Fractions and Decimals
What are Fractions?
A fraction is a part of a whole. It represents a number that is not a whole number. Fractions are written in the form of a⁄b, where 'a' is the numerator and 'b' is the denominator. The numerator tells us how many parts of the whole we have, and the denominator tells us how many equal parts the whole is divided into. For example, if a cake is divided into 8 equal slices and you eat 3 slices, you have eaten 3⁄8 of the cake. Here, 3 is the numerator and 8 is the denominator.
Types of Fractions
Fractions can be classified into different types based on their numerators and denominators:
- Proper Fractions: In a proper fraction, the numerator is smaller than the denominator. Examples: 1⁄2, 3⁄4, 7⁄10. These fractions represent a value less than 1.
- Improper Fractions: In an improper fraction, the numerator is greater than or equal to the denominator. Examples: 5⁄3, 7⁄7, 10⁄4. These fractions represent a value equal to or greater than 1.
- Mixed Fractions: A mixed fraction consists of a whole number and a proper fraction. It is another way to represent an improper fraction. Examples: 12⁄3 (which is equal to 5⁄3), 31⁄4 (which is equal to 13⁄4).
- Like Fractions: Fractions with the same denominator are called like fractions. Examples: 2⁄5, 3⁄5, 4⁄5.
- Unlike Fractions: Fractions with different denominators are called unlike fractions. Examples: 1⁄2, 2⁄3, 3⁄4.
- Equivalent Fractions: Fractions that represent the same value, even though they have different numerators and denominators, are called equivalent fractions. To find equivalent fractions, you can multiply or divide both the numerator and the denominator by the same non-zero number. Example: 1⁄2 is equivalent to 2⁄4, 3⁄6, 10⁄20.
Operations on Fractions
We can perform basic arithmetic operations on fractions:
Addition and Subtraction of Fractions
To add or subtract fractions, they must have a common denominator. If the fractions are like fractions (same denominator), simply add or subtract the numerators and keep the denominator the same. Example: 2⁄7 + 3⁄7 = (2+3)⁄7 = 5⁄7. Example: 5⁄9 - 2⁄9 = (5-2)⁄9 = 3⁄9 (which can be simplified to 1⁄3).
If the fractions are unlike fractions (different denominators), first find a common denominator. The easiest common denominator to find is the Least Common Multiple (LCM) of the denominators. Convert each fraction to an equivalent fraction with the common denominator, then add or subtract the numerators. Example: 1⁄2 + 1⁄3. The LCM of 2 and 3 is 6. Convert 1⁄2 to (1*3)⁄(2*3) = 3⁄6. Convert 1⁄3 to (1*2)⁄(3*2) = 2⁄6. Now, add: 3⁄6 + 2⁄6 = (3+2)⁄6 = 5⁄6.
Shortcut for adding/subtracting two unlike fractions a⁄b and c⁄d: a⁄b ± c⁄d = (ad ± bc)⁄(bd) Example: 1⁄2 + 1⁄3 = (1*3 + 1*2)⁄(2*3) = (3+2)⁄6 = 5⁄6.
Multiplication of Fractions
To multiply fractions, multiply the numerators together and multiply the denominators together. a⁄b × c⁄d = (a*c)⁄(b*d) Example: 2⁄3 × 4⁄5 = (2*4)⁄(3*5) = 8⁄15. You can simplify before multiplying if possible by cancelling out common factors between a numerator and a denominator. Example: 3⁄4 × 2⁄5. Here, 2 in the numerator and 4 in the denominator have a common factor of 2. 3⁄(2*2) × 2⁄5 = 3⁄2 × 1⁄5 = (3*1)⁄(2*5) = 3⁄10.
Division of Fractions
To divide by a fraction, you multiply by its reciprocal. The reciprocal of a fraction a⁄b is b⁄a. a⁄b ÷ c⁄d = a⁄b × d⁄c = (a*d)⁄(b*c) Example: 1⁄2 ÷ 1⁄3. The reciprocal of 1⁄3 is 3⁄1. So, 1⁄2 × 3⁄1 = (1*3)⁄(2*1) = 3⁄2.
What are Decimals?
A decimal is another way to represent a fraction, specifically fractions whose denominators are powers of 10 (like 10, 100, 1000, etc.). Decimals use a decimal point (.) to separate the whole number part from the fractional part. The digits to the right of the decimal point represent tenths, hundredths, thousandths, and so on. For example, 0.5 means 5 tenths, which is the same as the fraction 5⁄10 or 1⁄2. The number 3.14 means 3 whole units and 14 hundredths, which can be written as 3 + 14⁄100.
Place Value in Decimals
Understanding the place value of digits in a decimal is crucial:
| Thousands | Hundreds | Tens | Ones | Decimal Point | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|---|
| 1000 | 100 | 10 | 1 | . | 1⁄10 | 1⁄100 | 1⁄1000 |
Example: In the number 542.789 5 is in the hundreds place. 4 is in the tens place. 2 is in the ones place. 7 is in the tenths place (7⁄10). 8 is in the hundredths place (8⁄100). 9 is in the thousandths place (9⁄1000).
Converting Fractions to Decimals
To convert a fraction to a decimal, divide the numerator by the denominator. Example: Convert 3⁄4 to a decimal. Divide 3 by 4: 3 ÷ 4 = 0.75. Example: Convert 1⁄8 to a decimal. Divide 1 by 8: 1 ÷ 8 = 0.125.
