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Operations with Decimals

Operations with Decimals

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Understanding Decimals: An Introduction to Decimal Numbers

In mathematics, numbers can be classified as either whole numbers or non-whole numbers. While fractions can be used to represent values between whole numbers, there is another type of numerical representation called decimals. In this article, we will delve into the definition of decimals, explore how they are represented and used, and provide examples to aid in understanding this concept.

What are Decimals? A Definition and Explanation

A decimal is a numerical value that contains a decimal point, which is represented by a dot. This decimal point separates the whole number from its fractional part. The value of the fraction is always less than one, and the number of digits after the decimal point determines the number of decimal places. To better visualize this concept, imagine a number line with decimal numbers marked on it, similar to the one shown below.

Understanding Decimals Using Place Value Tables

Another useful tool for comprehending how decimals work is a place value table. This table helps identify the number of decimal places, which is crucial when performing operations involving decimals. The table is created by dividing the number into its whole and decimal parts, as illustrated below.

The digits before the decimal point represent the whole number while the digits after the decimal point represent the decimal part. For instance, in the number 12.45, 12 is the whole number component, and 0.45 is the decimal part. In this case, there are four decimals places, as there are four digits after the decimal point.

Converting Fractions and Percentages to Decimals

To convert a fraction to a decimal, use a calculator to divide the numerator by the denominator. To convert a percentage to a decimal, remove the % sign and divide the remaining number by 100. For example, 45% becomes 0.45, and 3.67% becomes 0.0367.

The Order of Operations with Decimals

When performing basic arithmetic operations with decimals, it is important to follow the order of operations. Remember the acronym PEMDAS: Parenthesis, Exponent, Multiplication/Division, Addition/Subtraction. This is especially crucial when dealing with mixed operations involving decimals.

Adding and Subtracting Decimals: Demonstration and Explanation

The process of adding and subtracting decimals is similar to that of whole numbers. We must start from the right and use the column method to maintain the correct place values and decimal places. The following examples illustrate this concept.

Example 1: Add 5.7 and 8.9

Solution: Write the numbers in column form, add zeros as placeholders to make the numbers the same length, then add using column addition. Finally, place the decimal point in the answer, lined up with the decimal points of the numbers being added.

5.7
+ 8.9
14.6

Example 2: Subtract 2.3 from 4.8

Solution: Write the numbers in column form, add zeros as placeholders to make the numbers the same length, then subtract using column subtraction. Again, place the decimal point in the answer, lined up with the decimal points of the numbers being subtracted.

4.8
- 2.3
2.5

Example 3: Add 6 and 4.3

Solution: This example involves the addition of a whole number and a decimal number. To make them the same length, add a zero to the end of the whole number, then follow the same steps as before.

6.0
+ 4.3
10.3

Example 4: Subtract 5 from 9.2

Solution: Similar to the previous example, this one involves subtracting a decimal number from a whole number. Add a zero to the end of the whole number to make them the same length, then follow the steps for column subtraction.

9.2
- 5.0
4.2

Multiplying Decimals: Rules and Guidelines

When multiplying decimals, it is important to follow certain rules to ensure accuracy. These include:

  • Remove the decimal point from the numbers being multiplied and perform the multiplication as usual.
  • Count the number of decimal places in each of the original numbers and add them together. The sum will be the number of decimal places in the final answer.
  • Finally, place the decimal point in the correct position in the answer, counting from the right.

Multiplying Decimals

When multiplying decimals, the final answer's place value is determined by the sum of the decimal places in the numbers multiplied in the previous step. To find the product of two decimals, follow these steps:

  • Ignore the decimal points and multiply the whole numbers.
  • Count the total number of decimal places in the two numbers multiplied. This sum is the number of decimal places in the final answer.
  • Place the decimal point in the product at the same position as the number calculated in step 2.

Let's see this technique in action with some examples:

Multiply 3.6 by 2.3

Ignoring the decimal places, we get 36 x 23 = 826. Since both numbers have one decimal place each, the sum is two. Therefore, the product of 36 x 23 must have two decimal places. Placing the decimal point, we get 3.6 x 2.3 = 8.26.

Multiply 5.7 by 8

Ignoring the decimal places, we get 57 x 8 = 456. The first number has one decimal place while the second number has none. The sum of their decimal places is one. Thus, the product of 57 x 8 must have one decimal place. Placing the decimal point, we get 5.7 x 8 = 45.6.

Multiply 2.165 by 9.1

Ignoring the decimal places, we get 2165 x 91 = 197015. The first number has three decimal places while the second number has one. The sum of their decimal places is four. Therefore, the product of 2165 x 91 must have four decimal places. Placing the decimal point, we get 2.165 x 9.1 = 19.7015.

When multiplying a decimal by a power of 10, simply move the decimal point to the right based on the number of zeros present in the power of 10.

For example, multiply 3.87 by 100

The number 100 has two zeros, so we move the decimal point of 3.87 two places to the right. Therefore, 3.87 x 100 = 387.

Another example, multiply 7.3956 by 1000

The number 1000 has three zeros, so we move the decimal point of 7.3956 three places to the right. Therefore, 7.3956 x 1000 = 7395.6.

