Decimals
In This Chapter
- Understanding the decimal system
- Adding and subtracting numbers with decimals
- Multiplying and dividing numbers with decimals
Like fractions, working with decimals is something many people find a bit difficult to handle. For any type of problem, the placement of decimals is a precise and sometimes confusing process. In this chapter, I throw some light on the concept of decimals and show you how to make it a little easier.
The Decimal System and Place Value
In the decimal number system, the position of the number determines its value—this is known as place value.
The principle of place value is that each place has a value 10 times the place to its right. Inversely, a position to the right is 10 times smaller than a value on the left. For example, in the number 47, the 4 is in the tens place, meaning it has a value of four 10s, while the 7 is in the ones place, meaning it has a value of seven 1s. Without place value, calculations would be extremely difficult, because place value helps us understand the meaning of a number. If people just used numbers randomly, no one would know which numbers people meant. You need place value to understand the order of numbers as well.
The decimal point is used to distinguish between whole numbers and parts of a whole. For example, 0.1 is one-tenth part of one, 0.01 is a one-hundredth part of one, and 0.001 is one-thousandth part of one.
The following table gives you a breakdown of the place values around a decimal point. As you can see, numbers get larger the farther left you go and smaller the farther right you go.
Adding Decimal Numbers
When adding numbers with decimals, keep the decimal points in a vertical line. You can do the problems from left to right or right to left. In some cases, you can make the process even easier by first adding without the decimals and then putting in the decimal point at the end.
Example 1
Solve the problem 4.34 + 3.42.
Step 1: For this problem, ignore the decimal point and add the two numbers.
Step 2: Put the decimal place back in the answer. Note that in these numbers, the decimal point is two places before the end digit. Therefore, you should put the decimal point two places before the end digit.
Solution: The answer is 7.76.
Example 2
Solve the problem 78.3 + 2.031 + 2.3245 + 9.2.
Step 1: Set up the problem vertically, with the decimal points in alignment.
Step 2: Add the numbers.
Solution: The answer is 91.8555.
QUICK TIP
You can also add zeroes to the numbers in example 2—making them 78.3000, 02.031, 02.3245, and 09.2000—so they all have the name number of digits both before and after the decimal.
Solve the problem 0.0004 + 6.32 + 1.008 + 3.452.
Step 1: Set up the problem vertically, with the decimal points in alignment.
Step 2: Add the numbers.
Solution: The answer is 10.7804.
Subtracting Decimal Numbers
The process for subtracting numbers with decimals isn’t that different from what you do when adding them. When subtracting decimal numbers, you first line up the decimal points—tens under tens, ones under ones, and so on. This makes it simpler to understand and solve the problem.
Example 1
Solve the problem 45 − 2.09.
Step 1: Set up the problem vertically, with the decimal points in alignment. Write 45 as 45.00 because 2.09 has two digits after the decimal points. Adding the zeroes makes it easier and safer to get the answer, because the ones are below the ones, the tens are under the tens, and so on.
Step 2: Subtract without considering the decimal points: 4,500 − 209 = 4,291. Put the decimal point in the answer again.
Solution: The answer is 42.91.
Example 2
Solve the problem 7.005 − 0.55.
Step 1: Set up the problem vertically, with the decimal points in alignment. Put a zero after 0.55 so it’s 0.550; this makes the subtraction easier and clearer.
Step 2: Subtract without considering the decimal points: 7,005 − 0,550 = 6,455. Put the decimal point in the answer again.
Solution: The answer is 6.455.
Example 3
Solve the problem 19.19 − 3.3.
Step 1: Set up the problem vertically, with the decimal points in alignment. Add zeroes around 3.3 so it’s 03.30.
Step 2: Subtract without considering the decimal points: 1,919 − 0,330 = 1,589. Put the decimal point in the answer again.
Solution: The answer is 15.89.
