Question
A picnic organiser needs at least 24 loaves of bread. A baker sells loaves for $6 each and gives one additional free loaf for every complete group of four loaves paid for. All free loaves are added only after the paid order is counted: they do not themselves earn more free loaves. The organiser must place one order with this baker and may keep any surplus bread.
What is the minimum number of loaves the organiser must pay for? How many loaves will arrive, how many will be spare, and what will the order cost?
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Hint 1
If you pay for \(n\) loaves, only complete blocks of four paid loaves produce a bonus. Express the number received using a floor.
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Solution
For \(n\) paid loaves, the number of qualifying groups of four is \(\lfloor n/4\rfloor\). Since the baker never treats free bread as additional paid bread, the delivered total is exactly
\[D(n)=n+\left\lfloor\frac n4\right\rfloor.\]
The demand requires \(D(n)\ge24\). Paying for 19 gives \(D(19)=19+4=23\), which is insufficient. Paying for 20 gives \(D(20)=20+5=25\), which meets demand with one extra loaf. Because \(D(n)\) is nondecreasing in the nonnegative integer \(n\), failure at 19 proves that no smaller order works. The minimum is therefore \(n=20\), costing \(20(\$6)=\boxed{\$120}\), and it delivers 25 loaves, leaving one spare.
The fifth free loaf appears only at paid quantity 20. Counting previous free loaves towards another four-loaf group would change the promotion's rules and incorrectly lower the paid count.
Assumptions
The order is made only with the specified baker, without mixing offers.
Only paid loaves count toward each complete group of four; free loaves do not recurse.
Whole loaves must be purchased and delivered; one or more surplus loaves are allowed.
Final Answer
Pay for 20 loaves, receive 25 in total (one spare), and spend $120.