Adding and Subtracting Decimals Study Guide (page 3)

Introduction to Adding and Subtracting Decimals

What would life be without arithmetic, but a scene of horrors?

— SYDNEY SMITH, English writer (1771–1845)

This second decimal lesson focuses on addition and subtraction of decimals. It ends by teaching you how to add or subtract decimals and fractions together.

You have to add and subtract decimals all the time, especially when dealing with money. This lesson shows you how and gives you some word problems to demonstrate how practical this skill is in real life as well as on tests.

Adding Decimals

There is a crucial difference between adding decimals and adding whole numbers; the difference is the decimal point. The position of this point determines the accuracy of your final answer; a problem solver cannot simply ignore the point and add it in wherever it "looks" best. In order to add decimals correctly, follow these three simple rules:

1. Line the numbers up in a column so their decimal points are aligned.
2. Tack zeros onto the ends of shorter decimals to keep the digits lined up evenly.
3. Move the decimal point directly down into the answer area and add as usual.

Example: 3.45 + 22.1 + 0.682

1.	Line up the numbers so their decimal points are even:
2.	Tack zeros onto the ends of the shorter decimals to fill in the "holes":
3.	Move the decimal point directly down into the answer area and add:

To check the reasonableness of your work, estimate the sum by using the rounding technique you learned in Lesson 6. Round each number you added to the nearest whole number, and then add the resulting whole numbers. If the sum is close to your answer, your answer is in the ballpark. Otherwise, you may have made a mistake in placing the decimal point or in the adding. Rounding 3.45, 22.1, and 0.682 gives you 3, 22, and 1. Their sum is 26, which is reasonably close to your actual answer of 26.232. Therefore, 26.232 is a reasonable answer.

Look at an example that adds decimals and whole numbers together. Remember: A whole number is understood to have a decimal point to its right.

Example: 0.6 + 35 + 0.0671 + 4.36

1.	Put a decimal point at the right of the whole number (35) and line up the numbers so their decimal points are aligned:
2.	Tack zeros onto the ends of the shorter decimals to fill in the "holes":
3.	Move the decimal point directly down into the answer area and add:

Subtracting Decimals

When subtracting decimals, follow the same initial steps as in adding to ensure that you're adding the correct digits and that the decimal point ends up in the right place.

Example: 4.8731 – 1.7

1.	Line up the numbers so their decimal points are aligned:
2.	Tack zeros onto the end of the shorter decimal to fill in the "holes":
3.	Move the decimal point directly down into the answer and subtract:

Subtraction is easily checked by adding the number that was subtracted to the difference (the answer). If you get back the other number in the subtraction problem, then your answer is correct. For example, let's check our last subtraction problem.

	Here's the subtraction:
1.	Add the number that was subtracted (1.7000) to the difference (3.1731):
2.	The subtraction is correct because we got back the other number in the subtraction problem (4.8731).

Checking your subtraction is so easy that you should never pass up the opportunity!

You can check the reasonableness of your work by estimating: Round each number to the nearest whole number and subtract. Rounding 4.873 and 1.7 gives 5 and 2. Since their difference of 3 is close to your actual answer, 3.1731 is reasonable.

Borrowing

Next, look at a subtraction example that requires "borrowing." Notice that borrowing works exactly the same as it does when you're subtracting whole numbers.

Example: 2 – 0.456

1. Put a decimal point at the right of the whole number (2) and line up the numbers so their decimal points are aligned: 2.
2. Tack zeros onto the end of the shorter decimal to fill in the "holes":

   0.456

   2.000

   0.456

3. Move the decimal point directly down into the answer and subtract after borrowing:

   

4. Check the subtraction by addition:

   

   Our answer is correct because we got back the first number in the subtraction problem.

Combining Addition and Subtraction

The best way to solve problems that combine addition and subtraction is to "uncombine" them; separate the numbers to be added from the numbers to be subtracted by forming two columns. Add each of the columns and you're left with two figures; subtract one from the other and you have your answer.

Example: 0.7 + 4.33 – 2.46 + 0.0861 – 1.2

1.	Line up the numbers to be added so their decimal points are aligned:
2.	Tack zeros onto the ends of the shorter decimals to fill in the "holes":
3.	Move the decimal point directly down into the answer and add:
4.	Line up the numbers to be subtracted so their decimal points are aligned:
5.	Tack zeros onto the end of the shorter decimal to fill in the "holes":
6.	Move the decimal point directly down into the answer area and add:
7.	Subtract the step 6 answer from the step 3 answer, lining up the decimal points, filling in the "holes" with zeroes, and moving the decimal point directly down into the answer area:

Working with Decimals and Fractions Together

When a problem contains both decimals and fractions, it's usually easiest to change the numbers to the same type, either decimals or fractions, depending on which you're more comfortable working with. Consult Lesson 6 if you need to review changing a decimal into a fraction and vice versa.

Example: Fraction-to-decimal conversion:

1.	Convert to its decimal equivalent:
2.	Add the decimals after lining up the decimal points and filling the "holes" with zeros:

Decimal-to-fraction conversion:

1.	Convert 0.37 to its fraction equivalent:
2.	Add the fractions after finding the least common denominator:

Both answers, 0.745 and , are correct. You can easily check this by converting the fraction to the decimal or the decimal to the fraction.