16. Decimals

# 16. Decimals ## Decimals, place value and fractions ### Introduction Welcome to our lesson on decimals! Decimals are a fundamental part of mathematics that help us represent parts of whole numbers. Think about money, measurements, or grades – decimals are everywhere! In this lesson, we will explore what decimals are, understand the value of each digit in a decimal number (place value), and learn how decimals are closely related to fractions. ### Key Concepts #### What are Decimals? Decimals are numbers that have a whole number part and a fractional part, separated by a decimal point. They are another way of writing fractions whose denominators are powers of 10 (like 10, 100, 1000, etc.). * **Whole Number Part:** The digits to the left of the decimal point represent whole numbers. * **Decimal Point:** A dot that separates the whole number part from the fractional part. * **Decimal Part:** The digits to the right of the decimal point represent parts of a whole. #### Place Value of Decimals Just like whole numbers, each digit in a decimal number has a specific place value. The place values to the right of the decimal point are fractions with denominators that are powers of 10. | Place Value Name | Value | Written as Fraction | Written as Power of 10 | | :--------------- | :---- | :------------------ | :--------------------- | | Hundreds | 100 | \[ \frac{100}{1} \] | \[ 10^2 \] | | Tens | 10 | \[ \frac{10}{1} \] | \[ 10^1 \] | | Ones | 1 | \[ \frac{1}{1} \] | \[ 10^0 \] | | **Decimal Point**| . | | | | Tenths | 0.1 | \[ \frac{1}{10} \] | \[ 10^{-1} \] | | Hundredths | 0.01 | \[ \frac{1}{100} \] | \[ 10^{-2} \] | | Thousandths | 0.001 | \[ \frac{1}{1000} \] | \[ 10^{-3} \] | **Example:** In the number 12.345: * 1 is in the Tens place (value of 10) * 2 is in the Ones place (value of 2) * 3 is in the Tenths place (value of \( 3 \times rac{1}{10} \) or 0.3) * 4 is in the Hundredths place (value of \( 4 \times rac{1}{100} \) or 0.04) * 5 is in the Thousandths place (value of \( 5 \times rac{1}{1000} \) or 0.005) #### Reading and Writing Decimals To read a decimal number: 1. Read the whole number part (to the left of the decimal point) as usual. 2. Say "and" for the decimal point. 3. Read the digits after the decimal point as a whole number. 4. State the place value of the last digit. **Example:** * 0.5 is read as "five tenths". * 3.27 is read as "three and twenty-seven hundredths". * 15.008 is read as "fifteen and eight thousandths". #### Converting Decimals to Fractions To convert a decimal to a fraction: 1. Write the digits after the decimal point as the numerator. 2. The denominator will be 10, 100, 1000, etc., depending on the place value of the last digit. * If the last digit is in the tenths place, use 10. * If the last digit is in the hundredths place, use 100. * If the last digit is in the thousandths place, use 1000. 3. Simplify the fraction to its lowest terms. **Example:** * Convert 0.6 to a fraction: The 6 is in the tenths place, so \[ 0.6 = \frac{6}{10} \]. Simplify: \[ \frac{6}{10} = \frac{3}{5} \]. * Convert 0.75 to a fraction: The 5 is in the hundredths place, so \[ 0.75 = \frac{75}{100} \]. Simplify: \[ \frac{75}{100} = \frac{3 \times 25}{4 \times 25} = \frac{3}{4} \]. #### Converting Fractions to Decimals To convert a fraction to a decimal, divide the numerator by the denominator. **Example:** * Convert \[ \frac{1}{2} \] to a decimal: Divide 1 by 2: \[ 1 \div 2 = 0.5 \]. * Convert \[ \frac{3}{4} \] to a decimal: Divide 3 by 4: \[ 3 \div 4 = 0.75 \]. * Convert \[ \frac{7}{10} \] to a decimal: Divide 7 by 10: \[ 7 \div 10 = 0.7 \]. ### Vocabulary * **Decimal:** A number that contains a decimal point, representing a part of a whole. * **Decimal Point:** A symbol ( . ) used to separate the whole number part from the fractional part of a decimal number. * **Place Value:** The value of a digit based on its position in a number. * **Tenths:** The first