Varsity Math, Week 139

Varsity Math 139

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Business decisions often require problem solving, as the coach points out with these puzzles.

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Pondering Productivity

Smith and Jones each manage egg farms. Chickens
Smith says: “1 1/7 of my hens lay 1 1/6 eggs in 1 1/5 days.”
Jones says: “1 1/5 of my hens lay 1 1/6 eggs in 1 1/7 days.”

(a) Is either farm more productive than the other in eggs per hen per day?
(b) Smith has 48 hens. How long must he wait to first get a whole number of eggs at the close of a day? How many eggs does his farm produce in that time?
(c) Jones has 300 hens. How many days must he wait to first get a whole number of eggs at the close of a day? How many eggs does his farm produce in that time?

Cornfield Planning

A farmer has a square plot of land that measures 100 feet on each side. She plans to grow corn in the plot, and she will install a fence around the corn. Fencing is expensive, so she wants to grow the corn in a shape that will maximize the ratio of the area of the cornfield to its perimeter.

What shape should the cornfield be?

momathlogohorizontalWSJLogo

Solutions to week 138

In Close to a Quart, the minimum number of transfers is 210. For Math Party, the children are 4, 4, and 9.

Close to a Quart answer explained:
By transferring water n = 2(a + b – 1) times, you can achieve water in one of the two containers in the amount of ae – bπ or aπ – be, making sure the expression used is positive. You must find the smallest integer value of a + b so that
.99 < ae – bπ < 1.01 or .99 < aπ - be < 1.01. By numerical search, you find 73π - 84e = 1.00059 and 57e – 49π = 1.004024 are the smallest values satisfying the above conditions. Thus a = 57 and b = 49 gives n = 210 as the minimum number of transfers.

Math Party answer explained:
The children are 4, 4, and 9 so their sum is 17 and their product is 144. Smith incorrectly guessed (3, 6, 8) for the ages. One year ago, the ages were (3, 3, 8) and Jones incorrectly guessed (2, 6, 6).

Recent Weeks

Week 138: Close to a Quart & Math Party, solutions to Room for One More & End Points

Week 137: Room for One More & End Points, solutions to Sevens and Elevens & Three-Digit Squares

Week 136: Sevens and Elevens & Three-Digit Squares, solutions to Swimming Workout & Batting Order

Week 135: Swimming Workout & Batting Order, solutions to Maximization & Multiplication Table

Week 134: Maximization & Multiplication Table, solutions to Maximizing Links & Seven Points

Links to all of the puzzles and solutions are on the Complete Varsity Math page.

Come back next week for answers and more puzzles.