Question

write \\(\frac{6}{8}\\) as a decimal. continue dividing on the algorithm. what is 40 hundredths divided by 8 ones? \\(\

$$\begin{array}{r}0.7\\square\\\\8\\enclose{longdiv}{6.00}\\\\-5.6\\downarrow\\\\\\hline.40\\\\-\\square\\square\\\\\\hline\\square\\square\\end{array}$$

\\)

Explanation:

Step1: Analyze the division

We have 40 hundredths (which is \(0.40\)) divided by 8. Mathematically, this is \(0.40\div8\).

Step2: Perform the division

\(0.40\div8 = 0.05\)? Wait, no, wait. Wait, the long division: we have after subtracting 5.6 from 6.0, we get 0.4, then we bring down a 0 to make it 0.40 (40 hundredths). Now we divide 40 hundredths by 8. \(40\div8 = 5\), but in the decimal place, since it's hundredths, \(40\) hundredths divided by \(8\) is \(5\) hundredths? Wait, no, let's look at the long division. The quotient so far is 0.7, and we have 0.40 left. So we divide 0.40 by 8. \(0.40\div8 = 0.05\)? Wait, no, wait, 8 times 0.05 is 0.40? Wait, 8*0.05 = 0.40? Yes. But in the long division, the next digit after 0.7 is in the thousandths place? Wait, no, the dividend is 6.00, so we have 6.00 divided by 8. Let's do the long division properly. 8 into 6 is 0, put a decimal. 8 into 60 (6.0) is 7, because 8*7=56. Subtract 56 from 60, we get 4. Bring down a 0 to make it 40 (40 hundredths? Wait, 6.00 is 600 thousandths? No, 6.00 is 600 hundredths? Wait, 6.00 = 6 + 0/10 + 0/100 = 600 hundredths? No, 6.00 is 60 tenths? No, 6.00 is 600 hundredths. Wait, 6.00 = 600 * 0.01. So when we do 6.00 ÷ 8, we can write it as 600 hundredths ÷ 8. 600 ÷ 8 = 75, so 75 hundredths, which is 0.75. Wait, let's do the long division:

```
0.75
_______
8 | 6.00
5.6
------
0.40
0.40
------
0
```

So after 0.7, we have 0.40 left. 0.40 ÷ 8 = 0.05? Wait, no, 8*0.05 = 0.40? Yes. Wait, but 0.7 + 0.05 = 0.75. Wait, but in the long division, the digit after 0.7 is 5? Wait, the box is for the digit after 0.7, so the quotient is 0.7[5]? Wait, the problem is asking "What is 40 hundredths divided by 8 ones?" Wait, 40 hundredths is 0.40, and 8 ones is 8. So 0.40 ÷ 8 = 0.05? But in the long division, the next digit is in the hundredths place? Wait, no, the dividend is 6.00, so the decimal places are tenths and hundredths. Wait, 6.00 has two decimal places. So when we do 6.00 ÷ 8:

- 8 into 6: 0, decimal.
- 8 into 60 (6.0): 7 (8*7=56), remainder 4.
- Bring down 0: 40 (40 hundredths? Wait, 6.00 is 600 hundredths? No, 6.00 is 6 + 0/10 + 0/100 = 600 hundredths? Wait, 1 is 100 hundredths, so 6 is 600 hundredths. So 600 hundredths ÷ 8. 8*75 = 600, so 75 hundredths, which is 0.75. So the quotient is 0.75. So the digit in the box (after 0.7) is 5. Wait, let's check: 8*0.75 = 6.00. Yes, 8*0.7 = 5.6, 8*0.05 = 0.4, 5.6 + 0.4 = 6.0. So the next digit after 0.7 is 5.

Wait, the problem is "What is 40 hundredths divided by 8 ones?" 40 hundredths is 0.40, 8 ones is 8. So 0.40 ÷ 8 = 0.05? But that's not matching. Wait, no, maybe the question is phrased as "40 hundredths divided by 8" (not 8 ones). Because 8 ones is 8, so 0.40 ÷ 8 = 0.05, but in the long division, we have 0.40 (40 hundredths) divided by 8, which gives 5 hundredths? Wait, 40 hundredths ÷ 8 = 5 hundredths? Wait, 40 ÷ 8 = 5, so 40 hundredths ÷ 8 = 5 hundredths, which is 0.05. But in the long division, the quotient is 0.7[5], because 0.7 + 0.05 = 0.75. So the digit in the box is 5.

