Word Problems with Decimals and Fractions: 6th Grade Advanced Practice

Master complex decimal and fraction operations through challenging word problems: multi-step solutions, real-world applications, and advanced concepts.

Advanced Decimal and Fraction Word Problems: 1 to 5
1 Complex Shopping Problem
Exercise 1
Emma bought 2.75 kg of apples for $3.20 per kg and 3/4 kg of oranges for $4.50 per kg. How much did she spend in total?
Definition:

Unit Rate Multiplication: Multiply quantity by unit price for each item, then add totals

Solution Method:
  1. Convert fraction to decimal: 3/4 = 0.75
  2. Calculate cost of apples: 2.75 × $3.20
  3. Calculate cost of oranges: 0.75 × $4.50
  4. Add both costs to find total
Apple Cost
$8.80
Orange Cost
$3.38
Total Cost
$12.18
Step 1: Convert fraction to decimal

3/4 = 0.75 kg

Step 2: Calculate apple cost

2.75 × $3.20 = $8.80

Step 3: Calculate orange cost

0.75 × $4.50 = $3.375 ≈ $3.38

Step 4: Add costs

$8.80 + $3.38 = $12.18

(2.75 × $3.20) + (0.75 × $4.50) = $12.18
Final answer:

Emma spent $12.18 in total

Applied rules:

• Unit Rate: Multiply quantity by unit price to find total cost

• Conversion: Convert fractions to decimals for multiplication

• Rounding: Round monetary amounts to nearest cent

2 Complex Measurement Problem
Exercise 2
A board is 8.5 meters long. First, 1/3 of the board is cut off, then 2.25 meters more is cut off. How much board remains?
Definition:

Fraction of a Decimal: Multiply fraction by decimal amount to find portion

Original Length
8.5 m
First Cut
- 2.83 m
Second Cut
- 2.25 m
Remaining
3.42 m
Step 1: Calculate first cut (1/3 of 8.5)

1/3 × 8.5 = 8.5 ÷ 3 = 2.833... ≈ 2.83 m

Step 2: Subtract both cuts from original

8.5 - 2.83 - 2.25 = 3.42 m

Step 3: Verify calculation

3.42 + 2.83 + 2.25 = 8.50 ✓

8.5 - (1/3 × 8.5) - 2.25 = 3.42 m
Final answer:

3.42 meters of board remains

Applied rules:

• Fraction of Decimal: Multiply fraction by decimal amount

• Sequential Operations: Perform operations in order

• Verification: Check by adding all parts to original

3 Advanced Money Problem
Exercise 3
Alex had $45.75. He spent 2/5 of his money on lunch, then spent $8.90 on snacks, and finally received 1/4 of his original amount as a gift. How much money does he have now?
Definition:

Multi-step Operations: Sequential operations with fractions and decimals

Lunch Cost
$18.30
After Lunch
$27.45
After Snacks
$18.55
Gift Added
$11.44
Final Amount
$29.99
Step 1: Calculate lunch cost (2/5 of $45.75)

2/5 × $45.75 = $91.50 ÷ 5 = $18.30

Step 2: Calculate money after lunch

$45.75 - $18.30 = $27.45

Step 3: Calculate money after snacks

$27.45 - $8.90 = $18.55

Step 4: Calculate gift amount (1/4 of original)

1/4 × $45.75 = $45.75 ÷ 4 = $11.4375 ≈ $11.44

Step 5: Calculate final amount

$18.55 + $11.44 = $29.99

Original: $45.75 - (2/5 × $45.75) - $8.90 + (1/4 × $45.75) = $29.99
Final answer:

Alex has $29.99 now

Applied rules:

• Sequential Operations: Perform operations in the order they occur

• Fraction of Amount: Multiply fraction by decimal to find portion

• Rounding: Round monetary amounts to nearest cent

4 Advanced Weight Problem
Exercise 4
A container initially holds 12.6 kg of sand. 3/8 of the sand is removed, then 1.75 kg is added, and finally 1/5 of the current amount is removed. What is the final weight?
Definition:

Sequential Fraction Operations: Multiple operations with fractions and decimals in sequence

Initial Weight
12.6 kg
After 1st Removal
7.88 kg
After Addition
9.63 kg
After 2nd Removal
7.70 kg
Step 1: Calculate first removal (3/8 of 12.6)

3/8 × 12.6 = 37.8 ÷ 8 = 4.725 kg removed

12.6 - 4.725 = 7.875 ≈ 7.88 kg remaining

Step 2: Add 1.75 kg

7.88 + 1.75 = 9.63 kg

Step 3: Calculate second removal (1/5 of 9.63)

