Lam Fung Academy· LF Academy
Primary 5 · 10th Lesson · Student Handout
Fraction word problems (remainder trap + multiple steps)
Unit three · fractionapplicationquestion · 65 minutes · 1-to-3 Online Lesson
Corresponding textbooks:"New Thinking in Primary School Mathematics (Second Edition)" Volume 5A Unit 3 + Modern Education Volume 5A Unit 11-12
Core Trap:🪤 T3 "Remaining Fraction" Baseline Quantity Change + T7 Multi-step Reasoning Chain
SSPA Related:🔴 High FrequencyThe two-word question in the score paper is compulsory, accounting for about 25-30% of the application questions
Prerequisite knowledge:Class 8 (Addition of different denominators), Class 9 (Multiplication of fractions - cross reduction method), application questions of addition and subtraction of fractions
Goal of this class:❶ Fraction "is what fraction of..." ❷ Remaining trap (change in baseline quantity) ❸ Multi-step reasoning chain ❹ Reverse reasoning (finding the original quantity with a known result)
Student Name:
Class:
Date:
Time Spent:
I.Warm-Up Questions（total 5 question，5 minutes）
🍕 分數大作戰
通分、約分、四則混合！避開分母陷阱。連續答對分數翻倍！
⚡ 開始挑戰 →
| # | Question | Difficulty | Working Space（Show full working） |
|---|
| 1 | Calculate23 × 12= ? (Cross reduction, Tang 9 basis) | Basic | |
| 2 | Calculate 1 −14= ? ("remaining" = 1 − used, P4 basis) | Basic | |
| 3 | A rope is 3 meters long and uses |||SEP|||. How many meters were used? ("Fraction of the total length" = multiplication)13. How many meters were used? ("Fraction of the total length" = multiplication) | Basic | |
| 4 | Xiao Ming has 12 candies. After eating |||SEP|||, how many are left? (Please tell me how many pills you have eaten first, and then ask for the rest)14, how many pills are left? (Please tell me how many pills you have eaten first, and then ask for the rest) | Basic | |
| 5 | Xiaomei has $60. After using |||SEP|||, how many dollars are left? Then use the "remaining |||SEP|||", how much more is used? (Note that the two reference amounts are different!)13, how many yuan are left? Then use the remaining14”, how much more was used? (Note that the two reference amounts are different!) | Advanced | |
II.Core Knowledge ＋ Worked Examples
Knowledge point 1: Fraction problem - "What fraction of..." 🔴 SSPA
Core formula:
① "C divided by B of A" = A ×CB(Keyword "of" = multiplication)
② "A is a fraction of B" = A ÷ B =AB(Comparative relationship)
③ Find "part" |||SEP|||: Total amount × fraction = partial amountFind "fraction" |||SEP|||: Partial quantity ÷ Total quantity = Fraction (not in depth in this chapter)
④ Find the "percentage": Part amount ÷ total amount = fraction (this chapter will not go into detail)
🪤 Trap detonation example
There are 40 people in the class, and35is a boy. How many boys are there?
❌ Common mistakes
40 ÷ 35 = 66.7？
"The" is multiplication, not division! 40 × 35 is right
✅ Correct solution
24 people
40 × 35 = 1205 = 24
🧠 Tip: "When you see "is", multiply, and when you see "yes", divide; the base quantity must be clearly seen, it may be different the first time and the second time."
⚠️ Key distinction: "C/B of A" → A × C/B (multiplication). "A is what fraction of B" → A ÷ B (division).
Knowledge Point - Worked Examples
| # | Question | Difficulty | Working Space |
|---|
| Example 1 | A bottle of water12liters, Xiao Ming drank |||SEP|||. How many liters did you drink?23. How many liters did you drink? | 🌱 | |
| Example 2 | A ribbon is34meters long, cut off |||SEP|||. How many meters were cut off?12. How many meters were cut off? | 🌱 | |
Knowledge point 2: "What fraction is left" - the benchmark quantity trap 🔴 SSPA must-take killer questions
This is the trap that most students fall into!
① The essence of the trap |||SEP|||: After the first use, the "remaining" is no longer the original total② The base amount of the second operation =
The remaining amount of the first time(not the original total!)Standard process
③ Step 1: Find the amount used for the first time (original total amount) × fraction ₁)：
Step 2: Find the amount left in the first time (original total amount − used)
Step 3: Find the amount used in the second time (amount left in the first time × fraction ₂)
Common but wrong |||SEP|||: directly use the original total amount × fraction ₂ (ignoring that the base amount has changed!)
