Whole Numbers and Place Value

1.1 Place Value and Number Forms

Understanding Place Value up to Millions and Billions

Place value represents the value of each digit in a number. The position of a digit determines its value. which determines its actual value. The places from right to left are: ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions, etc.

Example 1:
The number 4,567,891:
- 4 is in the millions place.
- 5 is in the hundred thousands place.
- 6 is in the ten thousands place.
- 7 is in the thousands place.
- 8 is in the hundreds place.
- 9 is in the tens place.
- 1 is in the ones place.
        

Example 2:
The number 7,432,195 has the following place values:
    - 7 → Millions 
    - 4 → Hundred Thousands 
    - 3 → Ten Thousands 
    - 2 → Thousands 
    - 1 → Hundreds 
    - 9 → Tens 
    - 5 → Ones 
        

        
Standard Form, Word Form, and Expanded Form
      
Example 1:
Number: 125,004
- Standard Form: 125,004
- Word Form: One hundred twenty-five thousand, four
- Expanded Form: 100,000 + 20,000 + 5,000 + 4

Example 2:
Number: 2,378,910
- Standard Form: 2,378,910
- Word Form: Two million, three hundred seventy-eight thousand, nine hundred ten
- Expanded Form: 2,000,000 + 300,000 + 70,000 + 8,000 + 900 + 10

Example 3:
Number: 45,602
- Standard Form: 45,602
- Word Form: Forty-five thousand, six hundred two
- Expanded Form: 40,000 + 5,000 + 600 + 2

Example 4:
Number: 987,321
- Standard Form: 987,321
- Word Form: Nine hundred eighty-seven thousand, three hundred twenty-one
- Expanded Form: 900,000 + 80,000 + 7,000 + 300 + 20 + 1

Example 5:
Number: 8,050,123
- Standard Form: 8,050,123
- Word Form: Eight million, fifty thousand, one hundred twenty-three
- Expanded Form: 8,000,000 + 50,000 + 100 + 20 + 3
        

        
Comparing and Ordering Large Numbers
        
When comparing numbers, start from the leftmost digit.
     
Example 1:
Compare 4,789,321 and 4,798,215:
- 4,789,321 < 4,798,215 (since 8 in ten thousands is less than 9 in ten thousands)

Example 2:
Compare 12,345,678 and 12,354,678:
- 12,345,678 < 12,354,678 (since 4 in hundred thousands is less than 5 in hundred thousands)

Example 3:
Compare 902,567 and 920,567:
- 902,567 < 920,567 (since 0 in hundred thousands is less than 2 in hundred thousands)

Example 4:
Compare 78,456,231 and 78,456,132:
- 78,456,231 > 78,456,132 (since 3 in hundreds is greater than 1 in hundreds)

Example 5:
Compare 5,678,912 and 5,687,123:
- 5,678,912 < 5,687,123 (because 7 in ten thousands is less than 8 in ten thousands)
        

1.2 Operations with Whole Numbers

Review of Addition, Subtraction, Multiplication, and Division

Example 1:
23,456 + 12,789 = 36,245
Example 2:
56,789 - 23,567 = 33,222
Example 3:
345 × 76 = 26,220
Example 4:
9,600 ÷ 12 = 800
Example 5:
1,234 + 5,678 - 3,456 = 3,456
        

Multi-Digit Multiplication and Long Division

Example 1:
512 × 34 = 17,408
Example 2:
7,209 × 45 = 324,405
Example 3:
84,672 ÷ 24 = 3,528
Example 4:
125,600 ÷ 50 = 2,512
Example 5:
91,245 × 19 = 1,733,655
        

Order of Operations (PEMDAS/BODMAS)

Example 1:
(3 + 5) × 4 = 32
Example 2:
12 ÷ (4 - 2) × 3 = 18
Example 3:
6 + 2 × (8 ÷ 4) = 10
Example 4:
(15 - 5) × 2 + 10 ÷ 5 = 22
Example 5:
100 ÷ (5 × 4) + 6 = 11
        

1.3 Estimation and Rounding

Rounding Whole Numbers to Specified Place Values

Rule: If the digit to the right of the rounding place is 5 or greater, round up. Otherwise, round down.

