Correct answer:
(First of all, the formulas listed in the solution in the photo actually have certain significance, but they are not as mentioned above. )
This kind of problem involves a basic relationship (formula): distance = speed × time; ————————①
According to this formula, other equations can be generalized, such as:
(distance of A-distance of B) = (A speed of A-speed of B) × time; —————————②
Simply put:
Distance difference = speed difference × time;
Let's talk about the difference between distance:
Draw the schematic diagram is very clear:
Obviously, the distance difference between A and B is two 8 kilometers, which is the origin of 8×2. ————③
In addition, the speed difference:
First of all, we already know their speed ratio:
Speed of a = speed of b×1.2;
Then the speed difference can be expressed as:
The speed of b × 1.2-B =( 1.2- 1)×B; ———————————④
This is the origin of 1.2- 1; In fact, this subtraction formula calculates not the difference of actual speed, but the ratio of speed difference;
Now, bring the formulas in ③ and ④ into ②, and you can get:
8× 2 = (1.2- 1 )× B speed× time; ——————————————————⑤
The speed of B × time is naturally the distance of b; Bring it into ⑤ and make a little modification:
8×2÷( 1.2- 1)= B distance; ————————————————————⑥
The left side of this formula 6 is the solution of 1 step in your photo. As I said before, the formula of step 1 is meaningful, but it calculates the distance of B, not the speed of B mentioned above. (Of course, the unit is not "km/h", but "km". )
With B's journey, we can naturally seek A's journey ... The most direct method: according to the above calculation:
Distance of a = distance of b+8× 2 = 80+16 = 96 km; ——————————————⑦
The second calculation method requires the use of attributes derived from Formula ①:
At the same time, the ratio of distance is equal to the ratio of their speed; ——————————⑧
So:
Distance from Party A = distance from Party B × 1.2 = 80× 1.2 = 96 km ———————————————————————————————————————————————————————————————————.
This formula corresponds to the second formula in your photo. Similarly, the formula is correct, but the meaning is wrong: it is not the speed of A, but the distance of A;
With two distances, the total distance between A and B is simple: equal to the sum of the two. -Solve the problem.
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Now I want to talk about the third and fourth formulas in your photo:
Number 3: 8× 2 ÷ (96-80) = 1
Now we know that 8×2 is the distance difference between A and B, and 96-80 is also the distance difference between them. When the two are divided, the result is definitely 1. Therefore, in fact, this formula has no practical significance.
The number 4: 96+80 itself is the sum of the distances between A and B, that is, the distance between A and B, so whether it is multiplied by 1 has no effect on the result. This formula doesn't make sense either.
Now you can see that the first three formulas in the photo are completely wrong in meaning. But why can you calculate the correct result? The reason is that the algorithm uses assumptions that have no influence on the results:
Suppose: the travel time of two cars from departure to encounter is 1 hour;
According to formula (1), we can know that:
When time = 1 (unit), the values of distance and speed are equal (but the units are different). So the two formulas 1 and 2 in the photo use the method of calculating the distance, but it is no problem to use the result as the speed. -The premise is that the time must be 1 hour.
For this reason, (96+80) × 1 in step 4 can also be understood as:
The sum of speed × time = the sum of distances;
The reason is clear, and the problem is clear: although the results are all right, the method in the photo is still not desirable. Because, we should not make assumptions about the driving time casually-unless you can prove that this assumption has no effect on solving the problem. However, for one thing, this is no longer a problem for junior high school students, and for another, it is unnecessary-wouldn't it be easier to calculate the distance directly according to my method?
In fact, in this case, the actual driving time is uncertain, that is, it cannot be calculated. You assume that it is 1 hour here, but I can also assume that it is 2 hours, and the distances A and B finally calculated are still correct. Of course, the speeds of A and B are not the current values.