovered); above 120 copies P is cheaper.
At 100 copies: P = 120 + 150 = ₹270, Q = ₹250 - not equal. The shopkeeper's claim is false; the correct break-even is 120 copies. He appears to have confused the ₹120 fixed charge with 120 copies.
2. Examine the shopkeeper's two statements about Plans Q and R, and compare them with the pair P and R.
Read the worked answer
Q: y = 2.5x (slope 2.5, y-intercept 0). R: y = 120 + 2.5x (slope 2.5, y-intercept 120).
Equal slopes with unequal intercepts → parallel lines with no common point, so no number of copies makes Q and R equal. Both his 'never equal' claim and his 'parallel' reasoning are correct.
Now compare P and R: y = 120 + 1.5x and y = 120 + 2.5x. Same intercept ₹120, different slopes, so they meet at x = 0, y = 120 - the plans are equal only when she makes no copies at all. That is a genuine solution of the pair of equations but meaningless in context; a good answer says so instead of quietly discarding it.
3. Recommend a plan for Ms. Rao, given that her usage is around 90 copies in normal months but about 200 in examination months.
Read the worked answer
Costs at x = 90: P = 120 + 135 = ₹255; Q = 2.5 × 90 = ₹225; R = 120 + 225 = ₹345. Cheapest: Q.
Costs at x = 200: P = 120 + 300 = ₹420; Q = ₹500; R = ₹620. Cheapest: P.
So no single plan is cheapest across her whole range - the recommendation must depend on volume. Plan R is dominated by Plan Q at every copy count and should be rejected outright.
A realistic comparison for, say, 6 normal months at 90 copies and 4 examination months at 200 copies:
P = 6(255) + 4(420) = 1,530 + 1,680 = ₹3,210
Q = 6(225) + 4(500) = 1,350 + 2,000 = ₹3,350
P saves ₹140 overall, but only if the heavy months really happen; if all months stay near 90 copies, Q saves ₹30 a month. State the assumption explicitly, because that is where the judgement lies.