Example:
Estimate 4,678 + 2,345 by rounding to the nearest thousand:
- 4,678 → 5,000
- 2,345 → 2,000
Estimated sum: 5,000 + 2,000 = 7,000 (actual sum = 7,023)
        

Overestimation: Round numbers up for a higher estimate.
Underestimation: Round numbers down for a lower estimate.

Example:
Estimate 679 – 248:
- Overestimate: 700 – 250 = 450
- Underestimate: 600 – 200 = 400
Actual answer: 431
        

Example 1: Round 5,678 to the nearest hundred → 5,700

Example 2: Round 98,345 to the nearest thousand → 98,000

Example 3: Round 432,198 to the nearest ten thousand → 430,000

Example 4: Round 7,845,678 to the nearest million → 8,000,000

Example 5: Round 23,456,789 to the nearest ten million → 20,000,000

        

Estimating Sums, Differences, Products, and Quotients

Example 1: Estimate 456 + 789 by rounding to the nearest hundred → 500 + 800 = 1,300

Example 2: Estimate 3,215 - 1,987 by rounding to the nearest thousand → 3,000 - 2,000 = 1,000

Example 3: Estimate 87 × 46 by rounding to the nearest ten → 90 × 50 = 4,500

Example 4: Estimate 9,876 ÷ 42 by rounding to the nearest ten → 9,880 ÷ 40 = 247

        

Using Estimation to Check the Reasonableness of Answers

Example 1: 5,672 + 3,291 ≈ 5,700 + 3,300 = 9,000 (Actual: 8,963, reasonable)

Example 2: 14,356 - 8,799 ≈ 14,000 - 9,000 = 5,000 (Actual: 5,557, reasonable)

Example 3: 246 × 39 ≈ 250 × 40 = 10,000 (Actual: 9,594, reasonable)

Example 4: 8,432 ÷ 42 ≈ 8,400 ÷ 40 = 210 (Actual: 201, reasonable)

        

1.4 Problem Solving with Whole Numbers

Solving Multi-Step Word Problems

Example 1: Sarah bought 3 packs of pencils with 12 pencils each. She gave 8 pencils to her friend. How many pencils does she have left? 
Solution: (3 × 12) - 8 = 36 - 8 = 28 pencils.


Example 2: A bookstore sold 128 books in the morning and 175 books in the afternoon. How many books were sold in total? 
Solution: 128 + 175 = 303 books.


Example 3: A school bus has 40 seats. If 28 students board the bus and 5 more get on at the next stop, how many empty seats are left? 
Solution: 40 - (28 + 5) = 40 - 33 = 7 seats.


Example 4: A bakery baked 360 cupcakes. They sold 125 in the morning and 97 in the afternoon. How many cupcakes are left? 
Solution: 360 - (125 + 97) = 360 - 222 = 138 cupcakes.


Example 5: A runner jogged 4 miles on Monday, 5 miles on Tuesday, and 6 miles on Wednesday. How many miles did he jog in total? 
Solution: 4 + 5 + 6 = 15 miles.

        

Using Strategies like Drawing Diagrams and Making Tables

Example 1: A farmer has 3 fields with 45, 60, and 75 trees. How many trees in total?
Strategy: Draw a diagram representing each field. Add: 45 + 60 + 75 = 180 trees.


Example 2: A family drove 120 miles on Friday, 135 miles on Saturday, and 150 miles on Sunday. How many miles in total?
Strategy: Use a table to track distances per day and sum: 120 + 135 + 150 = 405 miles.


Example 3: A zoo has 5 enclosures with 8, 12, 15, 10, and 14 animals. How many animals in total?
Strategy: Organize data in a table and add: 8 + 12 + 15 + 10 + 14 = 59 animals.


Example 4: A factory produces 250 toys per day. How many are made in a week?
Strategy: Use multiplication: 250 × 7 = 1,750 toys.


Example 5: A store sells 15 notebooks per day. How many are sold in 10 days?
Strategy: Multiply and check using a table: 15 × 10 = 150 notebooks.

        

2.1 Introduction to Fractions

Understanding Fractions as Parts of a Whole

Example 1: A pizza is divided into 8 equal slices. If you eat 3 slices, what fraction of the pizza have you eaten? 
Solution: 3/8 of the pizza.


Example 2: A class of 30 students has 12 girls. What fraction of the class are girls? 
Solution: 12/30, simplified to 2/5.