1/5 × 9.63 = 9.63 ÷ 5 = 1.926 kg removed

9.63 - 1.926 = 7.704 ≈ 7.70 kg final

((12.6 - (3/8 × 12.6)) + 1.75) - (1/5 × ((12.6 - (3/8 × 12.6)) + 1.75)) = 7.70 kg
Final answer:

The final weight is 7.70 kg

Applied rules:

• Sequential Operations: Perform operations in order from left to right

• Fraction of Current Amount: Calculate fraction of the current value after each operation

• Precision: Maintain decimal accuracy throughout calculations

5 Distance Rate Problem
Exercise 5
A car travels at 60.5 km/h for 2 hours, then at 3/4 of that speed for 1.5 hours. What is the total distance traveled?
Definition:

Distance Formula: Distance = Speed × Time

First Leg
121 km
Reduced Speed
45.38 km/h
Second Leg
68.07 km
Total Distance
189.07 km
Step 1: Calculate first leg distance

60.5 km/h × 2 h = 121 km

Step 2: Calculate reduced speed

3/4 × 60.5 = 181.5 ÷ 4 = 45.375 ≈ 45.38 km/h

Step 3: Calculate second leg distance

45.38 km/h × 1.5 h = 68.07 km

Step 4: Calculate total distance

121 + 68.07 = 189.07 km

(60.5 × 2) + ((3/4 × 60.5) × 1.5) = 189.07 km
Final answer:

The car traveled 189.07 km in total

Applied rules:

• Distance Formula: Distance = Speed × Time

• Fraction of Rate: Calculate fraction of original speed

• Multi-step Operations: Combine multiple calculations

Advanced Decimal and Fraction Word Problems: 6 to 10
6 Advanced Liquid Problem
Exercise 6
A tank contains 24.8 liters of water. 5/8 of the water is poured out, then 3.75 liters are added, and finally 1/3 of the current amount is evaporated. How much water remains?
Definition:

Sequential Operations with Fractions: Multiple operations in sequence with fractions and decimals

Initial Amount
24.8 L
After Pouring
9.3 L
After Adding
13.05 L
After Evaporation
8.7 L
Step 1: Calculate amount poured out

5/8 × 24.8 = 124 ÷ 8 = 15.5 L poured out

24.8 - 15.5 = 9.3 L remaining

Step 2: Add 3.75 L

9.3 + 3.75 = 13.05 L

Step 3: Calculate evaporation (1/3 of 13.05)

1/3 × 13.05 = 13.05 ÷ 3 = 4.35 L evaporated

13.05 - 4.35 = 8.7 L remaining

((24.8 - (5/8 × 24.8)) + 3.75) - (1/3 × ((24.8 - (5/8 × 24.8)) + 3.75)) = 8.7 L
Final answer:

8.7 liters of water remain

Applied rules:

• Sequential Operations: Perform operations in order from left to right

• Fraction of Current Amount: Calculate fraction of the current value after each operation

• Precision: Maintain decimal accuracy throughout calculations

7 Investment Growth Problem
Exercise 7
Maya invested $1,250.50. In the first year, her investment grew by 1/4 of its value. In the second year, it decreased by 0.15 of its value at the beginning of year 2. What is the value after two years?
Definition:

Compound Growth/Decrease: Sequential percentage changes based on current value

Initial Investment
$1,250.50
After Year 1
$1,563.13
After Year 2
$1,328.66
Step 1: Calculate growth in year 1

1/4 × $1,250.50 = $1,250.50 ÷ 4 = $312.625

$1,250.50 + $312.625 = $1,563.125 ≈ $1,563.13

Step 2: Calculate decrease in year 2

0.15 × $1,563.13 = $234.4695

$1,563.13 - $234.4695 = $1,328.6605 ≈ $1,328.66

($1,250.50 + (1/4 × $1,250.50)) - (0.15 × ($1,250.50 + (1/4 × $1,250.50))) = $1,328.66
Final answer:

The investment is worth $1,328.66 after two years

Applied rules:

• Sequential Calculations: Each year's change depends on previous year's value

• Fraction/Decimal Operations: Calculate portions of current amounts

• Rounding: Round monetary amounts to nearest cent

8 Complex Length Problem
Exercise 8
A rope is 15.75 meters long. First, 2/3 of its length is cut off. Then, 1/4 of the remaining piece is cut off. Finally, 0.85 meters is added. What is the final length?
Definition:

Sequential Fraction Operations: Multiple fraction operations applied to remaining amounts

Original Length
15.75 m
After 1st Cut
5.25 m
After 2nd Cut
3.94 m
After Addition
4.79 m
Step 1: Calculate first cut (2/3 of 15.75)

2/3 × 15.75 = 31.5 ÷ 3 = 10.5 m cut off

15.75 - 10.5 = 5.25 m remaining

Step 2: Calculate second cut (1/4 of 5.25)

