Two Printing Quotes and a Fixed Budget
The Eco Club of a school wants to print 200 copies of a 24-page magazine. Two printing shops have given quotes. Shop P charges a fixed ₹400 for plate-making plus ₹18 for every copy printed. Shop Q charges a fixed ₹150 plus ₹22 per copy. A third option, digital printing at ₹35 per copy with no fixed charge, is available only for orders of 40 copies or less. All three shops add 12% GST to the bill. The club has collected ₹5,000. The teacher-in-charge insists on at least 200 copies and has offered to make up any shortfall from the school fund, but the club president wants to know exactly how far the ₹5,000 will stretch if the whole bill, including GST, must stay within it. The treasurer also points out that the magazine is popular and the club may want about 400 copies next year.
1. For what order sizes is Shop Q cheaper than Shop P? Find the exact crossover point.
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Let the number of copies be n. Since GST of 12% is applied to both bills, it multiplies both sides equally and does not affect which shop is cheaper - compare pre-tax costs.
P: 400 + 18n Q: 150 + 22n
Set them equal: 400 + 18n = 150 + 22n → 250 = 4n → n = 62.5.
So the two shops are never exactly equal for a whole number of copies:
- n = 62: P = 400 + 1116 = ₹1,516; Q = 150 + 1364 = ₹1,514 → Q cheaper by ₹2.
- n = 63: P = 400 + 1134 = ₹1,534; Q = 150 + 1386 = ₹1,536 → P cheaper by ₹2.
Q is cheaper for 62 copies or fewer; P is cheaper for 63 copies or more. The small fixed-charge gap (₹250) is wiped out by Q's higher rate of ₹4 per copy.
2. If the entire bill including GST must stay within ₹5,000, what is the largest number of copies the club can order from Shop P?
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Write the condition with GST inside the bracket:
1.12 × (400 + 18n) ≤ 5000 400 + 18n ≤ 5000 ÷ 1.12 = 4464.29 18n ≤ 4064.29 n ≤ 225.79
Since n must be a whole number, n = 225.
Check: 400 + 18(225) = 400 + 4050 = ₹4,450; with GST, 1.12 × 4450 = ₹4,984 - within budget. Next value: 400 + 18(226) = ₹4,468; with GST, 1.12 × 4468 = ₹5,004.16 - over budget.
So the club can afford 225 copies, which is above the required 200 but short of 400. A common error is to divide ₹5,000 by ₹18, which ignores both the fixed charge and the GST.
3. The treasurer says: 'At 400 copies the cost per copy is lower than at 200 copies, so printing 400 is always better value.' Evaluate this reasoning.
Read the worked answer
The arithmetic behind the claim is correct, but the conclusion does not follow.
At Shop P, including GST:
- 200 copies: 1.12 × (400 + 3600) = 1.12 × 4000 = ₹4,480 → ₹22.40 per copy.
- 400 copies: 1.12 × (400 + 7200) = 1.12 × 7600 = ₹8,512 → ₹21.28 per copy.
The per-copy cost falls by ₹1.12 because the fixed ₹400 is spread over twice as many copies. But the total outlay rises by ₹4,032, and the club must find that money. Per-copy cost measures efficiency, not value.
Printing 400 is better only if the extra 200 copies are actually distributed or sold; otherwise the club pays ₹4,032 to store waste paper. Also, the ₹5,000 ceiling is already exceeded at 400 copies, so the treasurer's option would require extra funding, not just a smarter choice.