Lesson 20.2: Working Backwards and Logical Elimination
Applied concepts and practical problem solving for Working Backwards and Logical Elimination.
⏱️ Reading Time: 20 mins
• ❓ 50 Interactive Practice Questions
🌟
Real-Life Story: Mathematical Reasoning and Word Problems
Code Breaker Escape Room: Cracking mystery number locks using mathematical clues.
💡 Core Mathematical Concepts: Working Backwards and Logical Elimination
Focus areas: Reverse-engineering unknown starting values and logic puzzles..
| Concept / Term | Mathematical Meaning | Real-Life Application |
|---|---|---|
| Working Backwards | Start from the final result and invert each operation in reverse order | + becomes -, × becomes ÷ |
| Mystery Number Clues | Using properties of even/odd, divisibility, and prime numbers to isolate values | Eliminating non-matching possibilities |
| Systematic Listing | Testing possible combinations systematically to ensure no solution is missed | Logic grids and trial matrices |
📝 Step-by-Step Worked Examples
Worked Example 1: Working backwards example
I think of a number, multiply by 4, and subtract 10 to get 38. Reverse: (38 + 10) ÷ 4 = 48 ÷ 4 = 12.
Worked Example 2: Number riddle
I am an even number between 40 and 50. I am a multiple of 7. What number am I? 42.
⚠️ Common Misconceptions to Avoid
Common Misconception: Inverting operations in the wrong order
Correction: When working backwards, you must reverse both the operations AND their sequence order.
Common Misconception: Forgetting to check the original equation
Correction: Always substitute your final answer back into the forward problem to verify.
🎯
Self-Check Challenge
A magician says: 'Think of a number. Double it, add 14, and divide by 2. If the final result is 25, what was the starting number?'
Answer & Explanation:
Reverse operations: 25 × 2 = 50. 50 - 14 = 36. 36 ÷ 2 = 18. Starting number was 18.
🎯 Key Learning Takeaways
Extend understanding of Working Backwards and Logical Elimination.
Solve multi-step real-world problems systematically.
Verify solutions through mathematical reasoning.
📄 Printable Worksheets (5 Available)
Download ready-to-print PDFs for offline student practice with separate parent/teacher answer keys.
G5-M-CH20-WS05
A4 • 15 Qs
Mathematical Reasoning and Word Problems: Worksheet 5
Practice Assessment 5 • Central Botanical Gardens Context
G5-M-CH20-WS04
A4 • 15 Qs
Mathematical Reasoning and Word Problems: Worksheet 4
Practice Assessment 4 • Eco-Friendly Solar Park Context
G5-M-CH20-WS03
A4 • 15 Qs
Mathematical Reasoning and Word Problems: Worksheet 3
Practice Assessment 3 • Harbor Marine Terminal Context
G5-M-CH20-WS02
A4 • 15 Qs
Mathematical Reasoning and Word Problems: Worksheet 2
Practice Assessment 2 • Science & Discovery Museum Context
G5-M-CH20-WS01
A4 • 15 Qs
Mathematical Reasoning and Word Problems: Worksheet 1
Practice Assessment 1 • City Regional Airport Context
✍️ Lesson Practice Questions (50 Questions)
Solve these real-world challenges online, test your understanding, and reveal instant step-by-step explanations.
1
Short Answer
Hard
2 Marks
A grandfather's age is 6 times his grandson's age. In 12 years, the grandfather will be only 3 times as old as his grandson. What are their current ages?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
Grandfather is 48, grandson is 8
Step-by-Step Explanation:
Let grandson = g, grandfather = 6g. In 12 years: 6g + 12 = 3(g + 12) -> 6g + 12 = 3g + 36 -> 3g = 24 -> g = 8. Grandfather = 6 × 8 = 48.
2
Short Answer
Hard
2 Marks
In a barn, there are horses and ducks. Counting heads gives 35 heads; counting legs gives 110 legs. How many horses and how many ducks are in the barn?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
20 horses and 15 ducks
Step-by-Step Explanation:
If all 35 were ducks: 35 × 2 = 70 legs. Extra legs = 110 - 70 = 40 legs. Each horse contributes 2 extra legs: 40 ÷ 2 = 20 horses. Ducks = 35 - 20 = 15 ducks.
3
Short Answer
Hard
2 Marks
If the perimeter of a rectangle is fixed at 24 cm, which dimensions (length and width) yield the greatest possible area? Explain why.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A square of 6 cm × 6 cm gives the greatest area (36 cm²). As the rectangle becomes more square-like, the product of length and width is maximized for a fixed sum of sides.
Step-by-Step Explanation:
Equal side distribution maximizes area for any given perimeter boundary.
4
Short Answer
Hard
2 Marks
Explain why working backwards from the final result is an effective strategy for solving multi-step arithmetic mystery problems.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
Working backwards inverts each consecutive operation (addition becomes subtraction, multiplication becomes division), systematically unwrapping the mystery quantity without guessing.
Step-by-Step Explanation:
Inverse operations systematically undo transformations step-by-step to recover the starting value.
5
Short Answer
Medium
1 Mark
To divide $120 between two friends in a 2:1 ratio, a student calculates $120 ÷ 2 = $60 each. What is the correct division?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
$80 and $40
Step-by-Step Explanation:
A 2:1 ratio has 2 + 1 = 3 equal shares. $120 ÷ 3 = $40 per share. The shares are $80 (2 shares) and $40 (1 share).
