Question

kayla knows that \\(\frac{1}{16}\\) means “1 divided by 16.” she uses this to find the decimal equivalent for \\(\frac{1}{16}\\). enter a digit into each box to complete her work.
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$$\begin{array}{r}0.\\square\\square\\square\\square\\\\16\\overline{)1.0000}\\\\-\\quad0\\quad\\quad\\\\\\hline\\quad1\\ 00\\quad\\\\-\\quad\\ 96\\quad\\\\\\hline\\quad\\ \\ \\ \\square\\ 0\\\\-\\quad\\ 32\\\\\\hline\\quad\\ \\ \\ \\ \\ 80\\\\-\\quad\\ \\ \\square\\ 0\\\\\\end{array}$$

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Explanation:

Step1: Divide 1 by 16 (first decimal place)

1 divided by 16 is 0 with a remainder of 1. Bring down a 0 to make 10. 10 divided by 16 is 0, bring down another 0 to make 100. 16*6 = 96, so the first decimal digit (tenths place) is 0, second (hundredths) is 6? Wait, no, let's do long division properly. 1.0000 ÷ 16.

First, 16 into 10 (after decimal) is 0, 16 into 100: 16*6=96, so 6, remainder 4. So first decimal digit (tenths) is 0, hundredths is 6? Wait the first box after 0. is tenths, then hundredths, thousandths, ten - thousandths. Wait the long division shows:

16 ) 1.0000

First, 16 can't go into 1, so 0. Then 16 into 10 (bring down 0) is 0, 16 into 100 (bring down another 0): 16*6 = 96, so the first decimal digit (hundredths? Wait no, the quotient is 0.xxxx. Let's re - express:

1.0000 ÷ 16:

- 16 into 10 (1.0) is 0, remainder 10. Bring down a 0: 100.

- 16*6 = 96, so we write 6 in the hundredths place? Wait no, the decimal places: 0. (tenths) (hundredths) (thousandths) (ten - thousandths). Wait the first box after 0. is tenths: 16 into 10 (1.0) is 0, so tenths place is 0. Then 100 (after bringing down two 0s: 1.00) divided by 16: 16*6 = 96, so hundredths place is 6, remainder 4. Then bring down a 0: 40. 16*2 = 32, so thousandths place is 2, remainder 8. Then bring down a 0: 80. 16*5 = 80, so ten - thousandths place is 5, remainder 0.

Wait let's check the long division steps in the image:

First, 16 into 1 (1.0000) gives 0, subtract 0, bring down 0 (now 10), bring down another 0 (now 100). 16*6 = 96, subtract 96 from 100, remainder 4. Then bring down a 0 (now 40). Wait the next step has a box, then 0, and subtract 32. 16*2 = 32, so the box here (the third line, the digit before 0) is 4? Wait no, the remainder after 100 - 96 is 4, then we bring down a 0 to make 40. Wait the image shows:

After 100 - 96, we have a box (let's call it digit A) and 0, then subtract 32. So 16*2 = 32, so the number being subtracted is 32, so the number above is 40? Wait no, the number before 0 in that line is A, so the number is A0, and A0 - 32. So A0 must be 40? Wait 40 - 32 = 8. Then next, we have 80, subtract a number with 0 at the end, so 16*5 = 80, so the last box is 8? Wait no, let's go step by step.

1. Long division of 1 ÷ 16:

- Step 1: 16 into 1: 0, write 0 above, subtract 0, bring down 0 (now 10).

- Step 2: 16 into 10: 0, write 0 above (tenths place), subtract 0, bring down 0 (now 100).

- Step 3: 16 into 100: 16*6 = 96, write 6 above (hundredths place), subtract 96 from 100, remainder 4.

- Step 4: Bring down 0 (now 40). 16 into 40: 16*2 = 32, write 2 above (thousandths place), subtract 32 from 40, remainder 8.

- Step 5: Bring down 0 (now 80). 16 into 80: 16*5 = 80, write 5 above (ten - thousandths place), subtract 80 from 80, remainder 0.