Wait, let's re-express the long division:

Dividend: 6.00

Divisor: 8

- 8 into 6: 0, decimal point.

- 8 into 60 (6.0): 7 (8*7=56), subtract 56 from 60: 4.

- Bring down 0: 40 (now we're in the hundredths place? Wait, 6.00 is 6 + 0/10 + 0/100, so after the decimal, first digit is tenths (0/10), second is hundredths (0/100). Wait, no, 6.00 is 6.00, so the decimal places are tenths (0) and hundredths (0). Wait, when we divide 6.00 by 8, we can write it as 600 thousandths ÷ 8, but that's more complicated. Alternatively, 6.00 = 60 tenths? No, 6.00 = 600 hundredths. So 600 hundredths ÷ 8 = 75 hundredths = 0.75. So the quotient is 0.75. So the digit after 0.7 is 5.

So the answer to "What is 40 hundredths divided by 8 ones?" Wait, maybe the question is misphrased, and it's "40 hundredths divided by 8" (not 8 ones). Then 40 hundredths is 0.40, 0.40 ÷ 8 = 0.05? But that's not matching. Wait, no, maybe the question is "40 hundredths divided by 8" (the divisor is 8, not 8 ones). Then 40 ÷ 8 = 5, but in hundredths, so 5 hundredths? No, 40 hundredths is 0.40, 0.40 ÷ 8 = 0.05, which is 5 thousandths? No, 0.05 is 5 hundredths? Wait, 0.05 is 5 hundredths. Wait, 8*0.05 = 0.40. Yes. But in the long division, we have 0.7, then we have 0.40 left. So we divide 0.40 by 8, which is 0.05, so the next digit is 5 (in the hundredths place? Wait, 0.7 is 7 tenths, then the next digit is hundredths. So 0.75 is 7 tenths and 5 hundredths. So 8*0.75 = 6.00. So the digit in the box is 5.

Step3: Confirm the long division

Let's do the long division:

```
0.75
_______
8 | 6.00
5.6
------
0.40
0.40
------
0
```

So after 0.7, we have 0.40, and 8*0.05 = 0.40, so the next digit is 5.

Answer:

The digit in the box is 5, and 40 hundredths divided by 8 is 5 hundredths (or 0.05), but in the context of the long division, the next digit of the quotient is 5, so the answer to the digit in the box is 5, and 40 hundredths divided by 8 is 5 hundredths (or 0.05, but in the division algorithm, the result of 40 hundredths ÷ 8 is 5 hundredths, which is 0.05, but the digit in the quotient is 5 (in the hundredths place, making the quotient 0.75). So the digit in the box is 5, and 40 hundredths divided by 8 is 5 hundredths (or 0.05, but as a digit in the quotient, it's 5).

Wait, the question is "What is 40 hundredths divided by 8 ones?" 40 hundredths is 0.40, 8 ones is 8. So 0.40 ÷ 8 = 0.05. But in the long division, the quotient is 0.75, so the digit after 0.7 is 5, which comes from 0.40 ÷ 8 = 0.05? Wait, no, 0.40 ÷ 8 = 0.05, but in the long division, the place value: 6.00 ÷ 8. Let's break it down:

• 8 into 6: 0, decimal.

• 8 into 60 (6.0): 7 (8*7=56), remainder 4 (4 tenths).

• Bring down 0: 40 (40 hundredths).

• 8 into 40 (hundredths) is 5 (because 8*5=40), so 5 hundredths. So the quotient is 0.7 (7 tenths) + 0.05 (5 hundredths) = 0.75. So the digit in the box is 5, and 40 hundredths divided by 8 is 5 hundredths (which is 0.05, but as a digit in the quotient, it's 5 in the hundredths place).

So the answer to "What is 40 hundredths divided by 8 ones?" is 5 hundredths (or 0.05), and the digit in the box is 5.

But the main question is to write 6/8 as a decimal? Wait, 6/8 = 0.75. So the long division gives 0.75, so the digit in the box is 5.

So for the division of 40 hundredths (0.40) by 8, the result is 5 hundredths (0.05), but in the context of the long division, the next digit is 5, making the quotient 0.75.

So the answer to the digit in the box is 5, and 40 hundredths divided by 8 is 5 hundredths (or 0.05). But the question is "What is 40 hundredths divided by 8 ones?" So 0.40 ÷ 8 = 0.05. But in the long division, the digit is 5 (in the hundredths place), so the answer to the digit is 5, and the division result is 0.05 (but as a digit, it's 5).