There is such a knowledge point in IGCSE, how to convert circular decimals into fractions, domestic students may know it earlier, and some primary schools will know it. When they see the above problems, they will have conversion formulas in their minds.

Method one

As long as the method is faster, the idea is to remove the loop and turn it into a finite number.

Then subtract the above two to get:

It must be wrong, so how much is it?

In fact, the idea is the same, that is, to eliminate the cycle into a finite number, because it is still a two-digit cycle, then we are on both sides at the same time? × 100? ,

Finally, to sum up, we multiply the cycle number by the power of 10, subtract the cycle part, and finally divide by it to get the score.

Method 2

The idea of the second method is more direct, and we can directly calculate the score.

Anyway, let's do the math.

The first step is simple, we can write this cyclic decimal as? 0.2˙8˙=2× 10? 1+8× 10? 2+2× 10? 3+8× 10? 4+?

So we got two geometric series.

Using this method, we can directly count all the circulating decimals as component numbers, which is nothing more than using the summation formula of infinite proportional series several times.

Let's look at an AMC 12 problem.

Here we use the above two methods to solve it:

Method one k? Convert radix to 10 radix.

So,

Solve a quadratic equation with one variable, k= 16.

This method is the same as our idea of converting cyclic decimals into fractions by using infinite geometric series. So how do we solve it in the first way? That is, how to eliminate the circular part.

Method 2: eliminate the circulation part.

Notice that the number on the left is 10, and the number on the right is? k？ Decimal number.

So, subtract the above two expressions.

So far, the two methods mentioned above have solved the above problems.