Now let's map to the boxes:

The quotient is 0. 0 (tenths), 6 (hundredths), 2 (thousandths), 5 (ten - thousandths). Wait the first four boxes after 0. are tenths, hundredths, thousandths, ten - thousandths: 0, 6, 2, 5? Wait no, wait the long division in the image:

The first line of quotient: 0. [box1] [box2] [box3] [box4]

Then the long division steps:

16 ) 1.0 0 0 0

- 0

------

1 0

- 0

------

1 0 0

- 9 6

------

4 0 (wait no, in the image, after 100 - 96, the next line is [box5] 0, then - 32. So 40? Wait 40 - 32 = 8. Then next line is 8 0, - [box6] 0.

Wait let's re - examine the image's long division:

- Dividend: 1.0000, divisor 16.

- First, 16 into 1: 0, write 0, subtract 0, bring down 0 (now 10).

- 16 into 10: 0, write 0 (first box after 0.), subtract 0, bring down 0 (now 100).

- 16 into 100: 16*6 = 96, write 6 (second box), subtract 96, remainder 4. Bring down 0 (now 40). Wait the image shows after 100 - 96, the next line is [box5] 0, then - 32. So 40? Wait 40 - 32 = 8. So [box5] is 4? No, 40 - 32 = 8, so the number being subtracted is 32, so the number above is 40? Wait the line is [box5] 0 - 32. So [box5]0 must be 40, so [box5] is 4. Then remainder 8, bring down 0 (now 80). Then 80 - [box6]0 = 0, so [box6] is 8? No, 16*5 = 80, so [box6] is 8? Wait no, 16*5 = 80, so 80 - 80 = 0, so [box6] is 8? Wait no, 16*5 = 80, so the number being subtracted is 80, so [box6] is 8? Wait no, the line is - [box6]0, so [box6]0 is 80, so [box6] is 8?

Wait let's list the boxes:

1. First box (tenths place in quotient): 0 (because 16 into 10 is 0)

2. Second box (hundredths place): 6 (because 16*6 = 96, 100 - 96 = 4)

3. Third box (thousandths place): 2 (because 16*2 = 32, 40 - 32 = 8)

4. Fourth box (ten - thousandths place): 5 (because 16*5 = 80, 80 - 80 = 0)

5. Fifth box (the one above 0 in the line with - 32): 4 (because 40 - 32 = 8, so the number is 40)

6. Sixth box (the one in - [box6]0): 8 (because 80 - 80 = 0, so the number is 80)

Wait let's verify the decimal equivalent of 1/16. 1/16 = 0.0625. So the quotient is 0.0625. So the four boxes after 0. are 0, 6, 2, 5. The fifth box (in the line with - 32) is 4 (since 40 - 32 = 8), and the sixth box (in - [box6]0) is 8 (since 80 - 80 = 0).

Step1: Determine the first decimal digit (tenths place)

16 into 10 (after decimal) is 0, so the first box (tenths) is 0.

Step2: Determine the second decimal digit (hundredths place)

16 into 100: 16*6 = 96, so the second box (hundredths) is 6.

Step3: Determine the third decimal digit (thousandths place)

After subtracting 96 from 100, remainder is 4. Bring down a 0 to get 40. 16 into 40: 16*2 = 32, so the third box (thousandths) is 2.

Step4: Determine the fourth decimal digit (ten - thousandths place)

After subtracting 32 from 40, remainder is 8. Bring down a 0 to get 80. 16 into 80: 16*5 = 80, so the fourth box (ten - thousandths) is 5.

Step5: Determine the fifth box (in the line with - 32)

After subtracting 96 from 100, remainder is 4. Bring down a 0 to get 40. So the number in the line is 40, so the fifth box (the tens place of 40) is 4.

Step6: Determine the sixth box (in - [box6]0)

After subtracting 32 from 40, remainder is 8. Bring down a 0 to get 80. We subtract 80 (16*5) from 80, so the number is 80, so the sixth box (the tens place of 80) is 8.

Answer:

The digits in the boxes (from left to right, top to bottom for the quotient and then the other boxes):

First box (tenths): 0

Second box (hundredths): 6

Third box (thousandths): 2

Fourth box (ten - thousandths): 5

Fifth box (in the line with - 32): 4

Sixth box (in - [box]0): 8

So the quotient is 0.0625, the fifth box is 4, and the sixth box is 8.