Example 3: A chocolate bar is split into 10 pieces. If you take 4 pieces, what fraction do you have? 
Solution: 4/10, simplified to 2/5.


Example 4: There are 24 apples, and 6 are green. What fraction of the apples are green? 
Solution: 6/24, simplified to 1/4.


Example 5: A water tank is divided into 5 equal sections. If 2 sections are filled, what fraction of the tank is full? 
Solution: 2/5 of the tank is full.

        

Numerator and Denominator

Example 1: In the fraction 3/7, what is the numerator and denominator?
Solution: 3 is the numerator, 7 is the denominator.


Example 2: Identify the numerator and denominator in 5/9.
Solution: 5 is the numerator, 9 is the denominator.


Example 3: What are the numerator and denominator in 8/15?
Solution: 8 is the numerator, 15 is the denominator.


Example 4: In 11/20, what do the numerator and denominator represent?
Solution: 11 is the numerator (parts taken), 20 is the denominator (total parts).


Example 5: Identify the numerator and denominator in 4/13.
Solution: 4 is the numerator, 13 is the denominator.

        

Equivalent Fractions

Example 1: Find an equivalent fraction to 1/2.
Solution: 2/4, 3/6, 4/8, etc.


Example 2: What is an equivalent fraction to 3/5?
Solution: 6/10, 9/15, 12/20, etc.


Example 3: Convert 4/6 to an equivalent fraction.
Solution: 2/3 (simplified), 8/12, 12/18, etc.


Example 4: Write an equivalent fraction for 5/8.
Solution: 10/16, 15/24, 20/32, etc.


Example 5: Find an equivalent fraction to 7/9.
Solution: 14/18, 21/27, 28/36, etc.

        

Simplifying and Comparing Fractions

• Finding the greatest common factor (GCF):
    1. 15/25 -> GCF is 5, so simplified to 3/5.
    2. 12/18 -> GCF is 6, so simplified to 2/3.
    3. 24/36 -> GCF is 12, so simplified to 2/3.
    4. 50/75 -> GCF is 25, so simplified to 2/3.
    5. 45/60 -> GCF is 15, so simplified to 3/4.
    6. 28/56 -> GCF is 28, so simplified to 1/2.
    7. 18/54 -> GCF is 18, so simplified to 1/3.
    8. 32/48 -> GCF is 16, so simplified to 2/3.
    9. 56/84 -> GCF is 28, so simplified to 2/3.
    10. 30/90 -> GCF is 30, so simplified to 1/3.

• Simplifying fractions to lowest terms:
    1. 20/40 -> Simplified to 1/2.
    2. 36/72 -> Simplified to 1/2.
    3. 50/100 -> Simplified to 1/2.
    4. 45/90 -> Simplified to 1/2.
    5. 10/30 -> Simplified to 1/3.
    6. 15/30 -> Simplified to 1/2.
    7. 48/96 -> Simplified to 1/2.
    8. 18/54 -> Simplified to 1/3.
    9. 25/100 -> Simplified to 1/4.
    10. 12/36 -> Simplified to 1/3.

• Comparing fractions with like and unlike denominators:
    1. 1/2 and 3/4 -> 1/2 < 3/4.
    2. 5/8 and 3/8 -> 5/8 > 3/8.
    3. 2/3 and 5/6 -> 2/3 < 5/6.
    4. 1/3 and 2/5 -> 1/3 < 2/5.
    5. 7/10 and 5/10 -> 7/10 > 5/10.
    6. 3/4 and 4/5 -> 3/4 < 4/5.
    7. 9/10 and 7/10 -> 9/10 > 7/10.
    8. 1/4 and 2/3 -> 1/4 < 2/3.
    9. 5/12 and 7/12 -> 5/12 < 7/12.
    10. 2/5 and 3/5 -> 2/5 < 3/5.
        