place value to the right of the decimal point, representing one-tenth (\( \frac{1}{10} \)). * **Hundredths:** The second place value to the right of the decimal point, representing one-hundredth (\( \frac{1}{100} \)). * **Thousandths:** The third place value to the right of the decimal point, representing one-thousandth (\( \frac{1}{1000} \)). * **Fraction:** A way to represent parts of a whole or a collection, written as a numerator over a denominator. * **Numerator:** The top number in a fraction, indicating how many parts are being considered. * **Denominator:** The bottom number in a fraction, indicating the total number of equal parts in the whole. ### Examples **Example 1: Identifying Place Value** What is the place value of the digit '7' in the number 24.178? * The '7' is the second digit after the decimal point. The first digit is 'tenths', so the second is 'hundredths'. * **Answer:** The place value of '7' is hundredths. **Example 2: Writing Decimals from Words** Write "six and forty-two hundredths" as a decimal. * "Six" is the whole number part. * "and" means the decimal point. * "Forty-two hundredths" means 42 will end in the hundredths place. * **Answer:** 6.42 **Example 3: Converting Decimal to Fraction** Convert 0.125 to a fraction in its simplest form. * The last digit, 5, is in the thousandths place. * So, \[ 0.125 = \frac{125}{1000} \]. * Simplify by dividing both numerator and denominator by their greatest common divisor (125): \[ \frac{125 \div 125}{1000 \div 125} = \frac{1}{8} \] * **Answer:** \[ \frac{1}{8} \] **Example 4: Converting Fraction to Decimal** Convert \[ \frac{4}{5} \] to a decimal. * Divide the numerator (4) by the denominator (5). * \[ 4 \div 5 = 0.8 \] * **Answer:** 0.8 ### Exercises 1. **Identify Place Value:** State the place value of the underlined digit in each number. a) 5.**6**2 b) 13.0**4** c) 0.98**1** 2. **Write as Decimals:** Write the following in decimal form. a) Two and three tenths b) Fifteen and seven hundredths c) One hundred and twenty-three thousandths 3. **Convert Decimals to Fractions:** Convert each decimal to a fraction in its simplest form. a) 0.4 b) 0.85 c) 2.5 4. **Convert Fractions to Decimals:** Convert each fraction to a decimal. a) \[ \frac{1}{4} \] b) \[ \frac{7}{10} \] c) \[ \frac{3}{8} \] ### Assignment 1. Write the following numbers in words: a) 0.7 b) 12.06 c) 5.123 2. Complete the table by filling in the missing decimal or fraction: | Decimal | Fraction (Simplest Form) | | :------ | :----------------------- | | 0.2 | | | | \[ \frac{2}{5} \] | | 0.04 | | | | \[ \frac{3}{20} \] | | 1.75 | | 3. A recipe calls for 0.75 kg of flour. Express this amount as a fraction in its simplest form. 4. A runner completed a race in \[ \frac{9}{10} \] of an hour. Write this time as a decimal. ### Summary Today, we've learned that decimals are an important way to represent parts of a whole, extending our number system beyond whole numbers. We explored the concept of place value for digits after the decimal point (tenths, hundredths, thousandths) and how to read and write decimal numbers. Crucially, we discovered the close relationship between decimals and fractions, practicing how to convert between these two forms. Understanding decimals and their place value will be essential for future mathematical topics involving calculations with non-whole numbers. ## Approximating decimal, Operations with decimals ### Introduction Welcome to this lesson on decimals! Decimals are a fundamental part of mathematics that you encounter every day, from calculating money to measuring distances. In this lesson, we will explore how to make decimals simpler by approximating them, and how to perform basic operations like addition, subtraction, multiplication, and division with them. By the end of this lesson, you'll be confident in working with decimals in various situations. ### Key Concepts #### Approximating Decimals (Rounding