1/4 × 5.25 = 5.25 ÷ 4 = 1.3125 m cut off

5.25 - 1.3125 = 3.9375 ≈ 3.94 m remaining

Step 3: Add 0.85 m

3.94 + 0.85 = 4.79 m final length

((15.75 - (2/3 × 15.75)) - (1/4 × (15.75 - (2/3 × 15.75)))) + 0.85 = 4.79 m
Final answer:

The final length is 4.79 meters

Applied rules:

• Sequential Fraction Operations: Apply fractions to remaining amounts

• Precision: Maintain decimal accuracy throughout calculations

• Order of Operations: Perform operations in sequence

9 Advanced Comparison Problem
Exercise 9
A store sold 18.5 kg of apples and 3/4 as much oranges. If the price of apples is $2.40 per kg and oranges are $3.20 per kg, how much more revenue came from oranges than apples?
Definition:

Revenue Calculation: Revenue = Quantity × Price per unit

Apple Revenue
$44.40
Orange Quantity
13.88 kg
Orange Revenue
$44.42
Difference
$0.02
Step 1: Calculate apple revenue

18.5 × $2.40 = $44.40

Step 2: Calculate orange quantity

3/4 × 18.5 = 55.5 ÷ 4 = 13.875 ≈ 13.88 kg

Step 3: Calculate orange revenue

13.88 × $3.20 = $44.416 ≈ $44.42

Step 4: Calculate difference

$44.42 - $44.40 = $0.02

Orange revenue - Apple revenue = $0.02
Final answer:

Oranges generated $0.02 more revenue than apples

Applied rules:

• Revenue Formula: Revenue = Quantity × Price per unit

• Fraction of Amount: Calculate portion of original quantity

• Comparison: Find difference between two amounts

10 Budget Allocation Problem
Exercise 10
Sarah budgeted $200 for groceries. She spent 1/5 on produce, $45.75 on dairy, 0.3 of the remaining budget on meat, and $28.50 on pantry items. How much of her budget remains?
Definition:

Sequential Budget Allocation: Multiple operations to allocate budget portions

Budget
$200.00
Produce
- $40.00
After Produce
$160.00
After Dairy
$114.25
Meat
- $34.28
Final
$51.47
Step 1: Calculate produce cost

1/5 × $200 = $200 ÷ 5 = $40.00

Step 2: Calculate remaining after produce

$200 - $40 = $160.00

Step 3: Calculate remaining after dairy

$160 - $45.75 = $114.25

Step 4: Calculate meat cost (0.3 of remaining)

0.3 × $114.25 = $34.275 ≈ $34.28

Step 5: Calculate final remaining

$114.25 - $34.28 - $28.50 = $51.47

$200 - (1/5 × $200) - $45.75 - (0.3 × ($200 - (1/5 × $200) - $45.75)) - $28.50 = $51.47
Final answer:

$51.47 of the budget remains

Applied rules:

• Sequential Allocation: Calculate portions based on remaining budget

• Decimal Operations: Perform operations with decimals

• Verification: Check that all allocations sum to original budget

Advanced Decimal and Fraction Operations Infographic

Advanced Operations Guide

🔢
Fraction-Decimal Operations
Convert fractions to decimals for mixed operations

• To convert: divide numerator by denominator

• Common conversions: 1/2=0.5, 1/4=0.25, 3/4=0.75

• 1/3=0.333..., 1/5=0.2, 2/5=0.4

• Always align decimal points when adding/subtracting

Sequential Operations
Perform operations in sequence based on current value

• Each operation affects the result of the next

• Calculate fraction of current amount, not original

• Maintain precision throughout calculations

• Round only at the final step

Complex Operations Steps
1
Identify
2
Convert
3
Sequence
4
Calculate
5
Verify
Advanced Mistakes

• Calculating fraction of original instead of current amount

• Incorrect decimal alignment

• Premature rounding

• Forgetting order of operations

Advanced Tips

• Always convert fractions to decimals for mixed operations

• Keep extra decimal places during calculations

• Use estimation to check reasonableness

• Track each step of sequential operations

Complex Fraction-Decimal Relations
1/6 = 0.167
1/8 = 0.125
3/8 = 0.375
5/8 = 0.625
Master Complex Operations!
Advanced Concepts, Methods, and Rules
(a + b/c) × d = (a × d) + (b/c × d)
Distributive Property
Key definitions:

Unit Rate: A rate with denominator of 1, expressing quantity per single unit

Sequential Operations: Operations performed in sequence where each affects the next

