Reconciling a Stacked-Discount Invoice Against a Spending Cap
An office manager is reconciling a supply order before approving payment. The invoice lists 24 reams of paper at a list price of $4.85 per ream, with a store coupon reading “20% off your order, then an additional 10% off the reduced price.” Sales tax of 7% applies to the final discounted amount. The manager's supervisor approved a cap of $100 for this order. The register receipt shows a subtotal of $83.82, tax of $5.87, and a total of $89.69. The manager's own scratch work reads: “24 × 4.85 = 116.40; 30% off = 34.92; 116.40 − 34.92 = 81.48; plus 7% = 87.18,” and she cannot tell whether the receipt or her scratch work is correct. She also needs to know how much room remains under the $100 cap, and whether the order of the two discounts could have changed the result.
1. Determine the correct order total and the amount remaining under the $100 cap. Identify the specific error in the “30% off” scratch work and quantify how much it matters on this order.
Read the worked answer
List price: 24 × $4.85 = $116.40.
Apply discounts successively, each to the new base:
- After 20% off: 116.40 × 0.80 = $93.12
- After an additional 10%: 93.12 × 0.90 = $83.808, i.e. $83.81
Tax: 83.81 × 0.07 = $5.8667 → $5.87. Total = $89.68 (the receipt's $89.69 is a one-cent rounding difference from using the unrounded subtotal).
Cap room: 100.00 − 89.69 = $10.31.
Error: the scratch work adds the two rates (20 + 10 = 30). Successive discounts multiply: 0.80 × 0.90 = 0.72, which is 28% off, not 30%. The pre-tax gap is 116.40 × 0.02 = $2.33, or about $2.49 with tax - small here, but it grows with the size of the order.
2. A colleague insists the total would change if the 10% were applied first and the 20% second. Evaluate that claim and justify your answer without recomputing from the original price.
Read the worked answer
The claim is false. Both orders produce the same product of factors: 116.40 × 0.90 × 0.80 and 116.40 × 0.80 × 0.90 both equal 116.40 × 0.72 = $83.808. Multiplication is commutative, so the final discounted base is identical; only the intermediate dollar amount differs ($104.76 after the 10% first, versus $93.12 after the 20% first).
The confusion usually comes from comparing the dollar discount in each step rather than the multiplicative factor. A 10% cut taken from a larger base removes more dollars than the same 10% taken from a smaller base, but the second discount is applied to whatever base remains - so the two sequences land on the same price.
3. The manager expects to place the same order for 240 reams next quarter. Using proportional reasoning, estimate how far the “30% off” shortcut would drift from the correct pre-tax amount, and state whether the shortcut is safe for budgeting.
Read the worked answer
Scale the list price: 240 × $4.85 = $1,164.00.
Correct pre-tax: 1,164 × 0.72 = $838.08. Shortcut pre-tax: 1,164 × 0.70 = $814.80. Difference: $23.28 pre-tax, about $24.91 after 7% tax.
The gap scales linearly with order size, because it is always 2 percentage points of the list price (0.72 − 0.70). At 24 reams it was about $2.33; ten times the quantity gives ten times the error. The shortcut understates the amount owed, so a budget built on it would be too low - the failure mode is a shortfall at payment time, not a surplus. It is not safe for budgeting; use 0.72 as the combined factor.