2.3 Mixed Numbers and Improper Fractions

• Converting mixed numbers to improper fractions and vice versa:
    1. 3 1/2 -> 7/2
    2. 4 3/5 -> 23/5
    3. 6 1/4 -> 25/4
    4. 2 2/3 -> 8/3
    5. 5 1/6 -> 31/6
    6. 7 3/8 -> 59/8
    7. 9 1/3 -> 28/3
    8. 3 5/6 -> 23/6
    9. 4 7/10 -> 47/10
    10. 6 2/5 -> 32/5
    11. 1 4/9 -> 13/9
    12. 2 7/12 -> 31/12
    13. 8 5/7 -> 61/7
    14. 5 2/3 -> 17/3
    15. 4 1/8 -> 33/8
    16. 3 7/10 -> 37/10
    17. 2 3/4 -> 11/4
    18. 6 5/8 -> 53/8
    19. 7 2/5 -> 37/5
    20. 10 1/2 -> 21/2

• Representing mixed numbers and improper fractions visually:
    1. 3 1/2 -> (Three whole parts and half a part)
    2. 4 3/5 -> (Four whole parts and three-fifths of a part)
    3. 6 1/4 -> (Six whole parts and one-fourth of a part)
    4. 2 2/3 -> (Two whole parts and two-thirds of a part)
    5. 5 1/6 -> (Five whole parts and one-sixth of a part)
    6. 7 3/8 -> (Seven whole parts and three-eighths of a part)
    7. 9 1/3 -> (Nine whole parts and one-third of a part)
    8. 3 5/6 -> (Three whole parts and five-sixths of a part)
    9. 4 7/10 -> (Four whole parts and seven-tenths of a part)
    10. 6 2/5 -> (Six whole parts and two-fifths of a part)
    11. 1 4/9 -> (One whole part and four-ninths of a part)
    12. 2 7/12 -> (Two whole parts and seven-twelfths of a part)
    13. 8 5/7 -> (Eight whole parts and five-sevenths of a part)
    14. 5 2/3 -> (Five whole parts and two-thirds of a part)
    15. 4 1/8 -> (Four whole parts and one-eighth of a part)
    16. 3 7/10 -> (Three whole parts and seven-tenths of a part)
    17. 2 3/4 -> (Two whole parts and three-fourths of a part)
    18. 6 5/8 -> (Six whole parts and five-eighths of a part)
    19. 7 2/5 -> (Seven whole parts and two-fifths of a part)
    20. 10 1/2 -> (Ten whole parts and half a part)
        

Proper Fraction

1/2 → 0.5

Smaller than Improper Fractions and Mixed Numbers

Improper Fraction

5/3 → 1.6667

Larger than Proper Fractions but Smaller than Mixed Numbers

Mixed Number

2 1/2 → 2.5

Larger than Proper Fractions and Improper Fractions

Proper Fraction

1/2 → 0.5

Smaller than Improper Fractions and Mixed Numbers

Improper Fraction

5/3 → 1.6667

Larger than Proper Fractions but Smaller than Mixed Numbers

Mixed Number

2 1/2 → 2.5

Larger than Proper Fractions and Improper Fractions

What are Decimals?

Decimals are numbers that have a whole part and a fractional part, separated by a decimal point.
For example:
3.5 means 3 wholes and 5 tenths.
0.75 means 75 hundredths.
12.01 means 12 wholes and 1 hundredth.

Difference between Whole Numbers and Decimals

Whole Numbers: 0, 1, 2, 10 (no decimal)
Decimals: 2.5, 0.75 (have fractional part)

Explanation

Whole numbers are numbers without fractions or decimals (like 0, 1, 2).
Decimals are numbers with a decimal point showing values smaller than 1 (like 0.5, 2.75).

Place Value of Decimals

When you see a decimal number, every digit has a name and place depending on where it is. After the decimal point, we have tenths, hundredths, thousandths, and so on — these are like mini-chunks of one whole.

Steps to Understand

• Step 1: Identify where the number is after the decimal.
• Step 2: First digit after the dot is the tenths place.
• Step 3: Second is the hundredths, then thousandths, etc.
• Step 4: Say the number by calling out the place name.

Examples (15)

1. 0.1 → One tenth.
2. 0.01 → One hundredth.
3. 0.001 → One thousandth.
4. 3.5 → 5 is in the tenths place.
5. 3.05 → 5 is in the hundredths place.
6. 7.321 → 3 is tenths, 2 is hundredths, 1 is thousandths.
7. 0.9 → Nine tenths.
8. 1.04 → 4 is hundredths.
9. 8.06 → 6 is hundredths, 0 is tenths.
10. 2.007 → 7 is in the thousandths.
11. 9.999 → Each 9 is in a different place.
12. 5.50 → 5 is whole, first 5 is tenths, 0 is hundredths.
13. 0.100 → 1 tenth still — zeroes don’t change value.
14. 3.004 → 4 is thousandths.
15. 6.600 → That’s 6 and 6 tenths.

