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2.14: Estimation to Check Decimal Division

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Have you ever been to a science lab?

While at the science museum, Marc enjoyed looking at the scientists that were working in the lab. They were working with microscopes and tiny fragments of materials. Marc wasn't even sure what they were working on, but it was fascinating to see real scientists at work in a lab.

On the board there was this problem written.

.36007809 \div .0234

Marc is curious how to figure this out. He knows that there is a short - cut, but isn't sure what it is.

To figure this out, Marc can divide leading digits. This Concept will show you how to confirm decimal quotients by dividing leading digits.

Guidance

Estimation is a process by which we approximate solutions. Estimating either before or after solving a problem helps to generalize or confirm a solution.

Rounding decimals before division is one way to find an estimate. You can also simply divide the leading digits.

Remember how we estimated products by multiplying the leading digits?

Estimating quotients works the same way. Leading digits are the first two values in a decimal. To estimate a quotient, divide the leading digits exactly as you have been—move the decimal point in the divisor to make it a whole number, adjust the decimal point in the dividend accordingly, then divide, inserting the decimal point in the solution in line with its position in the dividend.

Estimate the quotient. 7.882 \div .4563

To estimate, we are going to work with only the leading digits, or the first two.

Let’s examine the decimals and reduce them to their leading digits.

7.882 \rightarrow 7.8 and .4563 \rightarrow .45.

Now let’s move the decimal point in the divisor and dividend.

In long-division form, we have {.45 \overline{ ) {7.8 \;}}} so we’re going to multiply both numbers by 100 and move the decimal points two places to the right: {45 \overline{ ) {780 \;}}}. Notice how we added a zero to the end of the dividend to make the move of the decimal point possible.

Now we can divide.

& \overset{ \quad 17.33}{45 \overline{ ) {780.00 \;}}}\\& \quad \underline{- \; 45 \;\;}\\& \qquad 330\\& \quad \underline{- \; 315\;\;}\\& \qquad \ \ 150\\& \quad \ \ \underline{- \; 135\;\;}\\& \qquad \quad \ 150\\& \qquad \ \underline{- \; 135}\\& \qquad \qquad 15

Notice how we added 2 extra zeros to the dividend to facilitate our division? We could keep adding zeros and keep dividing and our quotient would get longer and longer.

Can you notice a pattern in the quotient?

Each time we add a zero to the dividend, we come up with another 3 in the quotient. This is another repeating decimal. We can either round our answer to 17.33 or notate the repeating decimal by putting a line over the repeating part 17.\overline{3}.

Our answer is 17.33 or 17.\overline{3}.

Here is another one.

4.819 \div 1.245

First, let’s simplify these numbers to their leading digits.

4.819 = 4.8

1.245 = 1.2

Now we can set up the problem as a division problem.

{1.2 \overline{ ) {4.8 \;}}}

Next, we make the divisor into a whole number by multiplying by 10. We do the same thing in the dividend. Next, we can divide to find the quotient.

\overset{ \quad 4}{12 \overline{ ) {48 \;}}}

Our answer is 4.

Divide by using leading digits. You may round your answer to the nearest hundredth if necessary.

Example A

15.934 \div 2.57

Solution: 6

Example B

4.368 \div 3.12

Solution: 1.39

Example C

6.16 \div 1.12

Solution: 5.55

Here is the original problem once again.

While at the science museum, Marc enjoyed looking at the scientists that were working in the lab. They were working with microscopes and tiny fragments of materials. Marc wasn't even sure what they were working on, but it was fascinating to see real scientists at work in a lab.

On the board there was this problem written.

.36007809 \div .0234

Marc is curious how to figure this out. He knows that there is a short - cut, but isn't sure what it is.

First, we can identify the leading digits.

.36007809 = .36

.0234 = .02

Now we write our new division problem.

.36 \div .02

Next, we divide.

Our answer is 18.

Vocabulary

Here are the vocabulary words in this Concept.

Divisor
the number outside the division box. This is the number that is doing the dividing.
Dividend
the number being divided. It is the number inside the division box.
Quotient
the answer in a division problem.
Estimation
using rounding or dividing leading digits to find an approximate answer.

Guided Practice

Here is one for you to try on your own.

Divide by using leading digits. Round the quotient to the nearest tenth if necessary.

6.4256 \div 2.2453

Answer

First, we identify the leading digits.

6.4

1.2

Now we can set up a division problem and divide.

6.4 \div 2.2 = 2.9090909

Notice the repeating pattern in the quotient, but we can round to the nearest tenth.

2.9

This is our answer.

Practice

Directions: Estimate the quotient by dividing the leading digits. You can round to the nearest tenth when necessary.

1. 4.992 \div .07123

2. 1.8921 \div 6.0341

3. 26.2129 \div 1.5612

4. 1.00765 \div .33

5. 36.2129 \div 2.5612

6. .42129 \div .15612

7. 6.2129 \div 1.2612

8. 26.2129 \div 13.5612

9. 8.2129 \div 2.2612

10. 42.2129 \div 8.2612

11. 16.2129 \div 4.1612

12. 19.0029 \div 3.599

13. .45632 \div .09123

14. 8.765 \div 1.098

15. .145632 \div .701023

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Basic

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Date Created:

Nov 30, 2012

Last Modified:

Mar 17, 2014
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