If the denominator has only prime factors of 2 and 5, the decimal representation will terminate (end). Otherwise, it will be a repeating decimal. Example: 1⁄3 = 0.333... (repeating decimal). Example: 1⁄7 = 0.142857142857... (repeating decimal).
Converting Decimals to Fractions
To convert a terminating decimal to a fraction: 1. Write the decimal as a fraction with the decimal digits as the numerator and 1 followed by as many zeros as there are decimal places as the denominator. 2. Simplify the fraction. Example: Convert 0.75 to a fraction. 0.75 has two decimal places. So, write it as 75⁄100. Simplify by dividing both numerator and denominator by their greatest common divisor (25): 75 ÷ 25⁄100 ÷ 25 = 3⁄4. Example: Convert 0.125 to a fraction. 0.125 has three decimal places. So, write it as 125⁄1000. Simplify by dividing by 125: 125 ÷ 125⁄1000 ÷ 125 = 1⁄8.
Converting Repeating Decimals to Fractions: This is a bit more involved. Let the decimal be 'x'. Example: Convert 0.333... to a fraction. Let x = 0.333... Multiply by 10 (since one digit repeats): 10x = 3.333... Subtract the first equation from the second: 10x = 3.333... - x = 0.333... ---------------- 9x = 3 x = 3⁄9 = 1⁄3.
Example: Convert 0.121212... to a fraction. Let x = 0.121212... Multiply by 100 (since two digits repeat): 100x = 12.121212... Subtract: 100x = 12.121212... - x = 0.121212... ----------------- 99x = 12 x = 12⁄99. Simplify by dividing by 3: 12 ÷ 3⁄99 ÷ 3 = 4⁄33.
Operations on Decimals
Operations on decimals are similar to operations on whole numbers, with careful attention to the decimal point.
Addition and Subtraction of Decimals
1. Align the decimal points of the numbers vertically. 2. Add or subtract the numbers as if they were whole numbers. 3. Place the decimal point in the answer directly below the aligned decimal points. Example: Add 12.34 + 5.678 12.340 + 5.678 --------- 18.018 Example: Subtract 25.8 - 10.35 25.80 - 10.35 --------- 15.45
Multiplication of Decimals
1. Multiply the numbers as if they were whole numbers, ignoring the decimal points initially. 2. Count the total number of decimal places in all the numbers being multiplied. 3. Place the decimal point in the product so that it has the same number of decimal places as the total counted in step 2. Example: Multiply 2.5 × 3.1 Multiply 25 × 31 = 775. 2.5 has 1 decimal place. 3.1 has 1 decimal place. Total decimal places = 1 + 1 = 2. Place the decimal point 2 places from the right in 775: 7.75. Example: Multiply 0.4 × 0.02 Multiply 4 × 2 = 8. 0.4 has 1 decimal place. 0.02 has 2 decimal places. Total = 1 + 2 = 3. Place the decimal point 3 places from the right in 8: 0.008.
Division of Decimals
Dividing a decimal by a whole number: 1. Perform the division as if they were whole numbers. 2. Place the decimal point in the quotient directly above the decimal point in the dividend. Example: Divide 15.6 ÷ 3 5.2 ------- 3|15.6 15 --- 06 06 --- 0 So, 15.6 ÷ 3 = 5.2.
Dividing a decimal by a decimal: 1. Make the divisor a whole number by moving its decimal point to the right as many places as needed. 2. Move the decimal point in the dividend the same number of places to the right. 3. Perform the division as in the previous case. Example: Divide 12.48 ÷ 0.4 The divisor is 0.4. Move the decimal point one place to the right to make it 4. Move the decimal point in the dividend (12.48) one place to the right: 124.8. Now divide 124.8 ÷ 4. 31.2 ------- 4|124.8 12 --- 04 04 --- 08 08 --- 0 So, 12.48 ÷ 0.4 = 31.2.
Comparing Fractions and Decimals
To compare fractions and decimals, it's easiest to convert them to the same format (either all fractions or all decimals). Example: Compare 3⁄5 and 0.7. Method 1: Convert fraction to decimal. 3⁄5 = 3 ÷ 5 = 0.6. Now compare 0.6 and 0.7. Since 0.7 > 0.6, then 3⁄5 < 0.7. Method 2: Convert decimal to fraction. 0.7 = 7⁄10. Now compare 3⁄5 and 7⁄10. Find a common denominator (LCM of 5 and 10 is 10). 3⁄5 = (3*2)⁄(5*2) = 6⁄10. Compare 6⁄10 and 7⁄10. Since 7 > 6, then 7⁄10 > 6⁄10, so 0.7 > 3⁄5.
Key Takeaways for Fractions and Decimals
- Fractions represent parts of a whole (numerator⁄denominator).
- Decimals represent parts of a whole using a decimal point, based on powers of 10.
- Operations (addition, subtraction, multiplication, division) on fractions require understanding common denominators and reciprocals.
- Operations on decimals require careful alignment of decimal points and tracking decimal places.
- Conversions between fractions and decimals are essential for comparison and problem-solving.
- Remember: 0.5 = 1⁄2, 0.25 = 1⁄4, 0.75 = 3⁄4, 0.1 = 1⁄10.