④ Common but wrong: Directly use the original total amount × fraction ₂ (ignoring that the base amount has changed!)
🪤 Trap comparison questions (core of this class)
A rope is 24 meters long. Use13first, and then use the remaining |||SEP|||. How many meters did it take the second time?12. How many meters did it take the second time?
❌ Typical errors (80% students)
24 × 12= 12 meters
Just use the original total amount × 1/2, but the base amount has become 16 meters!
✅ Correct solution
8 meters
Step1: 24×13=8 (used) Step2: 24−8=16 (remaining) Step3: 16×12=8 (used for the second time)
🧠 Tip: "Every time you use it, count how much is left. The base number will be different the next time you use it!"
⚠️ The deadliest trap: "The remaining fraction" - the "base amount" must be re-confirmed before each operation!
Knowledge Point 2 Worked Examples
| # | Question | Difficulty | Working Space |
|---|
| Example 3 | There are 24 candies in a box. I ate13on the first day and "the remaining |||SEP|||" on the second day. How many pills did you take the next day? (Note that the base amount has changed)12”. How many pills did you take the next day? (Note that the base amount has changed) | 🌳 | |
| Example 4 | A barrel of oil is 10 liters, use25for the first time, and use the remaining13for the second time. How many liters were used the second time? | 🌳 | |
Knowledge point 3: Multi-step fraction word problems (continuous reasoning chain) 🔴 SSPA Advanced
Continuous surplus problem - the baseline amount must be recalculated at each step:
① Draw a flow chart |||SEP|||: Total amount → First step (used) → Remaining ₁ → Second step (used) → Remaining ₂ → ...Step-by-step calculation |||SEP|||: Do not skip steps, record "how much is used" and "how much is left" at each step
② Verification skills |||SEP|||: Add up the amount of each step + final remaining = total amount (check calculation): Don’t skip steps, record “how much is used” and “how much is left” at each step
③ Verification skills: Sum up the amounts used in each step + final remaining = total amount (check calculation)
①
Draw a flow chart
Total→Use ₁→Left ₁→Use ₂→Left ₂→...
②
Calculate step by step
Record used + remaining at each step
③
Confirm base amount
The base amount of points used each time is the remaining amount from the last time
④
Check calculation
Amount used in each step + final remaining amount = total amount
example
Example 5: Mom has $100, and uses14to buy vegetables, and the remaining13to buy fruits. How much did it cost to buy fruit?
example
Example 6: A bottle of oil23liters, use12for the first time, and use the remaining13for the second time. How many liters did you use the second time?
Knowledge Point 4: Reverse Reasoning - Find the Original Quantity from a Known Result 🔴 SSPA Paper 2 Killer Question
Reverse reasoning (reduction problem):
① Knowing the final remaining amount, find the original total amount
② Key: working backwards!
If13is used, the remaining amount = total amount × (1 −13) = total amount ×23
So: total amount = remaining amount ÷ (1 − fraction)
③ Simple formula |||SEP|||: original amount = remaining amount ÷ remaining fraction: Original amount = remaining amount ÷ remaining fraction
example
Example 7: Xiao Ming has some sugar, and after using |||SEP|||, there are 12 pieces left. How many grains were there? (Hint: What percentage is left?)13There are 12 pills left. How many grains were there? (Hint: What percentage is left?)
example
Example 8: A box of apples, after selling |||SEP|||, there are 30 apples left. How many were there? (Hint: remaining fraction = 1 −2530 are left. How many were there? (Hint: remaining fraction = 1 −25 = 35）
🧠 Tip (reverse reasoning): "It doesn't matter how much you use, find out what percentage is left. Divide the remaining percentage by the low amount, and get back to the original!"