Decimals) Approximating a decimal means making it simpler, often by reducing the number of decimal places, while keeping its value close to the original. This is also known as rounding. ##### Why do we approximate? * **Estimation:** To quickly get an idea of a value without needing exact precision. * **Practicality:** Sometimes exact precision isn't necessary or practical (e.g., measuring length to the nearest centimeter). * **Simplification:** To make numbers easier to work with or understand. ##### Rules for Rounding Decimals To round a decimal to a specific number of decimal places (d.p.): 1. **Identify the rounding digit:** This is the digit in the decimal place you are rounding to. 2. **Look at the digit immediately to its right:** This is the 'deciding digit'. 3. **Apply the rule:** * If the deciding digit is 5 or greater (5, 6, 7, 8, 9), **round up** the rounding digit. This means you add 1 to the rounding digit. * If the deciding digit is less than 5 (0, 1, 2, 3, 4), the rounding digit **remains the same**. 4. **Drop all digits to the right** of the rounding digit. ##### Examples of Rounding: * Rounding 3.472 to 1 decimal place: * Rounding digit is 4 (first d.p.). * Deciding digit is 7 (to the right of 4). * Since 7 is 5 or greater, round up 4 to 5. * Result: 3.5 * Rounding 12.349 to 2 decimal places: * Rounding digit is 4 (second d.p.). * Deciding digit is 9 (to the right of 4). * Since 9 is 5 or greater, round up 4 to 5. * Result: 12.35 * Rounding 0.821 to 1 decimal place: * Rounding digit is 8 (first d.p.). * Deciding digit is 2 (to the right of 8). * Since 2 is less than 5, 8 remains the same. * Result: 0.8 #### Operations with Decimals ##### Addition of Decimals To add decimals, follow these steps: 1. **Align the decimal points** vertically. This ensures that you are adding digits with the same place value. 2. Add zeros to the end of decimals if necessary, so they all have the same number of decimal places. 3. Add the numbers as you would whole numbers. 4. Place the decimal point in the sum directly below the decimal points in the numbers being added. ##### Subtraction of Decimals To subtract decimals, follow these steps: 1. **Align the decimal points** vertically. 2. Add zeros to the end of decimals if necessary, so they all have the same number of decimal places (especially important if the top number has fewer decimal places than the bottom). 3. Subtract the numbers as you would whole numbers, borrowing if needed. 4. Place the decimal point in the difference directly below the decimal points in the numbers being subtracted. ##### Multiplication of Decimals To multiply decimals, follow these steps: 1. Multiply the numbers as if they were whole numbers, ignoring the decimal points for now. 2. **Count the total number of decimal places** in both numbers being multiplied. 3. Place the decimal point in the product so that it has the same total number of decimal places counted in step 2. You might need to add leading zeros if the product has fewer digits than decimal places. ##### Division of Decimals To divide decimals, it's often easiest to make the **divisor** (the number you are dividing by) a whole number. 1. If the divisor is a decimal, move its decimal point to the right until it becomes a whole number. Count how many places you moved it. 2. Move the decimal point in the **dividend** (the number being divided) the *same number of places* to the right. Add zeros if necessary. 3. Place the decimal point in the quotient directly above the new position of the decimal point in the dividend. 