Multiplying Decimal Numbers
Knowing how to multiply numbers in relation to a decimal point can be frustrating. Where should the decimal go? What order do you calculate the numbers? Let me clear up any confusion for you by showing you some tips and tricks for multiplying numbers with decimals.
Multiplying by Powers of 10
Multiplying a number with a decimal by a power of 10 is very easy; it just involves moving the decimal point.
Say you have to multiply 7.86 by 10. Because 10 has one zero, you only need to move the decimal point one place to the right to get your answer: 78.6. And if you’re multiplying 7.86 by 100, you move the decimal point two places to right to account for the two zeroes in 100. This gives you an answer of 786. Simple, right?
The following are examples of what happens to a number with a decimal when it’s multiplied by different powers of 10. You should have no trouble calculating these in your head once you get comfortable with how you need to move the decimal.
Multiplying Vertically and Crosswise
It’s very easy to multiply numbers with decimals using the vertical and crosswise method.
If you recall, to do the vertically and crosswise method for two-digit numbers, you first multiply the right column. You then cross-multiply the numbers, and finish by multiplying the left column.
Multiplying a three-digit number by another three-digit number requires a couple more steps in the vertically and crosswise method—vertical, crosswise, star, crosswise, and vertical. See Chapter 1 if you need a visual, and keep these two versions in your head as you work the following examples.
Example 1
Solve the problem 7.3 × 1.4.
Step 1: Ignoring the decimals at first and following the vertically and crosswise pattern, begin by multiplying the right column: 4 × 3 = 12. Put down 2 and carry the 1 to the next step.
Step 2: Cross-multiply and add: (4 × 7) + (1 × 3) = 28 + 3 = 31. Add the carryover: 31 + 1 = 32. Put down 2 and carry the 3 to the next step.
Step 3: Multiply the left column: 1 × 7 = 7. Add the carryover: 7 + 3 = 10. Put down 10.
Step 4: Put the decimal in the sum. In each of the numbers, the decimal point is one place from the right; therefore, the decimal point should be two places from the right.
Solution: The answer is 10.22.
QUICK TIP
To put the decimal place in the correct spot, always count the number of places from the right in both the numbers and add them. Once you know the total number of decimal places, count over that many from the right and place the decimal point in the answer.
Example 2
Solve the problem 6.2 × 5.4.
Step 1: Ignoring the decimals and applying the vertically and crosswise formula, multiply the right column: 2 × 4 = 8. Put down 8.
Step 2: Cross-multiply and add: (4 × 6) + (5 × 2) = 24 + 10 = 34. Put down 4 carry over the 3 to the next step.
Step 3: Multiply the left column: 6 × 5 = 30. Add the carryover: 30 + 3 = 33. Put down 33.
Step 4: Put the decimal in the sum. In each of the numbers, the decimal point is one place from the right; therefore, the decimal point should be two places from the right.
Solution: The answer is 33.48.
Example 3
Solve the problem 3.42 × 71.5.
Step 1: Ignoring the decimals and applying the vertically and crosswise formula, multiply the right column: 2 × 5 = 10. Put down the 0 and carry over the 1.
Step 2: Cross-multiply the numbers in the right and middle columns and add them: (5 × 4) + (1 × 2) = 22. Add the carryover : 22 + 1 = 23. Put down 3 and carry over the 2 to the next step.
Step 3: Cross-multiply the numbers in the left and right columns, vertically multiply the numbers in the center, and add them: (5 × 3) + (7 × 2) + (1 × 4) = 33. Add the carryover: 33 + 2 = 35. Put down 5 and carry over the 3 to the next step.
Step 4: Cross-multiply the numbers in the left and center columns and add them: (1 × 3) + (7 × 4) = 31. Add the carryover: 31 + 3 = 34. Put down 4 and carry over the 3 to the next step.
Step 5: Multiply the left column: 7 × 3 = 21. Add the carryover: 21 + 3 = 24. Put down 24.
Step 6: Put the decimal in the sum. In 3.42, the decimal point is two places from the right; in 71.5, the decimal point is one place from the right. Therefore, the decimal should be three places from the right in the answer.