Reading and Writing Decimals

Reading decimals is just like reading whole numbers — but once you hit the dot, say "and" and then say the digits with their place value name. Writing them? Easy — use the numbers and the dot where the place splits.

Steps to Understand

• Step 1: Read whole number part.
• Step 2: Say "and" at the decimal point.
• Step 3: Read digits after the dot as one number.
• Step 4: Finish with the place value name.

Examples (15)

1. 0.1 → "One tenth"
2. 2.3 → "Two and three tenths"
3. 5.07 → "Five and seven hundredths"
4. 10.25 → "Ten and twenty-five hundredths"
5. 3.456 → "Three and four hundred fifty-six thousandths"
6. 4.0 → "Four" (nothing extra to say)
7. 7.01 → "Seven and one hundredth"
8. 0.09 → "Nine hundredths"
9. 6.78 → "Six and seventy-eight hundredths"
10. 9.99 → "Nine and ninety-nine hundredths"
11. 1.234 → "One and two hundred thirty-four thousandths"
12. 11.111 → "Eleven and one hundred eleven thousandths"
13. 8.008 → "Eight and eight thousandths"
14. 0.004 → "Four thousandths"
15. 3.333 → "Three and three hundred thirty-three thousandths"

Comparing and Ordering Decimals

To figure out which decimal is bigger, compare from left to right — whole numbers first, then tenths, hundredths, etc. If it’s the same so far, keep checking deeper places!

Steps to Understand

• Step 1: Line up the numbers by decimal point.
• Step 2: Compare whole numbers.
• Step 3: If equal, compare tenths.
• Step 4: Keep going until one is bigger.

Examples (15)

1. 0.5 vs 0.4 → 0.5 is bigger.
2. 3.7 vs 3.77 → 3.77 is bigger.
3. 6.02 vs 6.2 → 6.2 is bigger.
4. 1.1 vs 1.10 → Equal.
5. 0.09 vs 0.1 → 0.1 is bigger.
6. 2.5 vs 2.50 → Equal.
7. 5.005 vs 5.05 → 5.05 is bigger.
8. 7.9 vs 7.89 → 7.9 is bigger (think 0.90 vs 0.89)
9. 4.444 vs 4.4 → 4.444 is bigger.
10. 9.01 vs 9.001 → 9.01 is bigger.
11. 1.999 vs 2.0 → 2.0 is bigger.
12. 3.333 vs 3.330 → 3.333 is bigger.
13. 8.75 vs 8.705 → 8.75 is bigger.
14. 6.060 vs 6.06 → Equal.
15. 0.008 vs 0.009 → 0.009 is bigger.

Using Number Lines for Decimals

Number lines can be used to represent decimals. The whole numbers are marked at equal intervals, and the decimals are placed between them, showing the fractional values.
For example, on a number line between 0 and 1, 0.1, 0.2, 0.3, etc., can be marked.

Arranging Decimals in Ascending and Descending Order

To arrange decimals in ascending order (from smallest to largest), compare their place values.
To arrange in descending order (from largest to smallest), follow the same process but reverse the order.

Rounding to the Nearest Tenth, Hundredth, or Whole Number

Rounding is the process of adjusting a decimal to a certain place value.
To round to the nearest tenth, look at the digit in the hundredths place.
To round to the nearest hundredth, look at the digit in the thousandths place.
To round to the nearest whole number, look at the digit in the tenths place.

Rules for Rounding

When rounding a decimal number, consider the place value you are rounding to.
The general rule for rounding is:
- If the digit in the next place is 5 or greater, round up.
- If the digit is less than 5, round down.

Real-life Examples (e.g., Money, Measurements)

Decimals are commonly used in real life for measurements and money.
In money, prices are often rounded to the nearest cent.
For measurements, decimals are used to represent parts of a unit (e.g., inches, meters).

