# Decimals

# Understanding Decimals

Decimal Point: A decimal is a way to express parts of a whole. The decimal point is used to separate the whole number part from the fractional part. For example, in the decimal 3.141, the digit 3 is the whole number part, and .141 is the fractional part.

Place Value: Each digit in a decimal number has a specific place value based on its position relative to the decimal point. The place values to the right of the decimal point are powers of 10, with the first position being tenths, the second being hundredths, the third being thousandths, and so on.

In the number 3.141, the 3 is in the ones place, 1 is in the tenths place, 4 is in the hundredths place, and 1 is in the thousandths place.

# Adding & Subtracting Decimals

Aligning the decimal points is essential when adding or subtracting decimals. If necessary, add zeros to the right of the decimal point to ensure both numbers have the same number of decimal places. Then, perform the addition or subtraction with whole numbers, considering the decimal point.

## Multiplying Decimals

To multiply decimals, multiply the numbers as if they were whole numbers without considering the decimal point. Then, count the total decimal places in both factors and place the decimal point that many places from the right in the product.

For example, to multiply 3.25 and 2.7:

First, multiply as if the decimal point does not exist.

Then, counting the total number of decimal places in the factors, we place the decimal point 3 places from the right in the product.

## Dividing Decimals

To divide decimals, perform long division. First, move the decimal point in the dividend (the number being divided) to the right until it's a whole number. Simultaneously, move the decimal point in the divisor (the number doing the dividing) to the same number of places to the right. Divide as you would with whole numbers, and place the decimal point in the quotient.

For example, to divide 9.6 by 0.6:

### Step 1: Remove the Decimal Point

The first step is to eliminate the decimals by multiplying both the dividend (9.6) and the divisor (0.6) by 10. This is done to make the divisor a whole number.

Multiply 9.6 by 10 to get 96.

Multiply 0.6 by 10 to get 6.

So now, the problem is 96 divided by 6.

### Step 2: Divide as Usual

Now, divide 96 by 6 as you would with whole numbers.

96 ÷ 6 = 16.

### Step 3: Interpret the Answer

Since you removed the decimal by multiplying both numbers by 10, the answer remains the same as in whole number division.

So, 9.6 divided by 0.6 equals 16.

This method works because multiplying both the dividend and divisor by the same number (like 10 in this case) doesn’t change the result, just the form of the numbers.

## Rounding Decimals

Rounding decimals is reducing the number of digits in a decimal while maintaining a value close to the original. This is often done to simplify numbers for ease of use in calculations or when precision beyond a certain point is unnecessary.

### Steps to Round Decimals:

Identify the place value you want to round (e.g., tenths, hundredths, thousandths).

Look at the digit immediately to the right of the desired place value:

If this digit is 5 or greater, round up the digit in your desired place value by 1.

If this digit is less than 5, leave the digit in your desired place value unchanged.

Remove all digits to the right of the desired place value after rounding.

To make rounding decimals a breeze, just remember this simple rhyme:

## Rounding Decimals Examples:

Rounding to the nearest tenth:

Number: 3.46

Find your place, and look at the digit on the right of the tenths place (6).

Since 6 is greater than 5, round up the digit in the tenths place (4) to 5.

The rounded number is 3.5.

Rounding to the nearest hundredth:

Number: 7.123

Look at the digit in the thousandths place (3).

Since 3 is less than 5, the digit in the hundredths place (2) remains unchanged.

The rounded number is 7.12.

### Special Cases:

Rounding when the digit is exactly 5:

Generally, if the digit is exactly 5, you round up. For example, rounding 8.145 to the nearest hundredth gives 8.15.

Some conventions may involve rounding to the nearest even number if rounding to avoid bias, but the standard practice is to round up.

### Importance of Rounding Decimals:

Simplifies calculations: Rounding makes numbers easier to work with in everyday calculations, estimates, and reports.

Manages precision: In situations where overly precise numbers are unnecessary or unhelpful, rounding helps maintain clarity and focus on relevant figures.

Common in financial transactions: Rounding is often used in finance and billing to avoid dealing with long decimal places in monetary values.

### Practice Problem:

Round the following number to the nearest hundredth:

Number: 5.6789

Solution:

Look at the digit in the thousandths place (8).

Since 8 is greater than 5, round up the digit in the hundredths place (7) to 8.

The rounded number is 5.68.

## Practical Uses of Decimals

Decimals are used in various everyday situations, such as:

Money (e.g., prices, currency exchange).

Measurements (e.g., length, weight, volume).

Calculations (e.g., percentages, interest rates).

Scientific and engineering calculations.

Understanding and working with decimals is essential for many aspects of life and mathematics. Practice and application of decimal concepts will help you become proficient in using decimals in various contexts.