Solution: The answer is 244.530.
Dividing Decimal Numbers
Now that you’ve learned how to multiply decimal numbers, you can take on a more complex process—dividing decimal numbers. In the following sections, I give you some different scenarios for dividing decimals and ways to painlessly solve the problems.
Dividing by Powers of 10
The process for dividing by a power of 10 is exactly opposite to the process of multiplying by a power of 10. When you divide by a power of 10, instead of moving the decimal point to the right, you move it to the left to get your answer.
The number of places you move the decimal point depends on the number of zeroes in the power of 10. For example, in the problem 3.17 ÷ 10, the 10 has one zero; therefore, you move the decimal point one place to the left to get your answer: 0.317. If you were dividing by 100 instead, you’d move the decimal two places to the left, making the answer 0.0317.
The following are some examples of what happens to a number when it’s divided by different powers of 10. Like multiplying decimals, you should have little trouble calculating these in your hand once you understand the placement of the decimal in relation to the number of zeroes in the power of 10.
Dividing a Decimal Number by a Whole Number
The fastest and easiest way to divide a decimal number by a whole number is to remove the decimal point and treat them both as whole numbers. Once you get the sum, you then add the decimal back in the same place as it was in the dividend.
Example 1
Solve the problem 9.1 ÷ 7.
Step 1: Remove the decimal and divide the two numbers.
91 ÷ 7 = 13
Step 2: Put the decimal point in your answer, in the same place as the dividend. Because the decimal point in 9.1 is one place from the right, you should put the decimal one place from the right in your answer.
Solution: The answer is 1.3.
Solve the problem 5.26 ÷ 2.
Step 1: Remove the decimal and divide the two numbers.
526 ÷ 2 = 263
Step 2: Put the decimal point in our answer, right in the same place as it was in the dividend. Because the decimal point in 5.26 is two places from the right, you should put the decimal two places from the right in your answer.
Solution: The answer is 2.63.
SPEED BUMP
Because it’s so much simpler to divide without the decimal in place, you may forget about it entirely. Remember to add back the decimal at the end!
Dividing a Decimal Number by Another Decimal Number
So what do you do when you want to divide a decimal number by another decimal number? The trick is to convert the numbers you’re dividing by to whole numbers first by shifting the decimal points to the right.
Example 1
Solve the problem 567.29 ÷ 45.67.
Step 1: Because the decimal is two places to the right in each number, you shift the decimal points right two places to get the whole numbers. So 567.29 becomes 56,729 and 45.67 becomes 4,567.
Step 2: Divide the numbers.
56729 ÷ 4567 = 12.4215
Solution: The answer is 12.4215.
Solve the problem 7.625 ÷ 0.923.
Step 1: Because the decimal is three places to the right in each number, you shift the decimal points right three places to get the whole numbers. So 7.625 becomes 7,625 and 0.923 becomes 923.
Step 2: Divide the numbers.
7625 ÷ 923 = 8.2611
Solution: The answer is 8.2611.
Example 3
Solve the problem 52.3 ÷ 8.1.
Step 1: Because the decimal places are in different places in the numbers, multiply both numbers by 10 in order to remove the decimal points.
52.3 × 10 = 523
8.1 × 10 = 81
Step 2: Divide the two numbers. You don’t need to add the decimal back to the answer, because you multiplied both by the same base.
523 ÷ 81 = 6,456
Solution: The answer is 6.456.
The Least You Need to Know
- A decimal point is used to distinguish between whole numbers and parts of a whole.
- When adding and subtracting numbers with decimal points, ignore the decimal when doing your calculations until the very end. You can then place it back in the answer.
- To multiply two-digit or three-digit numbers with decimals, you can use the vertically and crosswise method.
- When multiplying a decimal number by a power of 10, you simply move the decimal point to the right based on the number of zeroes to get your answer; when dividing, you move the decimal point to the left.