# 2.2: Estimate Sums and Differences Using Front-End Estimation

**At Grade**Created by: CK-12

**Practice**Front-End Estimation

Jack and Thomas are going shopping for new school clothes. They each intend to buy two pairs of pants, three shirts, a new backpack and a lunchbox. They go to the local shopping mall and find all of their new supplies. The pants cost $49.99 each, the shirts are $24.95 each, the backpacks are on sale for $22.49, and the lunchbox is priced at $5.65. Since Jack only has $200 with him, he tries to figure out his bill quickly to see if he has enough money on him.

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

### Front End Estimation

**Front-end estimation** is a particular way of rounding numbers to estimate sums and differences. To use front- end estimation, add or subtract only the numbers in the greatest place value. Then add the decimals rounded to the nearest tenth.

Let’s look at an estimation problem.

Estimate the sum using front-end estimation: \begin{align*}4.9+3.8+7.2\end{align*}

First, look only at the whole numbers and add the digits in the ones places.

\begin{align*}4+3+7=14\end{align*}

Next, look at the decimals and round them to the nearest whole number.

\begin{align*}0.9+0.8+0.2\end{align*}

0.9 is approximately 1

\begin{align*}0.8 + 0.2 = 1\end{align*}

So \begin{align*}0.9+0.8+0.2\end{align*}

Then, add the two together.

\begin{align*}14+2=16\end{align*}

The answer is 16.

This is a good estimate but not an exact answer.

Let’s look at another problem using subtraction.

Estimate the difference using front-end estimation: \begin{align*}9.52-3.39\end{align*}

First, look only at the whole numbers. Subtract the digits in the ones places.

\begin{align*}9-3=6\end{align*}

Next, look at the decimals and round them to the nearest whole number.

\begin{align*}0.52-0.39\end{align*}

0.52 is approximately 0.5

0.39 is approximately 0.4.

So \begin{align*}0.5-0.4=0.1\end{align*}

Then, add the two together.

\begin{align*}6+0.1=6.1\end{align*}

The answer is 6.1.

This is a good estimate but not an exact answer.

### Examples

#### Example 1

Earlier, you were given a problem about Jack and Thomas getting their new school items.

Jack is trying to add up his bill quickly to check to see if he has enough money.

His bill would be: \begin{align*}2 \times $49.99 + 3 \times $24.95 + $22.49 + $5.65\end{align*}

First, add the digits in the ones and tens place. Note that there are 2 pairs of pants for $49.99 and three shirts for $24.95.

\begin{align*}49+49+24+24+24+22+5=197\end{align*}

Next, let’s look at the decimals. You want to make these decimals easier to add.

Again, remember that there are two pairs of pants and three shirts.

0.99 is approximately 1

0.95 is approximately 1

0.49 is approximately 0.5

0.65 is approximately 0.6

\begin{align*}1+1+1+1+1+0.5+0.6=6.1\end{align*}

Then, add the two together.

\begin{align*}197+6.1=203.1\end{align*}

The answer is 203.1

The estimated cost of the school supplies is $203.10 and Jack has $200. Therefore, Jack did not bring enough money.

#### Example 2

Estimate the following sum.

\begin{align*}4.01 + 6.27 + 18.12 + 11.30\end{align*}

First, look only at the whole numbers and add the digits in the ones places.

\begin{align*}4+6+18+11=39\end{align*}

Next, look at the decimals and round them to the nearest whole number.

0.01 is approximately 0.

0.27 is approximately 0.3.

0.12 is approximately 0.1.

0.30 is approximately 0.3.

So \begin{align*}0.3+0.1+0.3=0.7\end{align*}

Then, add the two together.

\begin{align*}39+0.7=39.7\end{align*}

The answer is 39.7.

#### Example 3

Estimate the following sum

\begin{align*}3.43+6.14+4.51\end{align*}

First, add the digits in the ones places.

\begin{align*}3+6+4=13\end{align*}

Next, let’s look at the decimals. You want to make these decimals easier to add.

0.43 is approximately 0.4

0.14 is approximately 0.1

0.51 is approximately 0.5

\begin{align*}0.4+0.1+.5=1\end{align*}

Then, add the two together.

\begin{align*}13+1=14\end{align*}

The answer is 14.

#### Example 4

\begin{align*}8.238-4.596\end{align*}

First, add the digits in the ones places: \begin{align*}8-4=4\end{align*}

Next, let’s look at the decimals. You want to make these decimals easier to subtract.

0.238 is approximately 0.2

0.596 is approximately 0.6

\begin{align*}0.2-0.6=-0.4\end{align*}

Then, add the two together.

\begin{align*}4+(-0.4)=3.6\end{align*}

The answer is 3.6.

#### Example 5

\begin{align*}4.53+6.32+7.02+3.45\end{align*}

First, add the digits in the ones places: \begin{align*}4+6+7+3=20\end{align*}

Next, let’s look at the decimals. You want to make these decimals easier to add.

0.53 is approximately 0.5

0.32 is approximately 0.3

0.02 is approximately 0

0.45 is approximately 0.4

\begin{align*}0.5+0.3+0.4=1.2\end{align*}

Then, add the two together.

\begin{align*}20+1.2=21.2\end{align*}

The answer is 21.2.

### Review

Find each estimate by using front-end estimation.

- \begin{align*}45.67 + 3.04 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
45.67+3.04=−−−−− - \begin{align*}55.10 + 5.6 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
55.10+5.6=−−−−− - \begin{align*}88.99 - 2.10 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
88.99−2.10=−−−−− - \begin{align*}80.09 - 12.78 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
80.09−12.78=−−−−− - \begin{align*}34.75 - 3.05 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
34.75−3.05=−−−−− - \begin{align*}5.67 + 3.87 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
5.67+3.87=−−−−− - \begin{align*}235.56 - 120.45 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
235.56−120.45=−−−−− - \begin{align*}17.8 + 12.3 + 5.3 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
17.8+12.3+5.3=−−−−− - \begin{align*}33.1 + 11.4 + 2.8 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
33.1+11.4+2.8=−−−−− - \begin{align*}18.11 + 25.4 + 2.1 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
18.11+25.4+2.1=−−−−− - \begin{align*}34.1 - 10.123= \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
34.1−10.123=−−−−− - \begin{align*}301.1 - 10.2345 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
301.1−10.2345=−−−−− - \begin{align*}12.234 + 15.1004 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
12.234+15.1004=−−−−− - \begin{align*}2.00987 + 5.0123 + 8.118 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
2.00987+5.0123+8.118=−−−−− - \begin{align*}3.0045 - 1.0008 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
3.0045−1.0008=−−−−−

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.2.

### Resources

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

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