Aligning Decimals for Operation

When performing operations such as addition or subtraction with decimals, always align the decimal points.
This ensures that the digits in each column correspond correctly.
For example:

Adding and Subtracting Decimals with the Same Number of Decimal Places

When adding or subtracting decimals with the same number of decimal places, simply align the decimal points and proceed as you would with whole numbers.
Example:
3.76 + 2.48
Step 1: Align the decimal points.
Step 2: Add 3.76 + 2.48.
Step 3: Perform the addition.
Answer: 6.24

Adding and Subtracting Decimals with Different Decimal Places

When adding or subtracting decimals with different decimal places, first align the decimal points, then add zeros to the shorter number if needed.
Example:
3.6 + 4.742
Step 1: Align the decimal points.
Step 2: Add zeros to 3.6 to make it 3.600.
Step 3: Add 3.600 + 4.742.
Answer: 8.342

Word Problems

Word problems often involve adding or subtracting decimals in real-life scenarios.
Example: Sarah bought a notebook for $3.75 and a pen for $1.50. How much did she spend in total?
Step 1: Align the decimal points.
Step 2: Add 3.75 + 1.50.
Step 3: Perform the addition.
Answer: $5.25

Multiplying Decimals by Whole Numbers

When multiplying a decimal by a whole number, treat the decimal as a whole number first, and then place the decimal point in the product based on the number of decimal places.
Example:
3.6 × 4
Step 1: Ignore the decimal and multiply 36 × 4.
Step 2: Perform the multiplication (36 × 4 = 144).
Step 3: Place the decimal in the product (since there is 1 decimal place in 3.6).
Answer: 14.4

Multiplying Decimals by Decimals

When multiplying decimals by decimals, first ignore the decimal points, multiply the numbers as whole numbers, and then place the decimal in the product based on the total number of decimal places.
Example:
2.5 × 3.6
Step 1: Ignore the decimals and multiply 25 × 36.
Step 2: Perform the multiplication (25 × 36 = 900).
Step 3: Count the total decimal places (1 decimal place in 2.5, 1 decimal place in 3.6).
Step 4: Place the decimal in the product (2 decimal places).
Answer: 9.00

Estimating Products

To estimate products of decimals, round the numbers to the nearest whole number or decimal place before multiplying.
Example:
Estimate 3.75 × 2.4.
Step 1: Round 3.75 to 4 and 2.4 to 2.
Step 2: Multiply the rounded numbers (4 × 2 = 8).
Answer: 8 (estimated product).

Real-life Problems (e.g., Cost, Area)

Real-life problems often involve multiplying decimals for cost calculations, area measurements, etc.
Example:
If one pencil costs $0.75, how much will 12 pencils cost?
Step 1: Multiply 0.75 × 12.
Step 2: Perform the multiplication (0.75 × 12 = 9.00).
Answer: $9.00

Dividing Decimals by Whole Numbers

To divide a decimal by a whole number, perform the division as if the decimal were a whole number, then place the decimal point in the quotient.
Example:
Divide 4.8 by 2.
Step 1: Remove the decimal and divide 48 by 2 (48 ÷ 2 = 24).
Step 2: Place the decimal point back in the result (4.8 ÷ 2 = 2.4).
Answer: 4.8 ÷ 2 = 2.4

Dividing Decimals by Decimals

To divide decimals by decimals, eliminate the decimal points by multiplying both the numerator and the denominator by 10, 100, or more as necessary, then perform the division.
Example:
Divide 3.6 by 0.6.
Step 1: Remove the decimals by multiplying both the numerator and denominator by 10 (36 ÷ 6).
Step 2: Divide 36 by 6 (36 ÷ 6 = 6).
Answer: 3.6 ÷ 0.6 = 6

Estimating Quotients

Estimating quotients involves rounding the numbers before dividing. This helps to make the calculation easier.
Example:
Estimate 7.5 ÷ 2.6.
Step 1: Round 7.5 to 8 and 2.6 to 3.
Step 2: Estimate 8 ÷ 3 (8 ÷ 3 = 2.67).
Answer: 7.5 ÷ 2.6 ≈ 2.67

Word Problems (e.g., Sharing, Rates)

Word problems can be solved by translating the text into a mathematical equation, performing the operations, and interpreting the result.
Example:
Sarah wants to share 6.4 apples among 4 friends. How many apples does each friend get?
Step 1: Divide 6.4 by 4 (6.4 ÷ 4 = 1.6).
Answer: Each friend gets 1.6 apples.

