Question

Экскаватором можно выкопать за 1 час канаву длиной 20 м. Одну канаву копали 10 ч ,а другую- 12 ч. Найти общую длину канав, которые выкопали за это время. Реши задачу разными способами. Сравни их и выбери самый удобный.

Answer 1

За 1 ч - 20 м
за 10 ч - ?
за 12 ч - ?
всего выкопали - ? м 
1 способ
10*20=200 м  длина 1 канавы
12*20=240 м длина 2 канавы
200+240=440 м общая длина
2 способ
10+12=22 ч - время, за которое выкопали две канавы
12*20=440 м - общая длина канав
 

Answer 2

Problem Analysis

We are given that an excavator can dig a trench with a length of 20 meters in 1 hour. One trench took 10 hours to dig, while another took 12 hours. We need to find the total length of the trenches that were dug in this time.

Solution 1: Using Proportions

Let's solve this problem using proportions. We know that the excavator can dig a trench with a length of 20 meters in 1 hour. Therefore, the rate at which the excavator digs is 20 meters per hour.

Let's assume the total length of the trenches dug in 10 hours is x meters. Using the proportion:

20 meters / 1 hour = x meters / 10 hours

Simplifying the proportion, we get:

20 * 10 = x

x = 200 meters

Similarly, let's assume the total length of the trenches dug in 12 hours is y meters. Using the same proportion:

20 meters / 1 hour = y meters / 12 hours

Simplifying the proportion, we get:

20 * 12 = y

y = 240 meters

To find the total length of the trenches dug in both 10 and 12 hours, we add the lengths:

Total length = x + y = 200 + 240 = 440 meters

Therefore, the total length of the trenches dug in 10 and 12 hours is 440 meters.

Solution 2: Using Algebra

Alternatively, we can solve this problem using algebra. Let's assume the length of the trench dug in 10 hours is x meters and the length of the trench dug in 12 hours is y meters.

We know that the excavator can dig a trench with a length of 20 meters in 1 hour. Therefore, the rate at which the excavator digs is 20 meters per hour.

Using the formula: distance = rate * time, we can write the following equations:

x = 20 * 10 y = 20 * 12

Simplifying the equations, we get:

x = 200 y = 240

To find the total length of the trenches dug in both 10 and 12 hours, we add the lengths:

Total length = x + y = 200 + 240 = 440 meters

Therefore, the total length of the trenches dug in 10 and 12 hours is 440 meters.

Comparison of Solutions

Both solutions yield the same result, which is a total length of 440 meters for the trenches dug in 10 and 12 hours. The first solution uses proportions, while the second solution uses algebra. Both methods are valid and provide the correct answer.

The choice between the two methods depends on personal preference and familiarity with the concepts. Some people may find proportions more intuitive, while others may prefer algebraic equations.