A chicken farmer has identified that a hen and also a fifty percent can place an egg and also a half in a day and a half.How many hens walk the farmer require to create one dozen eggs in six days?

This is a standard problem that many people get wrong because they reason that half of a hen can not lay an egg, and also a hen can not lay fifty percent an egg. However, us can acquire a satisfactory solution bytreating this together a completely mathematical problem where the numbers represent averages.

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To deal with the problem, we first need to uncover the price at i m sorry the hens lay eggs.The problem can be stood for by the complying with equation, wherein RATEis the variety of eggs created per hen·day:1½ hen × 1½ days × price = 1½ eggs

We convert this to fractions thus:3/2 hen × 3/2 days × price = 3/2 eggsMultiplying both political parties of the equation by 2/3, we get:1 hen × 3/2 work × rate = 1 eggMultiplying both political parties of the equation again through 2/3 and also solving because that RATE, us get:RATE = 2/3 eggs per hen·dayNow the we understand the rate at which hens lay eggs, we can calculate how countless hens (H)can create 12 egg in 6 days making use of the following equation:H hens × 6 job × 2/3 eggs every hen·day = 12 eggsSolving because that H, us get:H = 12 eggs /(6 days × 2/3 eggs every hen·day) = 12/4 = 3 hensTherefore, the farmer requirements 3 hens to produce 12 eggs in 6 days.

Steve Salazar Jr. Sent out the adhering to comment:

Maybe a tiny chart will help some of her readers know "The Chicken and Egg Problem".I likewise thought you were wrong until I go this tiny chart.

 notes hen egg job 1.5 eggs/1.5 hens/1.5 days 1.5 1.5 1.5 copy the hens will twin the eggs, yet not the days. 3 3 1.5 climate doubling the work will dual the eggs, but not the hens. 3 6 3 dual the days again (to with 6 days) will dual the eggs again, however not the hens. 3 12 6

Reader Dan sent out the following comment:

I think her answer is wrong because that The Chicken and Egg Problem.After a load of overcomplex calculations you arrive at the wrong answerof "Therefore, the farmer demands 3 hen to create 12 eggs in 6 days."The correct answer is 2.A hen and a half can put an egg and also a fifty percent in a day and a fifty percent = 1 hen lays 1 egg per day. To leveling it, just dual it:3 hen lay 3 egg in 3 days = 1 hen lays 1 egg in 1 day.So 2 hens lay 12 eggs in 6 days.

Unfortunately, Dan has actually made a wrong by multiplying everything by 2, thereby transforming the rate of egg production. In the initial problem, the rate in eggs per hen·day is:

1.5 eggs rate = ——————————————————— = 2/3 eggs/hen·day 1.5 hens × 1.5 daysBy the rule of algebra, you can multiply both the numerator and also denominator through 2, i beg your pardon is the exact same as multiply by one. So, girlfriend can twin the variety of eggs come 3 in the numerator, and, in the denominator, friend can twin the hen or twin the days, but you cannot double both. By multiplying everything by 2, Dan has actually really multiplied the rate by 2/(2×2) or 1/2,changing that from 2/3 to 1/3.

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Regarding the "Chicken and also the Egg" problem,I to be wondering if you might share v me why reader Dan"s math is false?In her own job-related of the equation, I have actually noticed that you neglected tomultiply the days of the equation through 2/3 in your 2nd step. Can you pleaseexplain to me why you neglected to execute so?

Let me attend to the 2nd question first. When we main point the equation by 2/3, we have:

2/3×(3/2 hens × 3/2 days × RATE) = 2/3×(3/2 eggs)1 hen × 3/2 days × rate = 1 eggBy the commutative legislations of algebra, over there is no difference between the equation above and: 3/2 hen × work × rate = 1 egg The image above may clarification why reader Dan made a mistake. He multiplied whatever times two, doubling the time period, and also doubling the variety of hens. He taken this to median thatthree hens will lay 3 eggs in three days, and also therefore that one hen lays one egg per day, which is wrong.The exactly postulation of the trouble is shown listed below where a hen and half lays one egg and a half in a day and also a half,and through doubling the time period, this very same hen and a fifty percent lays 3 egg in 3 days. This provides a rate: Domestication of chickens take it place an ext than 8,000 years ago in what is now Thailand and also Vietnam, the region in which red junglefowl are uncovered today. Chicken bones dating to around 7,500 years ago have been uncovered at Neolithic sites at the mouth the the Yellow river in China. Evidence of chicken domestication 4,000 years ago has also been uncovered in Pakistan.