Dividing Decimals

When dividing numbers, it's easier to divide by a whole number rather than a decimal. To divide decimals, follow these steps:

  • Convert the decimal divisor into a whole number by multiplying it by a power of 10.
  • Multiply the dividend by the same power of 10.
  • Divide the new numbers using long division.

Let's look at some examples:

Divide 4.62 by 0.12

First, we have 4.62 ÷ 0.12 where the dividend is 4.62 and the divisor is 0.12.

To convert the divisor into a whole number, we multiply it by 100, giving us 0.12 x 100 = 12.

We then multiply the same value to the dividend, giving us 4.62 x 100 = 462.

Therefore, we have 462 ÷ 12, which is the same as 4.62 ÷ 0.12.

Conducting long division, we get 38.5 as the answer.

Divide 5.525 by 5

In this problem, the divisor is already a whole number. However, the dividend is still in the form of a decimal.

To tackle this, we will conduct long division while ignoring the decimal point of the dividend.

We then place the decimal point of the answer directly above the decimal point of the dividend.

Thus, 5.525 ÷ 5 = 1.105.

Divide 3.432 by 1.04

Here, we have 3.432 ÷ 1.04 where the dividend is 3.432 and the divisor is 1.04.

To convert the divisor into a whole number, we multiply it by 100, giving us 1.04 x 100 = 104.

We then multiply the same value to the dividend, giving us 3.432 x 100 = 343.2.

Therefore, we have 343.2 ÷ 104, which is the same as 3.432 ÷ 1.04.

Once again, we will conduct long division while ignoring the decimal point of the dividend.

Finally, we place the decimal point of the answer directly above the decimal point of the dividend.

Thus, 3.432 ÷ 1.04 = 3.3.

When dividing decimals by powers of 10, simply move the decimal point to the left based on the number of zeros present in the power of 10.

For example, divide 2.34 by 10

There is one zero in the number 10, so we move the decimal point of 2.34 one place to the left. Therefore, 2.34 ÷ 10 = 0.234.

Another example, divide 17635.6 by 1000

There are three zeros in the number 1000, so we move the decimal point of 17635.6 three places to the left.

17635.6 ÷ 1000 = 17.6356.

Performing Operations with Decimals Using PEMDAS

When solving equations involving decimals, it is important to follow the order of operations (PEMDAS). Here are some examples to better understand this concept.

Evaluating Equations with Decimals

Calculating equations with decimals may seem daunting, but with a few simple conversions and following the correct order of operations, it can be easily solved. Let's take a look at how to approach these equations.

Converting Percentages to Decimals

The first step in solving an equation with decimals is to convert any percentage terms to decimals. This is done by dividing the percentage by 100. For example, 4.5% becomes 4.5 ÷ 100 = 0.045. From here, we can perform subtraction just as we normally would, giving us a final result of 0.078.

Converting Fractions to Decimals

If the equation involves a fraction, we can convert it to a decimal by dividing the numerator by the denominator. For instance, 3/4 would become 3 ÷ 4 = 0.75. We can then add or subtract as usual, giving us a final answer of 1.32.

Following the Order of Operations

When an equation contains multiple operations, the order of operations must be followed. This means starting with any operations within parentheses, then moving on to exponents, followed by multiplication and division, and finally addition and subtraction. For example:

  • First, we calculate the expression within the parentheses: (2.5 + 3.2) = 5.7
  • Next, we conduct the multiplication: 5.7 x 2 = 11.4
  • Finally, we perform the subtraction: 11.4 - 3.5 = 7.9

Similarly, if the equation involves a grouping, we must solve the numerator before the denominator. Then, we can add or subtract to find our answer.

Key Takeaways for Operations with Decimals

  • Follow the order of operations (PEMDAS) for solving equations with decimals
  • When adding or subtracting decimals, line up the decimal points and make sure the numbers are the same length, adding zeros if necessary
  • When multiplying decimals, remove the decimal points, multiply the numbers, and count the total number of decimal places in the given numbers
  • When dividing decimals, convert the divisor into a whole number and multiply the dividend by the same power of 10, then use long division to solve
  • When multiplying or dividing decimals by powers of 10, shift the decimal point to the right or left according to the number of zeros in the power of 10

Solving Equations with Decimals

To solve equations with decimals, we must first convert any fractions to decimals and then follow the PEMDAS rule. This will ensure that the correct order of operations is followed and the answer is accurate.

The Importance of Following the PEMDAS Rule

The order of operations for equations with decimals is just as important as it is for equations with whole numbers. Following the PEMDAS rule (Parenthesis, Exponents, Multiplication/Division, Addition/Subtraction) will lead to the correct answer every time.

Basic Operations with Decimals

Adding, subtracting, multiplying, and dividing decimals are all similar to operations with whole numbers. However, it is crucial to remember to follow the PEMDAS rule when working with decimals to ensure the correct solution is found.

Applying the Order of Operations to Decimals

The order of operations can involve decimals without any changes. The only key is to convert any fractions to decimals before starting the calculations.

Adding, Multiplying, and Dividing Decimals Effectively

When adding decimals, it is important to line up the decimal points and add as usual. When multiplying decimals, count the total number of decimal places in the given numbers and place the decimal point in the product accordingly. For division, convert the divisor into a whole number and move the decimal point in the dividend the same number of places to the right. Finally, use long division to solve.

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