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# 2.8: Decimal Subtraction Using Front-End Estimation

Difficulty Level: At Grade Created by: CK-12

Courtney’s house has a sour cherry tree in the backyard. Courtney loves picking the cherries and making jam with them. Her recipe for jam calls for 2.75 cups of sugar. Courtney just opened a new bag 3 pound bag of sugar that has 6.8571 cups of sugar in it. Courtney is planning to make one recipe’s worth of jam, but she wants to keep track of how much sugar she has in case she wants to make more jam. How can Courtney figure out approximately how much sugar she will have left after she makes one recipe’s worth of jam?

In this concept, you will learn how to estimate decimal differences using front-end estimation.

### Front End Estimation

Recall that when you estimate you are finding an approximate solution to a problem.

There are different methods for estimation. The front-end estimation method relies on focusing on only the first two digits of each number.

Here are the steps for front-end estimation with decimal number subtraction.

1.  Arrange the numbers vertically so that the decimal points line up.
2.  Subtract the two front (left-most) pairs of digits at the same time.

Here is an example.

Use front-end estimation to find the difference: 0.553730.27449.\begin{align*}0.55373 - 0.27449.\end{align*}

First, arrange the numbers vertically so that the decimal points line up.

0.55373

0.27449

Now, the two front pairs of digits are the digits in the tenths place and the digits in the hundredths place. Subtract these digits only. Note that you can ignore the ones place because those digits are both zero. Set up the subtraction problem.

0.55 0.27

Next, subtract. Start by borrowing from the 5 and subtracting 15 minus 7 to make 8. Then 4 minus 2 to make 2. Insert a decimal point into your answer under the decimal points in the original numbers.

0.55 0.27    0.28

The answer is 0.553730.27449\begin{align*}0.55373 - 0.27449\end{align*} is approximately 0.28.

Here is another example.

Use front-end estimation to find the difference: 4.0320.82\begin{align*}4.032-0.82\end{align*}.

First, arrange the numbers vertically so that the decimal points line up.

4.032

0.82

Now, the two front pairs of digits are the digits in the ones place and the digits in the tenths place. Subtract these digits only. Set up the subtraction problem.

4.0 0.8
Next, subtract. Start by borrowing from the 4 and subtracting 10 minus 8 to make 2. Then 3 minus 0 to make 3. Insert a decimal point into your answer under the decimal points in the original numbers.

4.0 0.8    3.2

The answer is 4.0320.82\begin{align*} 4.032 - 0.82\end{align*} is approximately 3.2.

### Examples

#### Example 1

Earlier, you were given a problem about Courtney and her cherry jam. Her new bag of sugar has 6.8571 cups of sugar in it and Courtney is planning to use 2.75 cups of sugar to make cherry jam. She needs to know approximately how many cups of sugar she will have left after making the jam.

First, Courtney should arrange the numbers vertically so that the decimal points line up.

6.8571

2.75

Now, the two front pairs of digits are the digits in the ones place and the digits in the tenths place. Courtney will subtract these digits only. Set up the subtraction problem.

6.8 2.7
Next, subtract. Start by subtracting 8 minus 7 to make 1. Then 6 minus 2 to make 4. Courtney should insert a decimal point into her answer under the decimal points in the original numbers.

6.8 2.7    4.1

The answer is 6.85712.75\begin{align*}6.8571 - 2.75\end{align*} is approximately 4.1. Courtney will have approximately 4.1 cups of sugar left after making her jam.

#### Example 2

Use front-end estimation to find the difference:0.222810.1946.\begin{align*} 0.22281 - 0.1946.\end{align*}

First, arrange the numbers vertically so that the decimal points line up.

0.22281

0.1946

Now, the two front pairs of digits are the digits in the tenths place and the digits in the hundredths place. Subtract these digits only. Set up the subtraction problem.

0.22 0.19

Next, subtract. Start by borrowing from the 2 and subtracting 12 minus 9 to make 3. Then 1 minus 1 to make 0. Insert a decimal point into your answer under the decimal points in the original numbers.

0.22 0.19    0.03
The answer is 0.222810.1946\begin{align*}0.22281 - 0.1946 \end{align*} is approximately 0.03 .

#### Example 3

Use front-end estimation to find the difference:4.67121.34.\begin{align*}4.6712 - 1.34.\end{align*}

First, arrange the numbers vertically so that the decimal points line up.

4.6712

1.34

Now, the two front pairs of digits are the digits in the ones place and the digits in the tenths place. Subtract these digits only. Set up the subtraction problem.

4.0 1.3

Next, subtract. Start by borrowing from the 4 and subtracting 10 minus 3 to make 7. Then 3 minus 1 to make 2. Insert a decimal point into your answer under the decimal points in the original numbers.

4.0 1.3    2.7

The answer is 4.67121.34\begin{align*}4.6712 - 1.34 \end{align*} is approximately 2.7 .

#### Example 4

Use front-end estimation to find the difference 6.7893.45.\begin{align*}6.789 - 3.45.\end{align*}

First, arrange the numbers vertically so that the decimal points line up.

6.789

3.45

Now, the two front pairs of digits are the digits in the ones place and the digits in the tenths place. Subtract these digits only. Set up the subtraction problem.

6.7 3.4
Next, subtract. Start by subtracting 7 minus 4 to make 3. Then 6 minus 3 to make 3. Insert a decimal point into your answer under the decimal points in the original numbers.

6.7 3.4    3.3

The answer is 6.7893.45\begin{align*}6.789 - 3.45 \end{align*} is approximately 3.3.

#### Example 5

Use front-end estimation to find the difference: 5.0232.346.\begin{align*} 5.023 - 2.346.\end{align*}

First, arrange the numbers vertically so that the decimal points line up.

5.023

2.346

Now, the two front pairs of digits are the digits in the ones place and the digits in the tenths place. Subtract these digits only. Set up the subtraction problem.

5.0 2.3
Next, subtract. Start by borrowing from the 5 and subtracting 10 minus 3 to make 7. Then 4 minus 2 to make 2. Insert a decimal point into your answer under the decimal points in the original numbers.

5.0 2.3    2.7
The answer is 5.0232.346\begin{align*}5.023 - 2.346\end{align*} is approximately 2.7.

### Review

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. 2.4361.267\begin{align*}2.436-1.267\end{align*}
6. 18.5279.431\begin{align*}18.527 - 9.431\end{align*}
7. 8.4147.3214\begin{align*} 8.414-7.3214\end{align*}
8. 12.6372.456\begin{align*}12.637-2.456\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. 10.57656.421192\begin{align*}10.5765- 6.421192\end{align*}
15. .5075412.45601\begin{align*} .5075412 -.45601\end{align*}

To see the Review answers, open this PDF file and look for section 2.8.

1. [1]^ License: CC BY-NC 3.0

## Date Created:

Dec 02, 2015

Dec 02, 2015
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