III. Lesson layered synchronization practice (total 15 questions)
Basic layer (total 5 questions, everyone must do)
| # | Question | Difficulty | Working Space |
|---|
| 9 | There are 36 people in the class,13is short-sighted. How many people are nearsighted? | 🌱 | |
| 10 | A ribbon is23meters long and uses |||SEP|||. How many meters were used?12. How many meters were used? | 🌱 | |
| 11 | There are 12 egg tarts in a box and I ate |||SEP|||. How many are left?13. How many are left? | 🌱 | |
| 12 | Dad’s monthly salary is $30,000, and he uses15to pay the rent. How much is the rent? | 🌱 | |
| 13 | A watermelon weighs 2 kg and is eaten |||SEP|||. How many kilograms did you eat?34. How many kilograms did you eat? | 🌱 | |
Advanced layer (total 5 questions, 🚶🚀 choose do)
| # | Question | Difficulty | Working Space |
|---|
| 14 | A barrel of oil is 5 liters, use15on the first day, and use the "remaining |||SEP|||" on the second day. How many liters will be used the next day?14”. How many liters will be used the next day? | 🌿 | |
| 15 | A rope is 3 meters long. Use13meters first, and then use the remaining12meters. How many meters are left at the end? | 🌿 | |
| 16 | I have 20 kilograms of rice at home. I eat14in the first week and "the remaining |||SEP|||" in the second week. How many kilograms did you eat in the second week?15”. How many kilograms did you eat in the second week? | 🌿 | |
| 17 | My sister had $240 and used13to buy books, and the remaining12to buy stationery. How much did you spend on stationery? | 🌳 | |
| 18 | For a cake, my father ate |||SEP|||, my mother ate "the remaining |||SEP|||", and my brother ate "the remaining |||SEP|||". What fraction of the whole cake did my brother eat?14, mother ate "the remaining13”, the younger brother ate “and the rest12”. What fraction of the whole cake did my brother eat? | 🌳 | |
🌳 challenge layer (total 5 questions, 🚀 choose do, SSPAKiller Questions)
| # | Question | Difficulty | Working Space |
|---|
| 19 | The warehouse has 180 bales of rice. Transport13on the first day, transport "the remaining |||SEP|||" on the second day, transport "the remaining |||SEP|||" on the third day. How many packages were shipped on the third day?14”, and on the third day the remaining ones were transported away12”. How many packages were shipped on the third day? | 🌳 | |
| 20 | A book has 240 pages. I watched14on the first day, "The rest of |||SEP|||" on the second day, and "The rest of |||SEP|||" on the third day. How many pages are left to read?13”, and on the third day I watched “The rest12”. How many pages are left to read? | 🏔️ | |
| 21 | A rectangle is long34meters and its width is long |||SEP|||. How many square meters is the area? (Find width first, then area)23. How many square meters is the area? (Find width first, then area) | 🌳 | |
| 22 | A bottle of juice contains 112liter. The first time I poured |||SEP|||, the second time I poured "the remaining |||SEP|||". How many liters are left?13, poured out for the second time "the rest12”. How many liters are left? | 🌳 | |
| 23 | Xiaomei has some stickers. I gave14to Xiao Hua, and then gave "the remaining |||SEP|||" to Xiao Ming, and finally there were 18 pieces left. How many stickers does Xiaomei have? (reverse reasoning)13"To Xiao Ming, there are 18 cards left. How many stickers does Xiaomei have? (reverse reasoning) | 🏔️ | |
IV.fractionapplicationquestionspecial topic training (total 8 questions, all must do)
| # | Question | Difficulty | Working Space |
|---|
| 24 | Residential buildings were built on14on one site, and shopping malls were built on the remaining |||SEP|||. What fraction of the floor space does the mall occupy? (Note: the base amount is the “remainder”!)25A shopping mall was built. What fraction of the floor space does the mall occupy? (Note: the base amount is the “remainder”!) | 🌳 | |
| 25 | The school has 800 students,38are primary school students (P1-P3), and the remaining25are middle school students (P4-P6). How many middle school students are there? | 🌳 | |
| 26 | One road is 2 kilometers long and has been repaired |||SEP|||. The remaining38will be repaired next week. How many kilometers will be repaired next week?15Will fix it next week. How many kilometers will be repaired next week? | 🌳 | |
| 27 | Garden25grows flowers, flowers34is roses. What percentage of the garden do roses occupy? (Two layers of "的" = continuous multiplication) | 🌳 | |
| 28 | There are 12 liters in a bucket of water. Use14first, then add13liters. How many liters are there now? (Mixed addition, subtraction and multiplication!) | 🌳 | |
| 29 | Xiao Ming had $150, so he used15to buy pencils, "the remaining |||SEP|||" to buy books, and "the last remaining |||SEP|||" to buy snacks. How much did you spend on snacks?13"Buy the book," the last remaining12”Buy snacks. How much did you spend on snacks? | 🏔️ | |
| 30 | A rectangular planter is 212meters long and12wide. How many square meters is the area? (Mixed fraction × fraction → turn to false first) | 🌳 | |
