Comprehensive Guide to Decimals
1. Introduction to Decimals
Decimals are a way of writing fractions with denominators of powers of 10. The decimal point separates the whole number part from the fractional part.
Examples:
- 3.75 = 3 + 7/10 + 5/100 = 3 + 0.7 + 0.05
- 0.125 = 1/10 + 2/100 + 5/1000 = 0.1 + 0.02 + 0.005
- 10.5 = 10 + 5/10 = 10 + 0.5
2. Types of Decimals
2.1 Terminating Decimals
These are decimal numbers that end after a certain number of digits.
Examples:
- 0.25 (ends after 2 decimal places)
- 3.75 (ends after 2 decimal places)
- 0.375 (ends after 3 decimal places)
2.2 Repeating Decimals
These are decimal numbers where a digit or sequence of digits repeats infinitely.
Examples:
- 0.333333... = 0.3̅ (the digit 3 repeats infinitely)
- 0.272727... = 0.27̅ (the sequence '27' repeats infinitely)
- 1.414141... = 1.41̅ (the sequence '41' repeats infinitely)
2.3 Non-terminating, Non-repeating Decimals
These decimals continue infinitely without repeating. They include irrational numbers like π and √2.
Examples:
- π = 3.14159265359...
- √2 = 1.41421356237...
- e = 2.71828182845...
3. Converting Between Fractions and Decimals
3.1 Fraction to Decimal
Method: Divide the numerator by the denominator.
Example: Convert 3/4 to a decimal
3 ÷ 4 = 0.75
Example: Convert 2/3 to a decimal
2 ÷ 3 = 0.666666... = 0.6̅
3.2 Decimal to Fraction
For Terminating Decimals:
Method: Write as a fraction with the denominator based on the place value of the last digit, then simplify.
Example: Convert 0.75 to a fraction
0.75 = 75/100 = 3/4 (after simplifying)
For Repeating Decimals:
Method 1: Algebraic approach for simple repeating patterns.
Example: Convert 0.3̅ to a fraction
Let x = 0.333333...
10x = 3.333333...
10x - x = 3.333333... - 0.333333...
9x = 3
x = 3/9 = 1/3
Method 2: For more complex repeating patterns.
Example: Convert 0.27̅ to a fraction
Let x = 0.272727...
100x = 27.2727...
100x - x = 27.2727... - 0.2727...
99x = 27
x = 27/99 = 3/11
4. Operations with Decimals
4.1 Addition and Subtraction
Method: Align the decimal points and add or subtract as with whole numbers.
Example: 3.75 + 2.8
3.75 + 2.80 ------ 6.55
Example: 5.63 - 2.45
5.63 - 2.45 ------ 3.18
4.2 Multiplication
Method: Multiply as with whole numbers, then place the decimal point by counting the total number of decimal places in the factors.
Example: 2.3 × 1.4
2.3 (1 decimal place) × 1.4 (1 decimal place) ------ 9.2 23. ------ 3.22 (2 decimal places in result)
4.3 Division
Method: Convert the divisor to a whole number by moving the decimal point, then move the decimal point in the dividend the same number of places. Divide as with whole numbers.
Example: 4.2 ÷ 0.6
First, convert 0.6 to 6 by moving the decimal point one place right.
Do the same with 4.2, becoming 42.
Now divide: 42 ÷ 6 = 7
Example: 3.75 ÷ 1.5
Convert to 37.5 ÷ 15
37.5 ÷ 15 = 2.5
5. Rounding Decimals
Method:
- Identify the place value to which you want to round.
- Look at the digit one place to the right of that place value.
- If it's less than 5, round down (keep the digit in the rounding place the same).
- If it's 5 or greater, round up (increase the digit in the rounding place by 1).
Example: Round 3.748 to the nearest tenth
The digit in the tenths place is 7.
The digit to its right is 4, which is less than 5.
So we round down: 3.748 rounds to 3.7
Example: Round 5.286 to the nearest hundredth
The digit in the hundredths place is 8.
The digit to its right is 6, which is 5 or greater.
So we round up: 5.286 rounds to 5.29
6. Comparing Decimals
Method:
- Compare the whole number parts first.
- If they're equal, compare the tenths places.
- If those are equal, compare the hundredths places, and so on.
- For easier comparison, you can add zeros to the right of the decimal to make the numbers have the same number of decimal places.
Example: Compare 3.42 and 3.4
Make them have the same number of decimal places: 3.42 and 3.40
The whole numbers (3) are the same.
The tenths (4) are the same.
The hundredths: 2 > 0
Therefore, 3.42 > 3.4
Example: Compare 2.7 and 2.07
Make them have the same number of decimal places: 2.70 and 2.07
The whole numbers (2) are the same.
The tenths: 7 > 0
Therefore, 2.7 > 2.07
7. Decimals and Percentages
Decimal to Percentage: Multiply by 100 and add a % symbol.
Examples:
- 0.25 = 0.25 × 100% = 25%
- 0.075 = 0.075 × 100% = 7.5%
- 1.5 = 1.5 × 100% = 150%
Percentage to Decimal: Divide by 100 (remove the % symbol and move the decimal point two places to the left).
Examples:
- 45% = 45 ÷ 100 = 0.45
- 7.5% = 7.5 ÷ 100 = 0.075
- 250% = 250 ÷ 100 = 2.5
8. Word Problems with Decimals
Problem: Janet bought a shirt for $24.99, pants for $35.50, and socks for $8.75. How much did she spend in total?
Solution:
Total cost = $24.99 + $35.50 + $8.75
24.99 35.50 + 8.75 ------ 69.24
Janet spent $69.24 in total.
Problem: A board is 2.5 meters long. If you cut off 0.75 meters, how much is left?
Solution:
Remaining length = 2.5 - 0.75
2.50 - 0.75 ------ 1.75
There is 1.75 meters of board remaining.
Problem: A recipe requires 2.25 cups of flour. If you want to make 3 batches, how much flour will you need?
Solution:
Total flour needed = 2.25 × 3
2.25 × 3 ------ 6.75
You will need 6.75 cups of flour.
Problem: A 2.5 kg bag of rice costs $8.75. What is the cost per kilogram?
Solution:
Cost per kilogram = Total cost ÷ Total weight
Cost per kilogram = $8.75 ÷ 2.5
Converting to division by a whole number: $87.5 ÷ 25 = $3.5
The rice costs $3.50 per kilogram.
9. Interactive Decimal Quiz
Test Your Decimal Knowledge
Answer the following questions to test your understanding of decimals.
1. Convert 3/8 to a decimal:
2. Calculate 2.7 + 1.35:
3. Calculate 4.2 × 0.5:
4. Round 3.467 to the nearest tenth:
5. Convert 0.45 to a percentage:
6. Which is larger: 0.52 or 0.498?
7. Calculate 5.6 ÷ 0.8:
8. What decimal represents 37.5%?
9. If a shirt costs $24.99 and you pay with $30, how much change should you receive?
10. Express 2/3 as a decimal:
