10 entertaining problems from an old arithmetic textbook

2024 Author: Malcolm Clapton | [email protected]. Last modified: 2023-12-17 03:44

These problems were included in LF Magnitsky's "Arithmetic" - 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. In how many days will a wife drink a keg alone?

Let's find a number that can be divisible by either 10 or 14. For example, 140. In 140 days a person will drink 10 barrels of kvass, and together with his wife - 14 barrels. 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.

Show answer Hide answer

2. On the hunt

A man went hunting with a dog. They were walking through 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 by both the hare and the dog at the same time.

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. Consequently, 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.

Show answer Hide answer

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?

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.

Show answer Hide answer

4. At the mill

There are three millstones in the mill. On the first of them 60 quarters of grain can be ground per day, on the second - 54 quarters, and on 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 it takes to grind the grain and how much for this do you need to pour it on each millstone?

The idle time of any of the three millstones increases the grinding time of the grain, so all three millstones must work the same amount of 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 the 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.

Show answer Hide answer

5.12 people

12 people are carrying 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?

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 in half - 0.5y, children in a quarter - 0.25z … Let's make the equation: 2x + 0.5y + 0.25z = 12. Multiply both sides by 4 to get rid of fractions: 2x × 4 + 0.5y × 4 + 0.25z × 4 = 12 × 4; 8x + 2y + z = 48.

Let's expand the equation in this way: 7x + y + (x + y + z) = 48. It is known that x + y + z = 12, we 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, this 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. So, there were five men, one woman, and children - 12 - 5 - 1 = 6.

Show answer Hide answer

6. Boys and apples

Three boys have some apples each. 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?

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. Therefore, they had 4 apples each, and the third had 16.

This means that before the second transfer, 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.

Show answer Hide answer

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 each peasant should 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?

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 Jacob has.

Let Yakov and Gerasim have x sheep each, then Ivan and Mikhail have 2 sheep each, 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 herd the sheep for one day, Ivan and Mikhail - for two days, and Peter - for four days.

Show answer Hide answer

8. Meeting travelers

One person goes to another city and walks 40 miles a day, and another person goes to meet him from another city and walks 30 miles a day. The distance between cities is 700 versts. How many days will the travelers meet?

In one day, travelers approach each other 70 miles. Since the distance between cities is 700 versts, they will meet in 700 ÷ 70 = 10 days.

Show answer Hide answer

9. Boss and employee

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?

If a person 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 was 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.

Show answer Hide answer

10. People on the team

The captain, when asked how many people he has in his team, answered: "There are 9 people, that is, ⅓ teams, the rest are on guard." How many are on guard?

In total, the team consists of 9 × 3 = 27 people. This means that there are 27 - 9 = 18 people on guard.

Show answer Hide answer