In a grapes puzzle, if the total number of grapes eaten over five days is 120 with an daily increase of 6 grapes, what was the number eaten on the first day?

In a grapes puzzle, if the total number of grapes eaten over five days is 120 with an daily increase of 6 grapes, what was the number eaten on the first day?

• A. 12
• B. 10
• C. 8
• D. 14

Answer: (A)

Think of this as an arithmetic sequence where each day the amount increases by a fixed amount. If the first day is some amount and the daily increase is six, the five daily amounts are: first day, first day plus six, first day plus twelve, first day plus eighteen, first day plus twenty-four. Adding them gives five times the first-day amount plus the sum of the increments, which is zero plus six plus twelve plus eighteen plus twenty-four — that total is sixty. So the overall total over five days is five times the first-day amount plus sixty. We know the grand total is one hundred twenty, so five times the first-day amount plus sixty equals one hundred twenty. Subtract sixty to get five times the first-day amount equals sixty, then divide by five to obtain the first-day amount as twelve. Therefore, the number eaten on the first day was twelve.

Think of this as an arithmetic sequence where each day the amount increases by a fixed amount. If the first day is some amount and the daily increase is six, the five daily amounts are: first day, first day plus six, first day plus twelve, first day plus eighteen, first day plus twenty-four. Adding them gives five times the first-day amount plus the sum of the increments, which is zero plus six plus twelve plus eighteen plus twenty-four — that total is sixty. So the overall total over five days is five times the first-day amount plus sixty. We know the grand total is one hundred twenty, so five times the first-day amount plus sixty equals one hundred twenty. Subtract sixty to get five times the first-day amount equals sixty, then divide by five to obtain the first-day amount as twelve. Therefore, the number eaten on the first day was twelve.