Question

В коробке лежат карандаши, всего их меньше 30. Если карандаши разложить в ряды по 5 штук в каждом, то один ряд окажется неполным. Если эти карандаши переложить в ряды по 4 штуки в каждом, то в неполном ряду окажется на 2 карандаша больше, чем при раскладывании по 5 штук в ряд. Сколько всего карандашей? Запишите решение и ответ.

Answer 1

Пусть было х рядов
если по 4 в ряду, то будет 4х+3  (один ряд неполный)
если по 5. то 5х+(3-2)=5х+1
общее количество одинаково, поэтому
5х+1=4х+3
х=2 - ряда
4*2+3=11 - карандашей
5*2+1=11 - решение верное
Ответ: 11 карандашей

Answer 2

Problem Analysis

We are given that there are fewer than 30 pencils in a box. When the pencils are arranged in rows of 5, one row is incomplete. When the pencils are arranged in rows of 4, the incomplete row has 2 more pencils than when arranged in rows of 5. We need to determine the total number of pencils in the box.

Solution

Let's assume the total number of pencils in the box is x.

When the pencils are arranged in rows of 5, one row is incomplete. This means that the total number of pencils is not divisible by 5. Therefore, we can express the total number of pencils as:

x = 5n + r, where n is the number of complete rows and r is the number of pencils in the incomplete row.

When the pencils are arranged in rows of 4, the incomplete row has 2 more pencils than when arranged in rows of 5. This means that the number of pencils in the incomplete row is r + 2. Therefore, we can express the total number of pencils as:

x = 4m + (r + 2), where m is the number of complete rows when arranged in rows of 4.

We can set up the following equation based on the above expressions:

5n + r = 4m + (r + 2)

Simplifying the equation, we get:

5n = 4m + 2

To find the possible values of n and m, we can substitute different values for n and check if the equation is satisfied.

Let's start with n = 1: - When n = 1, 5n = 5. - Substituting n = 1 in the equation, we get 5 = 4m + 2. - Solving the equation, we find m = 1.

Therefore, when n = 1 and m = 1, the equation is satisfied.

Now, let's substitute these values in the expressions for x:

For the arrangement in rows of 5: x = 5n + r = 5(1) + r = 5 + r

For the arrangement in rows of 4: x = 4m + (r + 2) = 4(1) + (r + 2) = 4 + r + 2 = 6 + r

Since both expressions represent the same total number of pencils, we can equate them:

5 + r = 6 + r

Simplifying the equation, we find:

5 = 6

This equation is not true, which means our assumption that n = 1 and m = 1 is incorrect.

Let's try another value for n:

When n = 2: - When n = 2, 5n = 10. - Substituting n = 2 in the equation, we get 10 = 4m + 2. - Solving the equation, we find m = 2.

Therefore, when n = 2 and m = 2, the equation is satisfied.

Now, let's substitute these values in the expressions for x:

For the arrangement in rows of 5: x = 5n + r = 5(2) + r = 10 + r

For the arrangement in rows of 4: x = 4m + (r + 2) = 4(2) + (r + 2) = 8 + r + 2 = 10 + r

Since both expressions represent the same total number of pencils, we can equate them:

10 + r = 10 + r

This equation is true, which means our assumption that n = 2 and m = 2 is correct.

Therefore, the total number of pencils in the box is x = 10 + r.

To find the value of r, we can substitute the values of n and m in the equation 5n = 4m + 2:

5(2) = 4(2) + 2

Simplifying the equation, we find:

10 = 10

This equation is true, which means our assumption is correct.

Therefore, the total number of pencils in the box is x = 10 + r.

To find the value of r, we can substitute the values of n and m in the equation 5n = 4m + 2:

5(2) = 4(2) + 2

Simplifying the equation, we find:

10 = 10

This equation is true, which means our assumption is correct.

Therefore, the total number of pencils in the box is x = 10 + r.

Since we know that the total number of pencils is less than 30, we can substitute different values for r and check if the total number of pencils is less than 30.

Let's start with r = 1: - When r = 1, x = 10 + 1 = 11.

Let's check if x = 11 is less than 30.

11 < 30

This is true, so x = 11 is a valid solution.

Therefore, the total number of pencils in the box is 11.

Answer

The total number of pencils in the box is 11.

Solution Verification

To verify the solution, let's check if the given conditions are satisfied:

1. When the pencils are arranged in rows of 5, one row is incomplete. - If we arrange 11 pencils in rows of 5, we get 2 complete rows and 1 incomplete row with 1 pencil. - Therefore, this condition is satisfied.

2. When the pencils are arranged in rows of 4, the incomplete row has 2 more pencils than when arranged in rows of 5. - If we arrange 11 pencils in rows of 4, we get 2 complete rows and 1 incomplete row with 3 pencils. - The incomplete row has 3 pencils, which is 2 more than the incomplete row when arranged in rows of 5. - Therefore, this condition is satisfied.

Since both conditions are satisfied, we can conclude that the solution is correct.

Summary

The total number of pencils in the box is 11.