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# 2.8: Decimal Subtraction Using Front-End Estimation

Difficulty Level: At Grade Created by: CK-12
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Practice Decimal Subtraction Using Front-End Estimation

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Remember the track question from the last Concept? Take a look.

The girls had won the Championship and had been re - thinking their times over a sundae.

Here are the times.

53.87 The winning time last year

53.73 The girls' winning time this year

What is the difference between the two times?

You could simply subtract, but can you use front - end estimation to calculate the difference?

This Concept is all about how to use front - end estimation to find differences. Pay attention and you can do this too at the end of the Concept.

### Guidance

You probably use estimation—the process of finding an approximate solution to a problem—without even thinking about it. When you figure out how much money you might need to bring to the movies, you use estimation. When you figure out how long you might wait for the bus, or how far it is to the corner store, you’re estimating values.

There are many different techniques for estimating, some of which provide “better” estimates than others. Rounding numbers before performing an operation is one way to find an estimate.

Front-end estimation is another way to estimate.

Good question! Let’s review the steps to using front-end estimation.

The steps to front-end estimation when estimating decimal differences are a little different.

Because subtraction often requires regrouping, we need to subtract the two leftmost digits at the same time. All you do is subtract the two front end—or leftmost—digits. If we were working with whole numbers, we would then need to insert zeros for the other place values. When working with decimals, however, we don’t need to do this. Where there are blank spaces at the end of decimals, zeros are assumed so put them in anyway.

Here are the steps:

Front-End Estimation for Subtraction:

--Subtract the two (leftmost) digits at the same time

Make a few notes on front-end estimation then continue.

Use front-end estimation to find the difference. 0.55373 - 0.27449

We’re going to use front-end estimation to subtract 0.27449 from 0.55373. Let’s line up our decimal points and take a closer look at the numbers. Then we’ll perform front-end estimation.

0.55373

0.27449

Front-end estimation tells us to subtract the two front or leftmost digits at the same time, so we want to subtract the tenths and hundredths places.

0.550.270.28\begin{align*}& \quad 0.55\\ &\underline{- \; 0.27}\\ & \quad 0.28\end{align*}

The original numbers extended to the hundred-thousandth place, you should put them in to convince yourself that you could keep writing zeros to infinity but the value of the decimal will remain the same, so we don’t need to show this.

Use front-end estimation to find the difference, 4.032 - 0.82

Let’s begin by lining up the decimal places—adding zeros as necessary—take a close look at the numbers. Then we’ll perform front-end estimation.

4.038

0.820

Front-end estimation tells us to subtract the two front or leftmost digits at the same time, so we want to subtract the ones and the tenths places.

4.00.83.2\begin{align*}& \quad 4.0\\ & \underline{- \; 0.8}\\ & \quad 3.2\end{align*}

One of the numbers extended to the thousandth place but we hide the unnecessary zeros in our answer. Where decimals end, it is assumed that zeros extend on to infinity.

Use front-end estimation to find the following differences.

#### Example A

4.67121.34=\begin{align*}4.6712 - 1.34 = \underline{\;\;\;\;\;\;\;\;}\end{align*}

Solution: 3.3\begin{align*}3.3\end{align*}

#### Example B

6.7893.45=\begin{align*}6.789 - 3.45 = \underline{\;\;\;\;\;\;\;\;}\end{align*}

Solution: 3.3\begin{align*}3.3\end{align*}

#### Example C

5.0232.346=\begin{align*}5.023 - 2.346 = \underline{\;\;\;\;\;\;\;\;}\end{align*}

Solution: 2.7\begin{align*}2.7\end{align*}

Now back to the original problem. Here it is once again.

The girls had won the Championship and had been re - thinking their times over a sundae.

Here are the times.

53.87 The winning time last year

53.73 The girls' winning time this year

What is the difference between the two times?

To find the difference using front - end estimation, we simply use the first two digits of each value. But because they are the same and would equal 0, we are going to use the tenths place and the hundredths place.

.87

.73

.14\begin{align*}.14\end{align*}

The difference between the two winning times is .14\begin{align*}.14\end{align*}.

### Vocabulary

Here are the vocabulary words in this Concept.

Difference
a key word that means subtraction.
Estimate
to find an approximate answer to a problem.
Rounding
one method of estimating where a number is changed according to the place value that it is closest to.
Front-end Estimation
a method of estimating. With subtraction, you only subtract the first two places at the same time.

### Guided Practice

Here is one for you to try on your own.

Use front - end estimation to find this difference.

.22281.1946\begin{align*}.22281 - .1946\end{align*}

To work with this problem, we simply subtract the first two places after the decimal point. We are going to work with the tenths and hundredths places.

.22

.19

.22.19=.03\begin{align*}.22 - .19 = .03\end{align*}

### Practice

Directions: Use front-end estimation to estimate the following differences.

1. 4.572.34\begin{align*}4.57 - 2.34\end{align*}

2. 2.1231.136\begin{align*}2.123 - 1.136\end{align*}

3. 8.9132.047\begin{align*}8.913 - 2.047\end{align*}

4. 8.76512.345\begin{align*}8.7651 - 2.345\end{align*}

5. 2.4361.567\begin{align*}2.436 - 1.567\end{align*}

6. 18.1279.431\begin{align*}18.127 - 9.431\end{align*}

7. 8.2147.3214\begin{align*}8.214 - 7.3214\end{align*}

8. 12.1372.456\begin{align*}12.137 - 2.456\end{align*}

9. .671.41\begin{align*}.671 - .41\end{align*}

10. .643.123\begin{align*}.643 - .123\end{align*}

11. .912.367\begin{align*}.912 - .367\end{align*}

12. .456.023\begin{align*}.456 - .023\end{align*}

13. .48218.1927\begin{align*}.48218 - .1927\end{align*}

14. 6.07656.421192\begin{align*}6.0765 - 6.421192\end{align*}

15. .5075412.45601\begin{align*}.5075412 - .45601\end{align*}

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