Multi-Step Fraction Word Problems - Sixth Grade Mathematics
Solving complex multi-step fraction word problems requires dissecting realistic scenarios, tracking changing wholes, and applying operations in logical sequence.
The Anatomy of Multi-Step Problems
Single-step problems require one calculation to reach a conclusion. Multi-step problems require you to use the result of one computation as the input for the next. The most critical skill is monitoring the reference whole: does a subsequent fraction apply to the original total, or to the remainder?
+-------------------------------------------------------------------------+
| DIAGNOSING FRACTION WORD PROBLEMS |
+-------------------+-----------------------+-----------------------------+
| Scenario Type | Phrasing Clue | Mathematical Action |
+-------------------+-----------------------+-----------------------------+
| Fixed Whole | "... spent 1/4 of his | Fractions are added |
| | total money on food | directly because both share |
| | and 1/3 of his total | the same original whole: |
| | money on clothes..." | 1/4 + 1/3 = 7/12 total spent|
+-------------------+-----------------------+-----------------------------+
| Changing Whole | "... spent 1/4 of his | Find remainder first! |
| (Fraction of | money, then spent 1/3 | Remainder = 1 - 1/4 = 3/4. |
| Remainder) | OF THE REMAINDER..." | Next spend = 1/3 * 3/4 = 1/4|
+-------------------+-----------------------+-----------------------------+
Using Tape Diagrams to Model Fractions of Remainders
Visual tape diagrams make multi-step remainder problems intuitive by showing the physical reduction of the whole.
Tape Diagram for: "Elena had $120. She spent 1/3 on books,
then 1/4 of the remaining money on lunch."
Step 1: Divide whole ($120) into 3 equal thirds:
+-------------------+-------------------+-------------------+
| Books ($40) | Remaining ($40) | Remaining ($40) |
+-------------------+-------------------+-------------------+
Spent on books = 1/3 of 120 = $40.
Remaining money = $80.
Step 2: Partition the remaining $80 into 4 equal fourths ($20 each):
+---------+---------+---------+---------+
| Lunch | Leftover ($60 total) |
| ($20) | |
+---------+---------+---------+---------+
Spent on lunch = 1/4 of $80 = $20.
Total remaining = $80 - $20 = $60.
The Four-Phase Problem Solving Architecture
Approaching complex word problems systematically prevents computational disorientation.
Step 1: Annotate & Clarify --> Circle numbers, underline question, state units.
Step 2: Model & Sequence --> Draw tape diagram or outline intermediate goals:
Goal 1: Find remainder after first action.
Goal 2: Calculate second fractional portion.
Goal 3: Subtract to find final balance.
Step 3: Execute Operations --> Compute with common denominators or multiplication.
Step 4: Check Reasonableness-> Verify against original total and logical limits.
Comprehensive Case Studies
Examining complete case studies demonstrates how to unpack multi-layered scenarios.
Case Study A: The Recipe Batch Problem
A bakery recipe requires 2 and 1/4 cups of flour for 1 batch of cookies.
The baker has a 10-cup bag of flour. How many full batches can the baker
make, and how much flour will be left over?
1. Understand: Divide total flour (10) by flour per batch (2 and 1/4).
2. Convert: 2 and 1/4 = 9/4. Whole number 10 = 10/1.
3. Compute: 10 / (9/4) = (10/1) * (4/9) = 40/9.
4. Interpret: 40 / 9 = 4 and 4/9 batches.
The baker can make 4 full batches.
5. Leftover flour: 4 batches require 4 * (9/4) = 36/4 = 9 cups.
Remaining flour = 10 - 9 = 1 cup of flour remaining!
Exercises
Exercise 1. Liam had 60 dollars in savings. He spent 2/5 of his savings on a video game, and then spent 1/3 of his REMAINING money on a wireless mouse. How much money does Liam have left?
Exercise 2. A school bus travels at an average speed of 42 miles per hour. If the driver travels for 2 and 3/4 hours, how many miles does the bus cover?
Exercise 3. A container was filled with 15 and 3/4 gallons of cider. The school sold 8 and 1/2 gallons in the morning and 4 and 2/3 gallons in the afternoon. How many gallons of cider remain in the container?
Solutions
Solution to Exercise 1. First, calculate the amount spent on the video game: 2/5 of 60 = (2/5) * 60 = (2 * 60) / 5 = 120 / 5 = 24 dollars. Subtract to find the remaining money: 60 - 24 = 36 dollars. Next, calculate the amount spent on the mouse: Liam spent 1/3 of the remaining 36 dollars: (1/3) * 36 = 12 dollars. Finally, subtract to find the money left: 36 - 12 = 24 dollars. Liam has 24 dollars left.
Solution to Exercise 2. To find total distance, multiply speed by time: Distance = Speed * Time. Distance = 42 * (2 and 3/4). Convert 2 and 3/4 to an improper fraction: (2 * 4 + 3)/4 = 11/4. Write 42 as 42/1: (42/1) * (11/4). Cross-simplify 42 and 4 by dividing both by 2: 42 becomes 21, and 4 becomes 2. Multiply across: (21 * 11) / (1 * 2) = 231 / 2. Convert to a mixed number: 231 / 2 = 115 with a remainder of 1. The bus travels 115 and 1/2 miles.
Solution to Exercise 3. First, calculate the total cider sold by adding morning and afternoon sales: 8 and 1/2 + 4 and 2/3. Find the common denominator for 2 and 3, which is 6: 1/2 = 3/6, and 2/3 = 4/6. Add whole numbers: 8 + 4 = 12. Add fractions: 3/6 + 4/6 = 7/6 = 1 and 1/6. Combine: 12 + 1 and 1/6 = 13 and 1/6 gallons sold. Now subtract total sold from the starting container volume: 15 and 3/4 - 13 and 1/6. Find LCD of 4 and 6, which is 12: 3/4 = 9/12, and 1/6 = 2/12. Subtract whole numbers: 15 - 13 = 2. Subtract fractions: 9/12 - 2/12 = 7/12. There are 2 and 7/12 gallons of cider remaining in the container.