4. Divide as you would with whole numbers. ### Vocabulary * **Decimal:** A number that contains a decimal point, representing parts of a whole. * **Decimal Point:** The dot that separates the whole number part from the fractional part of a decimal number. * **Decimal Place:** The position of a digit to the right of the decimal point (e.g., tenths, hundredths, thousandths). * **Approximate:** To estimate or round a number to a less precise but easier-to-use value. * **Round Up:** To increase the rounding digit by one when the deciding digit is 5 or more. * **Round Down:** To keep the rounding digit the same when the deciding digit is less than 5. * **Estimate:** An approximate calculation or judgment of the value, number, quantity, or extent of something. * **Sum:** The result of adding two or more numbers. * **Difference:** The result of subtracting one number from another. * **Product:** The result of multiplying two or more numbers. * **Quotient:** The result of dividing one number by another. * **Divisor:** The number by which another number is divided. * **Dividend:** The number being divided. ### Examples #### Example 1: Rounding Round 5.763 to: a) 1 decimal place b) 2 decimal places **Solution:** a) To 1 decimal place: Look at the first decimal digit (7) and the digit next to it (6). Since 6 is \(\ge 5\), we round up 7 to 8. So, 5.763 rounded to 1 d.p. is 5.8. b) To 2 decimal places: Look at the second decimal digit (6) and the digit next to it (3). Since 3 is \(< 5\), we keep 6 as it is. So, 5.763 rounded to 2 d.p. is 5.76. #### Example 2: Addition Calculate \(12.5 + 3.78 + 0.6\) **Solution:** Align the decimal points and add: ``` 12.50 3.78 + 0.60 ------- 16.88 ``` #### Example 3: Subtraction Calculate \(25.3 - 8.45\) **Solution:** Align the decimal points, add a zero to 25.3, and subtract: ``` 25.30 - 8.45 ------- 16.85 ``` #### Example 4: Multiplication Calculate \(4.2 \times 1.5\) **Solution:** 1. Multiply as whole numbers: \(42 \times 15 = 630\). 2. Count decimal places: 4.2 has 1 d.p., 1.5 has 1 d.p. Total: \(1+1 = 2\) d.p. 3. Place decimal in product: \(6.30\). So, \(4.2 \times 1.5 = 6.30\) or \(6.3\). #### Example 5: Division Calculate \(15.3 \div 0.3\) **Solution:** 1. Make the divisor a whole number: Move the decimal in 0.3 one place to the right to get 3. 2. Move the decimal in the dividend the same number of places: Move the decimal in 15.3 one place to the right to get 153. 3. Now divide \(153 \div 3\). \[ 153 \div 3 = 51 \] So, \(15.3 \div 0.3 = 51\). ### Exercises 1. Round the following decimals to 2 decimal places: a) 7.843 b) 0.129 c) 15.006 d) 9.995 2. Perform the following additions: a) \(3.6 + 8.15\) b) \(1.23 + 4.5 + 0.789\) 3. Perform the following subtractions: a) \(10.5 - 3.25\) b) \(28.1 - 12.98\) 4. Perform the following multiplications: a) \(6.2 \times 3\) b) \(0.4 \times 0.7\) c) \(2.15 \times 1.4\) 5. Perform the following divisions: a) \(9.6 \div 3\) b) \(12.4 \div 0.4\) c) \(0.81 \div 0.09\) ### Assignment 1. Round 24.5678 to: a) 1 decimal place b) 3 decimal places 2. A shop sells three items with prices: \(\$12.99\), \(\$7.50\), and \(\$3.25\). What is the total cost if you buy one of each? 3. If a runner completes a lap in 68.75 seconds and another runner completes it in 65.9 seconds, what is the difference in their times? 4. A car travels at an average speed of 65.5 km/h for 2.5 hours. How far did it travel? 5. A piece of wood is 8.4 meters long. If it is cut into equal pieces, each 0.7 meters long, how many pieces can be cut? 6. Evaluate: \((1.5 + 2.3) \times 0.25 - 0.1\) ### Summary In this lesson, we covered two important aspects of working with decimals: approximating them and performing basic arithmetic operations. We learned that to **approximate (round)** a decimal, we look at the digit to the right of the desired decimal place and either round up or keep the digit the same. For **addition and subtraction**, the key is to align the decimal points vertically. For **multiplication**, we multiply as whole numbers and then count the total decimal places in the factors to place the decimal in the product. Finally, for **division**, we simplify by making the divisor a whole number before performing the division. Mastering these skills will greatly help you in various mathematical and real-life situations.