Question

ngoc wants to find the decimal equivalent of \\(\frac{29}{9}\\), so she divides. study ngocs work shown here, and then answer the questions below.

(image of long division: 9 \\(\overline{)29.00000}\\), quotient so far 3.2, subtraction steps: 29 - 27 = 2, bring down 0 to get 20; 20 - 18 = 2)

1. continue ngocs work to find the next three digits in the decimal equivalent of \\(\frac{29}{9}\\).

3.2□□□

1. what do you notice about the work and the decimal equivalent of \\(\frac{29}{9}\\)?

• each of the digits after the decimal point will be □ because this number is a dropdown.
• the difference between 20 and the product of these digits and 9 is always □.

Explanation:

Step1: Analyze the division process

We have the fraction $\frac{29}{9}$, and the long - division process starts with dividing 29 by 9. We know that $9\times3 = 27$, and $29 - 27=2$. Then we bring down a 0 to get 20. $9\times2 = 18$, and $20 - 18 = 2$.

Step2: Find the next digits

Since we have a remainder of 2 after the first few steps, when we continue the long - division, we will keep getting the same remainder. So, after 3.2, when we divide 20 (by bringing down a 0) by 9, $9\times2=18$, remainder 2. So the next digit is 2, and this pattern will repeat. So the next three digits are 2, 2, 2.

Step3: Analyze the decimal properties

• For the first blank: Each of the digits after the decimal point will be 2 because when we do the long - division of $\frac{29}{9}$, we keep getting a remainder of 2. When we bring down a 0, we have 20, and $20\div9 = 2$ with a remainder of 2. This means the decimal is a repeating decimal, and the repeating digit is 2.
• For the second blank: This number is a repeating decimal because the remainder keeps repeating (we keep getting a remainder of 2, leading to the same digit 2 in the decimal part).
• For the third blank: The difference between 20 and the product of these digits (2) and 9 is $20-(2\times9)=20 - 18 = 2$, and this difference is always 2.

Answer:

The next three digits are 2, 2, 2.

• First blank: 2; Second blank: a repeating decimal; Third blank: 2.