Solving Real-life Problems with Decimals in Context

Real-life problems often involve decimals for calculations related to money, measurements, and other daily activities.
Example:
Sarah buys 2.5 pounds of apples at $1.50 per pound. How much does she pay?
Step 1: Multiply 2.5 by 1.50 (2.5 × 1.50 = 3.75).
Answer: Sarah pays $3.75 for the apples.

Multi-step Word Problems

Multi-step problems require performing more than one operation, such as addition, subtraction, multiplication, or division.
Example:
Sarah buys 3.5 kg of apples at $2 per kg and 2.3 kg of bananas at $1.5 per kg. How much does she spend?
Step 1: Multiply 3.5 by 2 (3.5 × 2 = 7).
Step 2: Multiply 2.3 by 1.5 (2.3 × 1.5 = 3.45).
Step 3: Add 7 and 3.45 (7 + 3.45 = 10.45).
Answer: Sarah spends $10.45 in total.

Estimating and Checking Answers

Estimating involves rounding numbers to make mental calculations easier. After solving, check the answer to ensure it is reasonable.
Example:
Estimate the sum of 14.7 and 6.8.
Step 1: Round 14.7 to 15 and 6.8 to 7.
Step 2: Add 15 and 7 (15 + 7 = 22).
Answer: The estimated sum is 22. Check the actual sum: 14.7 + 6.8 = 21.5.

Choosing the Correct Operation

In some problems, you need to decide whether to add, subtract, multiply, or divide based on the situation.
Example:
Sarah has $20 and buys 4 apples for $1.25 each. How much money does she have left?
Step 1: Multiply 1.25 by 4 (1.25 × 4 = 5).
Step 2: Subtract 5 from 20 (20 - 5 = 15).
Answer: Sarah has $15 left.

Decimal Games and Puzzles

Decimal games and puzzles help reinforce decimal concepts in a fun and engaging way. Use games to practice adding, subtracting, multiplying, and dividing decimals.
Example: A game where you match decimal pairs that equal the same value, like 0.5 + 0.5 = 1.
Answer: Matching games can make learning decimals more interactive!

Mixed Operations with Decimals

In mixed operations, we use addition, subtraction, multiplication, and division in a single problem.
Example:
Add 3.5 + 2.3, then subtract 1.2 from the result.
Step 1: Add 3.5 and 2.3 (3.5 + 2.3 = 5.8).
Step 2: Subtract 1.2 from 5.8 (5.8 - 1.2 = 4.6).
Answer: The final result is 4.6.

Assessment Tests and Review Sheets

Review sheets and tests help assess your understanding of decimal concepts and operations. Answer the following practice questions to test your knowledge.
Example: What is 4.5 + 2.3? Step-by-step:
Step 1: Add 4.5 and 2.3 (4.5 + 2.3 = 6.8).
Answer: The result is 6.8.

Understanding Equivalence

Equivalence means two values are the same, though represented in different ways. For decimals and fractions, equivalence means that the value represented by both forms is the same.
Example:
0.5 = 1/2 (both represent the same value).
Step 1: Write the fraction or decimal.
Step 2: Convert one form into the other (fraction to decimal or decimal to fraction).
Answer: 0.5 is equivalent to 1/2.

Converting Fractions to Decimals (Division Method)

To convert a fraction to a decimal, divide the numerator by the denominator.
Example:
Convert 3/4 to a decimal.
Step 1: Divide 3 by 4 (3 ÷ 4 = 0.75).
Answer: 3/4 = 0.75

Converting Decimals to Fractions (Place Value Method)

To convert a decimal to a fraction, use the place value of the decimal.
Example:
Convert 0.75 to a fraction.
Step 1: Count the decimal places (0.75 has two decimal places).
Step 2: Write the decimal as a fraction over 100 (75/100).
Step 3: Simplify the fraction (75/100 = 3/4).
Answer: 0.75 = 3/4

Using Decimals in Units (Meters, Liters, Kilograms)

Decimals are often used to represent fractional units in measurements such as meters, liters, and kilograms.
Example:
If a tank holds 25.6 liters of water, the measurement is represented as 25.6 liters.
Answer: 25.6 liters

Understanding Decimal Currency (Dollars and Cents)

Decimal currency refers to the use of decimals in representing amounts of money, where one dollar equals 100 cents.
Example:
$5.75 means 5 dollars and 75 cents.
Step 1: Identify the dollar amount (5).
Step 2: Identify the cents amount (75).
Answer: $5.75 = 5 dollars and 75 cents

Explanation:

A ratio is a way to compare two quantities. It shows how much of one thing there is compared to another. Ratios are typically written in one of three forms: a:b, a/b, or "a to b". Ratios can be used to describe relationships between numbers or objects.