Compound Change: Multiple increases or decreases applied to current values

Proportional Reasoning: Understanding relationships between quantities

Repeating Decimals: Decimals with digits that repeat infinitely

Advanced Problem-Solving Methodology:
  1. Analyze: Identify all operations and their sequence
  2. Convert: Change all values to same format (decimals preferred)
  3. Plan: Determine order of operations and intermediate steps
  4. Execute: Perform calculations with precision
  5. Verify: Check that each step makes sense and final answer is reasonable
Tip 1: Convert fractions to decimals for easier mixed operations.
Tip 2: Keep extra decimal places during calculations to maintain precision.
Tip 3: For sequential operations, each fraction is applied to the current value, not the original.
Tip 4: Use estimation to verify that your answer is reasonable.
Common errors: Calculating fractions of original amount instead of current, premature rounding, incorrect decimal alignment, forgetting order of operations.
Success strategies: Work systematically, verify each step, use consistent format for all values.
Advanced Rules:

• Fraction of Current: When applying a fraction sequentially, use the current value, not the original

• Decimal Precision: Maintain extra decimal places during calculations

• Order of Operations: Perform operations in the order they occur

• Conversion Priority: Convert fractions to decimals for mixed operations

• Verification: Check that your answer makes sense in context

Advanced Decimal and Fraction Operations Summary

Definitions & Concepts

Unit Rate: A rate with denominator of 1, expressing quantity per single unit

Sequential Operations: Operations performed in sequence where each affects the next

Compound Change: Multiple increases or decreases applied to current values

Proportional Reasoning: Understanding relationships between quantities

Repeating Decimals: Decimals with digits that repeat infinitely

Core Rules & Principles

• When applying fractions sequentially, use the current value, not the original

• Always convert fractions to decimals for mixed operations

• Maintain extra decimal places during calculations for precision

• Perform operations in the order they occur (left to right)

• The decimal point in the answer goes directly below the aligned decimal points

Methods & Procedures

1. Analyze: Identify all operations and their sequence

2. Convert: Change all values to same format (decimals preferred)

3. Plan: Determine order of operations and intermediate steps

4. Execute: Perform calculations with precision

5. Verify: Check that each step makes sense and final answer is reasonable

Examples & Applications

Simple: 2.5 + 1/2 = 2.5 + 0.5 = 3.0

Intermediate: 3/4 × 8.8 = 0.75 × 8.8 = 6.6

Advanced: 10 - (1/3 × 10) = 10 - 3.33 = 6.67

Real-world: 20% of $45.50 = 0.20 × $45.50 = $9.10

Complex: Sequential operations with fractions and decimals

Tips & Tricks

• Convert fractions to decimals for easier mixed operations

• Keep extra decimal places during calculations

• Draw vertical lines to align decimal points

• Use estimation to check reasonableness

• Track each step of sequential operations

Memory Aids

• "Fractions to decimals: divide top by bottom"

• "Current value matters in sequential operations"

• "Decimal points line up like soldiers in a row"

• "Precision comes from maintaining decimal places"

• "Check your answer makes sense"

Questions & Answers

Question: I get confused when there are multiple operations with fractions and decimals. How do I know what to do first?

Answer: Follow this systematic approach:

  1. Convert all fractions to decimals for consistency
  2. Identify the sequence of operations in the problem
  3. Perform operations in order from left to right
  4. Keep extra decimal places during calculations

Example: For "1/2 of 8.4 plus 0.3 of 12":
Step 1: Convert: 0.5 × 8.4 + 0.3 × 12
Step 2: Calculate: 4.2 + 3.6 = 7.8

Always read the problem carefully to identify the sequence of operations!

Question: How can I help my child practice advanced decimal and fraction operations at home?

Answer: Focus on real-world applications that involve multiple operations:

  • Shopping: Calculate discounts, taxes, and totals with fractional percentages
  • Cooking: Adjust recipes with fractional amounts and metric measurements
  • Finance: Track investments, interest, and budget allocations
  • Science: Calculate measurements with precision and conversions

Practice with problems that involve multiple steps and require keeping track of values throughout the process.

Encourage your child to show all work and verify answers by working backwards.

Question: When do I use the current value versus the original value for fractions in multi-step problems?

Answer: This is a key concept! Use the current value when the problem says "of the current amount," "of what remains," or "of the new value."

Example 1 (current value):
"Start with $100, remove 1/4, then remove 1/3 of what remains"
Step 1: Remove 1/4 of $100 = $25, leaving $75
Step 2: Remove 1/3 of $75 = $25, leaving $50

Example 2 (original value):
"Start with $100, remove 1/4, then remove 1/3 of the original"
Step 1: Remove 1/4 of $100 = $25, leaving $75
Step 2: Remove 1/3 of $100 = $33.33, leaving $41.67

Look for key phrases to determine which value to use!