Great Picnic Puzzle

Number Theory Level 2

When they started off on the great annual picnic every wagon in town was pressed into service, each one carrying the same number of people.

Half way to the picnic ground ten wagons broke down, so it was necessary for each of the remaining wagons to carry one more person.

When they started for home it was discovered that fifteen more wagons were out of commission, so on the return trip there were three persons more in each wagon than when they started out in the morning.

Now who can tell how many people attended the great annual picnic?

The answer is 900.

2 solutions

Michael Mendrin
Aug 1, 2018

Let $(p, w, r)$ be the number of people, number of wagon at the start, and number of riders in each wagon. Then we solve the three simultaneous equations

$r=\dfrac{p}{w}$

$r+1=\dfrac{p}{w-10}$

$r+3=\dfrac{p}{w-25}$

which gets us $(p,w,r)=(900,100,9)$

Ilya Rudnitskiy
Jul 31, 2018

There must have been 900 picnickers who would be seated 9 to a wagon if there were 100 vehicles, or 10 to a wagon after 10 of the wagons had broken.

When they started for home with 75 wagons, it was necessary for 12 persons to ride in each wagon (3 more than the 9 per wagon in the morning).

Algebra solution:

wp = (w-10)(p+1) = (w-25)(p+3)

Now solve!

Expand the brackets: wp = wp +w -10p -10 = wp +3w -25p -25 Subtract wp from each: 0 = w -10p -10 = 3w -25p -75 Multiply middle part by 3: 0 = 3w -30p -30 = 3w -25p -75 Subtract middle from last: 0 = 3w -30p -30 = 3w -25p -75 - (3w -30p -30) Simplify: 0 = 5p -45 p = 9

Great Picnic Puzzle

The answer is 900.

2 solutions

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