Jill says that 12.6......... Is less than 12.63. Explain her error?

Jill's error lies in the incorrect understanding of decimal numbers. When comparing decimal numbers, each digit to the right of the decimal point has a decreasing value. In this case, 12.6....... is the same as 12.6, while 12.63 is greater than 12.6.

To explain further, let's break down the decimal numbers:

12.6:
- The 6 in the tenths place represents 6/10 or 0.6
- There are no digits after the tenths place

12.63:
- The 3 in the tenths place represents 3/10 or 0.3
- The 6 in the hundredths place represents 6/100 or 0.06
- When combined, the number is 12.63

As we can see, 12.63 is greater than 12.6 because it has an additional hundredths place digit. Alternatively, you can also compare the two numbers by aligning the digits:

12.6
12.63

Comparing digit by digit from left to right, we see that 12.63 is greater since it has a 3 in the hundredths place, while 12.6 does not. Therefore, Jill's statement that 12.63 is less than 12.6 is incorrect.

To explain Jill's error, let's examine the numbers she mentioned: 12.6 and 12.63.

At first glance, it may seem that 12.6 is indeed less than 12.63 because 6 is less than 63. However, Jill's error lies in the fact that she did not consider the decimal points and the numbers after them.

In the decimal system, numbers to the right of the decimal point represent fractions or parts of a whole. Each digit after the decimal point has a decreasing value. The first digit after the decimal point is tenths (1/10), the second is hundredths (1/100), the third is thousandths (1/1000), and so on.

In this case, we can see that 12.6 represents 12 whole units and 6 tenths (12 + 0.6) while 12.63 represents 12 whole units and 63 hundredths (12 + 0.63).

Since 63 hundredths is greater than 6 tenths, we can conclude that 12.63 is greater than 12.6.

Therefore, Jill's error is that she disregarded the digits after the decimal point, mistakenly considering only the whole numbers. By ignoring the decimal values, she incorrectly concluded that 12.6 is less than 12.63.

12.6666666666..... is greater than 12.63