Translate an ancient math problem. Classical Chinese math problem solving

Translated into modern Chinese to the effect of:

When the swing is still hanging, the pedal height from the ground is 1 foot. Now a two-step distance is swayed, and someone records that the pedal is 5 feet above the ground. Ladies and ladies compete for swings, laughing and laughing all day long; The craftsmen are curious about how long the swing rope is.

Answer: The chord length of the fan-shaped swing that can be calculated is root number (100 + 16) = 2 root number (29).

From the perpendicular line of the center of the circle, the tangent angle of the chord is equal to half of the angle of the center of the circle, and the obtained right triangle is similar to the triangle where the tangent angle of the chord is located, and the radius (i.e., the length of the rope) can be obtained from the proportional relationship of the corresponding line segments.

l=r=ruler.

That is, the length of the rope is a ruler.

Ancient Dimensions: The length of the cable is: six feet seven inches.

Ancient scale (one foot).

quotient centimeters. Sengoku centimeters.

Western Han centimeters. New centimeters.

Eastern Han centimeters. Three Kingdoms Wei centimeters.

Three Kingdoms Wu centimeters.

Western Jin centimeters. After cooling cm.

North cool centimeters. South Dynasty centimeters.

Northern Wei centimeters. Sui centimeters.

Don centimeters. Northern Song Dynasty centimeters.

Southern Song centimeters. Ming centimeters.

Clear centimeters.

"Nine Chapters of Arithmetic": "Today, there are women who are good at weaving, doubling every day, weaving five feet in five days, asking how much they weave in the day? ”

There is a woman who is very good at weaving, the number of weaving per day is twice that of the previous day, and she weaves 5 feet in 5 days.

This is a proportional series problem.

"Today, there are good deeds to do a hundred steps, and those who are not good to do sixty steps, and those who are not good to do a good deeds are a hundred steps ahead, and those who are good deeds are chasing after them, asking how many steps can be reached? ”

That is, today there is a person who is good at walking and a person who is not good at walking has walked 60 steps. Now people who are not good at walking walk 100 steps first, let people who are good at walking catch up, and ask how many steps to catch up?

Solution: Suppose the good doer walks 100 steps a minute, the bad doer walks 60 steps, and the good doer catches up with the bad doer in x minutes.

So: 100 + 60x = 100x

If x=100*2, 5=250 steps, then the good deed will catch up after 250 steps.

Answer: Two hundred and fifty steps.

The technique says: Put 100 steps for those who do good, reduce 60 steps for those who do not do good, and take the remaining 40 steps as the law. Take one hundred steps for those who do good, and take one hundred steps for those who do not do good. It's a step away.

Now there is a person who walks a 100 steps in a unit of time, and a person who is not good at walking B walks 60 times in a unit. Now B goes 100 cloth first, A chases the right B, and asks how long it will take to catch up. Solution:

Set the required x unit time. Then 100x=100+60x, the solution yields x=

The "100 steps" mentioned in the title and the original title are not the 100 steps we have taken arbitrarily now. In ancient times, "step" was a unit of length, 1 step = 5 feet, about the current meter. We must pay attention to this when solving this problem.

As for the "good doers" and "bad doers", it is easier to understand that they refer to two pedestrians who walk fast and those who walk slowly.

This question can be expressed in layman's terms.

There are two people, A who can walk and B who can't walk, and at the same time, A walks 100 steps and B can only walk 60 steps. Now, B has taken 100 steps before A starts chasing B. Q: How many steps does A have to take to catch up with B?

In "Nine Chapters of Arithmetic", a very interesting solution to this problem is given. The solution to the "Shuwen" and the answer in the book are:

100 steps for those who do good deeds minus 60 steps for those who do not do good, and the remaining 40 steps for the Dharma; Take one hundred steps for those who do good, and take one hundred steps for those who do not do good. It's a step away. Answer: Two hundred and fifty steps. ”

The "law" in the text is the name of the ancients for the "divisor", and "real" is the name of the ancients for the "dividend". "One step by the truth" is to divide the obtained "real (dividend)" by the "law (divisor)" to get the step required by the question.

This solution of the ancients, if expressed in the current formula, can be.

100-60=40………As a "law" (divisor).

100×100=10000………as "real" (dividend).

10000 40 = 250 (step) ......Summing up these steps, it is.

100 100 (100-60) = 10000 40 = 250 (steps).

Answer: A good doer walks 250 steps to catch up with a bad doer.

Why is this calculated?

Based on the basic quantitative relationship of "chasing the problem".

Distance Separated (Speed Difference) = Catch-up Time.

It can be seen that the time it takes for the good doers to catch up with the bad doers is.

100 (100-60) = units of time).

And in this "unit of time", the number of steps that a good deer needs to take is.

100 steps).

That's the answer to this question.

If it is listed as a composite calculation, it can be.

The earliest to put forward and describe this mathematical problem was the topic of "things do not know the number" in the mathematical work "Sun Tzu's Arithmetic" during the Northern and Southern Dynasties. The title of this "things don't know how to count" goes like this:

There are some things that are not known in number. If you count it in threes, there are two left; If five of five are counted on the ground, there are three left at the end; If seven and seven are counted in the ground, there will be two left. Q: How many of these things are there? ”

Not as you understand it. In fact, 70 is divisible by 5 and 7 but divided by 3, 1 by 21, divisible by 3 and 7, but divisible by 5 by 1, 15 divisible by 3 and 5 but divided by 7 by 1. In the problem, this number is divided by 3 by 2, then 70 is multiplied by 2, 5 is divided by 3, then 21 times 3, 7 divided by 2, then 15 times 2, and add.

Depending on the situation, minus the multiples of the least common multiple. Subtract 105 by 2 times to get 23.

This system algorithm was obtained after the study of Qin Jiushao, a mathematician in the Southern Song Dynasty.

This is known as the Chinese remainder theorem.