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10 entertaining problems from an old arithmetic textbook
Recreation / / December 29, 2020
These tasks were included in "Arithmetic" by L. F. Magnitsky is a textbook that appeared at the beginning of the 18th century. Try to solve them!
1. Keg of kvass
One person drinks a keg of kvass in 14 days, and together with his wife he drinks the same keg in 10 days. How many days will a wife drink a keg alone?
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Find a number that can be divisible by either 10 or 14. For example, 140. In 140 days, a person will drink 10 kegs of kvass, and together with his wife - 14 kegs. This means that in 140 days the wife will drink 14 - 10 = 4 kegs of kvass. Then she will drink one keg of kvass in 140 ÷ 4 = 35 days.
2. On the hunt
The man went hunting with a dog. They were walking in the forest, and suddenly the dog saw a hare. How many jumps will it take to catch up with the hare, if the distance from the dog to the hare is 40 dog jumps and the distance that the dog travels in 5 jumps, the hare runs in 6 jumps? It is understood that the races are done simultaneously by the hare and the dog.
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If the hare makes 6 jumps, then the dog will make 6 jumps, but the dog in 5 jumps out of 6 will run the same distance as the hare in 6 jumps. Therefore, in 6 jumps, the dog will approach the hare at a distance equal to one of its jumps.
Since at the initial moment the distance between the hare and the dog was equal to 40 dog jumps, the dog will catch up with the hare in 40 × 6 = 240 jumps.
3. Grandchildren and nuts
The grandfather says to his grandchildren: “Here are 130 nuts for you. Divide them in two so that the smaller part, enlarged by 4 times, is equal to the larger part, reduced by 3 times. " How to split nuts?
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Let x of nuts be the smallest part, and (130 - x) is the largest part. Then 4 nuts is a smaller part, increased by 4 times, (130 - x) ÷ 3 - a large part, decreased by 3 times. By condition, the smaller part, increased by 4 times, is equal to the larger part, reduced by 3 times. Let's make an equation and solve it:
4x = (130 - x) ÷ 3
4x × 3 = 130 - x
12x = 130 - x
12x + x = 130
13x = 130
x = 10
This means that the smaller part is 10 nuts, and the larger one is 130 - 10 = 120 nuts.
4. At the mill
There are three millstones in the mill. On the first one per day you can grind 60 quarters of grain, in the second - 54 quarters, and in the third - 48 quarters. Someone wants to grind 81 quarters of grain in the shortest time on these three millstones. In what is the shortest time you can grind the grain and how much for this do you need to pour it on each millstone?
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The idle time of any of the three millstones increases the grinding time of the grain, so all three millstones must work the same time. In a day, all millstones can grind 60 + 54 + 48 = 162 quarters of grain, but you need to grind 81 quarters. This is half of 162 quarters, so the millstones must run 12 hours. During this time, the first millstone needs to grind 30 quarters, the second - 27 quarters, and the third - 24 quarters of the grain.
5. 12 people
12 people carry 12 loaves of bread. Each man carries 2 loaves, each woman carries half a loaf, and each child carries a quarter. How many men, women and children were there?
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If we take men for x, women for y, and children for z, we get the following equality: x + y + z = 12. Men carry 2 loaves - 2x, women - 0.5y for half, children - 0.25z for a quarter. Let's make the equation: 2x + 0.5y + 0.25z = 12. Let's multiply both sides by 4 to get rid of fractions: 2x × 4 + 0.5y × 4 + 0.25z × 4 = 12 × 4; 8x + 2y + z = 48.
We expand the equation in this way: 7x + y + (x + y + z) = 48. It is known that x + y + z = 12, substitute the data into the equation and simplify it: 7x + y + 12 = 48; 7x + y = 36.
Now, by the selection method, you need to find x satisfying the condition. In our case, it is 5, because if there were six men, then all the bread would be distributed among them, and children and women would not get anything, and this contradicts the condition. Substitute 5 into the equation: 7 × 5 + y = 36; y = 36 - 35 = 1. This means that there were five men, one woman, and children - 12 - 5 - 1 = 6.
6. Boys and apples
Three boys have some apples. The first of the guys gives the other two as many apples as each of them has. Then the second boy gives the other two as many apples as each of them now has. In turn, the third gives each of the other two as many apples as each has at that moment.
After that, each of the boys has 8 apples. How many apples did each child have in the beginning?
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At the end of the exchange, each boy had 8 apples. According to the condition, the third boy gave the other two as many apples as they had. Consequently, they had 4 apples each, and the third had 16.
This means that before the second transmission, the first boy had 4 ÷ 2 = 2 apples, the third - 16 ÷ 2 = 8 apples, and the second - 4 + 2 + 8 = 14 apples. Thus, from the very beginning, the second boy had 7 apples, the third had 4 apples, and the first had 2 + 7 + 4 = 13 apples.
7. Brothers and sheep
Five peasants - Ivan, Peter, Yakov, Mikhail and Gerasim - had 10 sheep. They could not find a shepherd to graze them, and Ivan says to the others: "Let us, brothers, graze ourselves in turn - for as many days as each of us has sheep."
For how many days should each peasant be a shepherd, if it is known that Ivan has twice as few sheep as Peter, Jacob has twice as few as Ivan; Mikhail has twice as many sheep as Jacob, and Gerasim four times less than Peter?
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It follows from the condition that both Ivan and Mikhail have twice as many sheep as Jacob; Peter has twice as much as Ivan's, and, therefore, four times more than Jacob's. But then Gerasim has as many sheep as Yakov has.
Let Yakov and Gerasim have x sheep each, then Ivan and Mikhail have 2 sheep each, and Peter - 4. Let's make the equation: x + x + 2 x + 2x + 4x = 10; 10x = 10; x = 1. This means that Yakov and Gerasim will tend the sheep for one day, Ivan and Mikhail - for two days, and Peter - for four days.
8. Meeting of travelers
One person is walking to another city and passes 40 miles a day, and another person comes to meet him from another city and walks 30 miles a day. The distance between the cities is 700 versts. How many days will the travelers meet?
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In one day, travelers approach each other 70 miles. Since the distance between the cities is 700 versts, they will meet in 700 ÷ 70 = 10 days.
9. Owner and worker
The owner hired an employee with the following condition: for each working day, he is paid 20 kopecks, and for each non-working day, 30 kopecks are deducted. After 60 days, the employee has not earned anything. How many working days were there?
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If a man worked without absenteeism, then in 60 days he would have earned 20 × 60 = 1,200 kopecks. For each non-working day, 30 kopecks are deducted from him and he does not earn 20 kopecks, that is, for each absenteeism he loses 20 + 30 = 50 kopecks.
Since the employee did not earn anything in 60 days, the loss for all non-working days amounted to 1,200 kopecks, that is, the number of non-working days is 1,200 ÷ 50 = 24 days. The number of working days is therefore 60 - 24 = 36 days.
10. People in the team
When asked how many people he has in his team, the captain replied: “There are 9 people, that is commands, the rest are on guard. " How many are on guard?
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The team consists of 9 × 3 = 27 people. This means that there are 27 - 9 = 18 people on guard.
What was the most difficult task? Share in the comments!
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