| 31 | Brother, I have some candy. After he ate |||SEP|||, he had 40 pills left. How many grains were there? (reverse reasoning)15Afterwards, 40 pills remain. How many grains were there? (reverse reasoning) | 🌳 | |
V.🏔️ Ultimate challenge area (total 3 question)
| # | Question | Difficulty | Working Space |
|---|
| 🏔️1 | A book, read13on the first day, and read "the remaining |||SEP|||" on the second day, with 40 pages left. How many pages does this book have? (Ultimate reverse reasoning - two-level reduction)35”, with 40 pages left. How many pages does this book have? (Ultimate reverse reasoning - two-level reduction) | 🏔️ | |
| 🏔️2 | The water tank contains 20 liters of water. Use15on the first day, use "the remaining |||SEP|||" on the second day, use "the remaining |||SEP|||" on the third day, and use "the remaining |||SEP|||" on the fourth day. How many liters were used on the fourth day? How many liters are left at the end? (Four-step reasoning chain)14” and use the rest on the third day13” and used the remaining ones on the fourth day.12”. How many liters were used on the fourth day? How many liters are left at the end? (Four-step reasoning chain) | 🏔️ | |
| 🏔️3 | Three children share a box of chocolates. A took |||SEP|||, B took "the remaining |||SEP|||", C took "the remaining |||SEP|||", and finally there were 6 pills left in the box. How many pills are in a box? (Triple reduction reasoning)14, B took "the remaining13”, C took “the rest12”, and finally there were 6 pills left in the box. How many pills are in a box? (Triple reduction reasoning) | 🏔️ | |
VI.Class afterhomework
Basic must-do questions (total 5 questions)
| # | Question | Difficulty | Working Space |
|---|
| H1 | A rope is 5 meters long and uses |||SEP|||. How many meters were used?15. How many meters were used? | 🌱 | |
| H2 | A bottle of juice12liters, drank12in it. How many liters did you drink? | 🌱 | |
| H3 | There are 24 biscuits in a box and I ate |||SEP|||. How many pieces are left?23. How many pieces are left? | 🌱 | |
| H4 | A potted flower is12meters high. After watering, it grows taller than the original |||SEP|||. How many meters tall is it now? (Note: I have grown 1/2 of my original height)12. How many meters tall is it now? (Note: I have grown 1/2 of my original height) | 🌿 | |
| H5 | My sister had $120, so she used13to buy books, and then used the "remaining |||SEP|||" to buy handmade materials. How much did it cost to buy handmade materials?14"Buy handmade materials. How much did it cost to buy handmade materials? | 🌳 | |
Advanced choose doquestion (total 3 questions, 🚀 choose do)
| # | Question | Difficulty | Working Space |
|---|
| H6 | A ribbon is 114meter long and uses |||SEP|||. How many meters were used?13. How many meters were used? | 🌳 | |
| H7 | One month of14is sunny, the remaining13is rainy, and the remaining12is cloudy. What fraction of the month is cloudy? (Three levels of residual reasoning) | 🏔️ | |
| H8 | A water tank contains several liters of water. After using |||SEP|||, 18 liters remain. How many liters of water did the water tank originally contain? (reverse reasoning)25Afterwards, 18 liters remain. How many liters of water did the water tank originally contain? (reverse reasoning) | 🌳 | |
VII. The Lessoncorecommon errorsummary
✅ Self-examination in this hall (tick after completion)
☐ I know the pitfalls of solving each knowledge point
☐ I can complete 🌱basic questions independently
☐ I can challenge 🌿advanced questions
☐ I remember the formula
🎯 Review of Learning Objectives - After completing this lesson you should be able to:
☐ Identify all trap types in our hall
☐ Solve 🌱basic questions independently (100% correct)
☐ Challenge🌿Advanced questions (80%+ correct)
☐ Explain the lesson formula to classmates
| # | common error | Correct Approach |
|---|
| 1 | "What fraction is left" is directly multiplied by the original total amount | Calculate the true remaining amount first, then use the remaining amount × fraction. Baseline amounts may vary each time! |
| 2 | Skip intermediate calculations in multi-step problems | Calculate step by step, record "how much is used" and "how much is left" every time, without skipping steps. |
| 3 | "A's C/B" uses division incorrectly | "'s" = multiplication: A × C/B. Division is only used if "A is a fraction of B". |
| 4 | Wrong direction when reasoning backwards | Reduction formula: original quantity = remaining quantity ÷ (1 − fraction used). Not a ride! |
| 5 | Calculate mixed numbers without converting them to improper fractions | For any multiplication or division involving mixed numbers, the first step = converting improper fractions. |
| 6 | Unit confusion – fractions vs actual quantities | "1/3" may be a fraction (without unit) or an actual quantity (with unit), please read the context clearly! |
| 7 | Missing sentences/missing units/missing calculations in application questions | Must write "Answer:..." + unit. Calculation: Amount used in all steps + final remaining amount = total amount. |
Lam Fung Academy · LF Academy · We don't teach math. We teach trap avoidance.