Reconciling a Hostel Mess Bill with Mixed Fractions and Decimals
A hostel mess has 24 resident students and charges ₹2,450 per student per month. During a reconciliation, the warden finds that 9 students have paid the full fee, 7 have paid three-fourths of the fee, 5 have paid 0.6 of the fee, and the remaining 3 have paid nothing. Separately, the cook submits a provision bill: 18¾ kg rice at ₹52.40 per kg, 7.5 kg dal at ₹118 per kg, and a credit note of ₹325.50 for returned oil. The warden's rough notebook says 'about ₹38,000 collected' and the cook's rough total is 'around ₹1,600'. The principal wants to know whether the rough figures hide a real shortage, and whether the difference is caused by rounding or by an actual arithmetic error. All rates include local taxes, and no other charges apply.
1. Calculate the exact fee collection. Is the warden's 'about ₹38,000' an overestimate or underestimate, and by how much?
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Full-fee group: 9 × 2450 = ₹22,050. Three-fourths group: each pays 0.75 × 2450 = ₹1,837.50, so 7 × 1837.50 = ₹12,862.50. 0.6 group: each pays 0.6 × 2450 = ₹1,470, so 5 × 1470 = ₹7,350. Non-payers add ₹0. Exact total = 22,050 + 12,862.50 + 7,350 = ₹42,262.50. The warden's ₹38,000 is an underestimate by 42,262.50 − 38,000 = ₹4,262.50. That is too large to dismiss as ordinary rounding.
2. Compute the net provision bill exactly. If the cook is handed ₹1,600, what change should be returned, and does the rough total create a cash risk?
Read the worked answer
Rice: 18.75 × 52.40 = ₹982.50. Dal: 7.5 × 118 = ₹885.00. Credit note: −₹325.50. Net bill = 982.50 + 885.00 − 325.50 = ₹1,542.00. The cook's 'around ₹1,600' is ₹58 higher than the exact bill. If ₹1,600 is handed over, change of ₹58 is due. Paying ₹1,600 without recording the change would leave a ₹58 cash shortfall, so the rough total is not safe as a payment instruction.
3. Suppose each student's payment is rounded to the nearest ₹100 before totalling. Find the rounded total, the net error, and identify which group contributes the largest rounding error.
Read the worked answer
Round each payment to the nearest ₹100: Full fee: 2450 → 2500; 9 × 2500 = 22,500. Exact was 22,050, so error = +₹450. Three-fourths: 1837.50 → 1800; 7 × 1800 = 12,600. Exact was 12,862.50, so error = −₹262.50. 0.6 fee: 1470 → 1500; 5 × 1500 = 7,500. Exact was 7,350, so error = +₹150. Rounded total = 22,500 + 12,600 + 7,500 = ₹42,600. Net error = 42,600 − 42,262.50 = +₹337.50. The largest single distortion is the full-fee rounding (+₹450). The warden's separate ₹38,000 guess cannot be explained by nearest-₹100 rounding, because rounded categories give ₹42,600, not ₹38,000.