Lecture 9 Ratio and Proportion

Division of two numbers is also called the ratio of two numbers.

I. Nature of ratio and proportion

Property 1: If a: b = c: d, then (a+c): (b+d) = a: b = c: d;

Property 2: If a: b = c: d, then (a-c): (b-d) = a: b = c: d;

Property 3: If a: b = c: d, then (a+x c): (b+x d) = a: b = c: d; (x is a constant)

Property 4: If A: B = C: D, then ad = b (that is, the outer product equals the inner product).

Proportional ratio: If ab=k(k is a constant), A is said to be proportional to B;

Inverse ratio: If ab=k(k is a constant), A and B are said to be inversely proportional.

Second, the proportion and the embodiment of proportion in travel issues

On the issue of travel, because there is speed =

So:

When the walking speed of a group of objects is equal, the walking distance ratio is equal to the inverse ratio of the corresponding time;

When a group of objects walk the same distance, the walking speed ratio is equal to the inverse ratio of the corresponding time;

When the walking time of a group of objects is equal, the walking speed ratio is equal to the corresponding distance.

1 ratio. A and b are 8: 5. After subtracting 34 from each number, A is twice that of B. Try to find these two numbers.

Analysis and solutions

Method 1: Let A be 8x and B be 5x, so there are (8x-34): (5x-34) = 2: 1, and x= 17, so A is 136 and B is 85.

Method 2: Because the number of reductions is the same, the difference between A and B remains unchanged. The initial time difference accounts for 3 copies, and the later time difference accounts for 1 copy as much as B. That is to say, the reduction of 34 copies accounts for 3- 1=2 copies, so the initial ` 1 copy is 342= 17.

In recent years, the train speed has been greatly improved. 1427 train leaves Beijing West Railway Station for Anqing West Railway Station. It takes 60 minutes to drive to the whole distance of 5/ 1 1 and then walk 56 kilometers, but it arrives at Anqing West Station 2 hours earlier than before. How many kilometers is it from Beijing West Railway Station to Anqing West Railway Station?

6. It is known that three kinds of mixtures consist of three components, A, B and C. The first one only contains two components, A and B, with a weight ratio of 3: 5; The second one only contains component B and component C, and the weight ratio is I: 2; The third one only contains component A and component C, and the weight ratio is 2: 3. What proportion of these mixtures can make the weight ratio of A, B and C in the obtained mixture 3: 5: 2?

Analysis shows that the weight ratio of the first mixed species A and B is the same as the final mixture, which is 3: 5. Therefore, the weight ratio of the second and third mixtures A and B is adjusted to 3: 5, and then A and B in the second and third mixtures and A and B in the first mixture are regarded as a single substance.

The second mixture does not contain A and the third mixture does not contain B, so the third mixture contains 3 times A and 5 times B, that is, the weight ratio of the second mixture to the third mixture is 5: 1.5.

So at this time, the content of C is 52+ 1.53= 14.5, and the content of C in the final mixture is twice that of 3A/5B. There is 14.52- 1=6.25, so the first mixture is 6.25.

That is, the ratio of the first, second and third mixture is 6.25: 5:1.5 = 25: 20: 6.

7. There are *** 1 100 male and female employees, all of whom can complete the same work in the same number of days; If the number of male workers and the number of female workers are reversed, it will take 36 days for all female workers to complete the work completed by all male workers in 25 days. Q: How many male and female workers are there?

The analysis and solution directly set the number of male and female workers, and then solve it through the equation, the process will be more complicated.

Let's assume that the starting male employee is 1 and the female employee is K. There are 1 male employee equivalent to K female employees. After the number of male workers and female workers is reversed, the male worker is K, which is equivalent to the female worker k2, and the female worker is i 。

;