There are two main types of ratios:

• Part-to-Part Ratio: This compares one part of a whole to another part of the same whole.
• Part-to-Whole Ratio: This compares one part of a whole to the entire whole.

Examples:

• The ratio of 2 red balls to 3 blue balls is written as 2:3.
• If there are 4 boys and 6 girls in a class, the ratio of boys to girls is 4:6 (which can be simplified to 2:3).
• In a recipe, if you use 2 cups of sugar for every 5 cups of flour, the ratio of sugar to flour is 2:5.
• A bag contains 10 marbles, of which 3 are blue and 7 are green. The ratio of blue to green marbles is 3:7.
• A school has 12 teachers and 18 students in a club. The ratio of teachers to students is 12:18 (which simplifies to 2:3).

Explanation:

Writing ratios involves expressing relationships between two quantities in various forms: as a colon (a:b), a fraction (a/b), or with words (a to b). It’s essential to understand how to correctly write and simplify ratios, especially when comparing different quantities.

Examples:

• If there are 4 apples and 5 oranges, write the ratio as 4:5, 4/5, or "4 to 5".
• A ratio of 6 boys to 8 girls can be written as 6:8, 6/8, or "6 to 8". Simplified, this becomes 3:4.
• If a recipe calls for 3 cups of sugar and 2 cups of flour, the ratio of sugar to flour is written as 3:2, 3/2, or "3 to 2".
• In a class, 10 students speak French, and 15 students speak Spanish. The ratio of French to Spanish speakers is 10:15 or "10 to 15", which simplifies to 2:3.
• If there are 7 boys and 5 girls in a group, the ratio of boys to girls is 7:5, 7/5, or "7 to 5".

Explanation:

Equivalent ratios represent the same relationship between two quantities. These ratios can be obtained by multiplying or dividing both terms by the same number. Equivalent ratios express the same proportion, even if the numbers are different.

Examples:

• 2:3 is equivalent to 4:6 because both ratios have the same relationship (multiplying both terms by 2 gives 4:6).
• The ratio 5:10 is equivalent to 1:2 (dividing both terms by 5 gives 1:2).
• 3:4 is equivalent to 6:8 (multiplying both terms by 2 gives 6:8).
• 8:12 is equivalent to 2:3 (dividing both terms by 4 gives 2:3).
• The ratio 15:20 is equivalent to 3:4 (dividing both terms by 5 gives 3:4).

Explanation:

A rate is a specific type of ratio that compares two quantities with different units. Rates are commonly used to describe things like speed, price, or other comparisons that involve different units of measurement (such as miles per hour or cost per item).

Examples:

• If a car travels 100 miles in 2 hours, the rate is 100 miles per 2 hours, or 50 miles per hour.
• A worker completes 8 tasks in 4 hours. The rate is 8 tasks per 4 hours, or 2 tasks per hour.
• A store sells 3 apples for $1.50. The rate is $1.50 per 3 apples, or $0.50 per apple.
• If a printer can print 200 pages in 5 minutes, the rate is 200 pages per 5 minutes, or 40 pages per minute.
• A cyclist travels 30 miles in 3 hours. The rate is 30 miles per 3 hours, or 10 miles per hour.

Explanation:

Calculating rates involves dividing one quantity by another. To find the rate, divide the first quantity by the second. This is often used to determine things like speed, cost per item, or productivity.

Examples:

• If a car travels 150 miles in 3 hours, the rate of speed is 150 miles ÷ 3 hours = 50 miles per hour.
• A factory produces 500 items in 10 hours. The rate of production is 500 ÷ 10 = 50 items per hour.
• A shop sells 24 items in 4 days. The rate of sales is 24 ÷ 4 = 6 items per day.
• If a worker earns $200 for 5 hours of work, the rate of pay is $200 ÷ 5 = $40 per hour.
• A runner completes 4 laps around a track in 20 minutes. The rate is 4 laps ÷ 20 minutes = 0.2 laps per minute.