6
Short Answer
Medium
1 Mark
A student solves: 'If 3 cats catch 3 mice in 3 minutes, how many cats catch 100 mice in 100 minutes?' The student answers '100 cats.' Explain the flaw and state the correct answer.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
3 cats
Step-by-Step Explanation:
Rate of 3 cats is 1 mouse per minute. In 100 minutes, those same 3 cats catch 100 mice. 3 cats are needed.
7
Fill in the Blanks
Medium
1 Mark
Work backwards to find X: ((X + 6) × 3) - 9 = 45. X = [ ? ].
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
12
Step-by-Step Explanation:
45 + 9 = 54; 54 ÷ 3 = 18; 18 - 6 = 12.
8
Fill in the Blanks
Medium
1 Mark
Solve the logic puzzle: [ ? ] + [ ? ] = 18 and [ ? ] × [ ? ] = 77. The two numbers are 7 and [ ? ].
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
11
Step-by-Step Explanation:
7 + 11 = 18 and 7 × 11 = 77.
9
Short Answer
Medium
1 Mark
A water tank empties at a rate of 18 liters per minute while a pipe fills it at 25 liters per minute. What is the net increase in water volume after 20 minutes?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
140 liters
Step-by-Step Explanation:
Net fill rate = 25 - 18 = 7 L/min. In 20 minutes: 7 × 20 = 140 liters.
10
Short Answer
Medium
1 Mark
Together, Ethan and his sister have 64 marbles. Ethan has 12 more marbles than his sister. How many marbles does his sister have?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
26 marbles
Step-by-Step Explanation:
Subtract the difference: 64 - 12 = 52. Divide equally: 52 ÷ 2 = 26 marbles for the sister (Ethan has 26 + 12 = 38).
11
Short Answer
Medium
1 Mark
At the Central Botanical Gardens, an event organizer arranges 144 chairs into rows. The number of chairs in each row is 2 more than the number of rows. How many rows of chairs are there?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
10 rows or 12 chairs
Step-by-Step Explanation:
10 rows × 14 = 140; 12 rows × 12 = 144. If square: 12 rows of 12 chairs.
12
Multiple Choice
Easy
1 Mark
Which two consecutive whole numbers have a product of 156?
A
12 and 13
B
11 and 12
C
13 and 14
D
10 and 15
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
12 and 13
Step-by-Step Explanation:
12 × 13 = 156.
13
Multiple Choice
Easy
1 Mark
A mystery number is multiplied by 4, and then 12 is added. The result is 52. What is the mystery number?
A
10
B
12
C
8
D
11
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
10
Step-by-Step Explanation:
Work backwards: 52 - 12 = 40. 40 ÷ 4 = 10.
14
Multiple Choice
Easy
1 Mark
If 5 identical notebooks cost $15, how much will 12 of the same notebooks cost?
A
$36
B
$30
C
$40
D
$48
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
$36
Step-by-Step Explanation:
Unit price = $15 ÷ 5 = $3. For 12 notebooks = 12 × $3 = $36.
15
Multiple Choice
Easy
1 Mark
A farmer has 15 cows and 25 chickens. How many total legs are there on the farm?
A
110 legs
B
80 legs
C
100 legs
D
120 legs
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
110 legs
Step-by-Step Explanation:
(15 cows × 4 legs = 60) + (25 chickens × 2 legs = 50) = 110 legs.
16
Short Answer
Hard
2 Marks
A grandfather's age is 6 times his grandson's age. In 12 years, the grandfather will be only 3 times as old as his grandson. What are their current ages?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
Grandfather is 48, grandson is 8
Step-by-Step Explanation:
Let grandson = g, grandfather = 6g. In 12 years: 6g + 12 = 3(g + 12) -> 6g + 12 = 3g + 36 -> 3g = 24 -> g = 8. Grandfather = 6 × 8 = 48.
17
Short Answer
Hard
2 Marks
In a barn, there are horses and ducks. Counting heads gives 35 heads; counting legs gives 110 legs. How many horses and how many ducks are in the barn?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
20 horses and 15 ducks
Step-by-Step Explanation:
If all 35 were ducks: 35 × 2 = 70 legs. Extra legs = 110 - 70 = 40 legs. Each horse contributes 2 extra legs: 40 ÷ 2 = 20 horses. Ducks = 35 - 20 = 15 ducks.
18
Short Answer
Hard
2 Marks
If the perimeter of a rectangle is fixed at 24 cm, which dimensions (length and width) yield the greatest possible area? Explain why.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A square of 6 cm × 6 cm gives the greatest area (36 cm²). As the rectangle becomes more square-like, the product of length and width is maximized for a fixed sum of sides.
Step-by-Step Explanation:
Equal side distribution maximizes area for any given perimeter boundary.
19
Short Answer
Hard
2 Marks
Explain why working backwards from the final result is an effective strategy for solving multi-step arithmetic mystery problems.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
Working backwards inverts each consecutive operation (addition becomes subtraction, multiplication becomes division), systematically unwrapping the mystery quantity without guessing.
Step-by-Step Explanation:
Inverse operations systematically undo transformations step-by-step to recover the starting value.
20
Short Answer
Medium
1 Mark
To divide $120 between two friends in a 2:1 ratio, a student calculates $120 ÷ 2 = $60 each. What is the correct division?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
$80 and $40
Step-by-Step Explanation:
A 2:1 ratio has 2 + 1 = 3 equal shares. $120 ÷ 3 = $40 per share. The shares are $80 (2 shares) and $40 (1 share).
21
Short Answer
Medium
1 Mark
A student solves: 'If 3 cats catch 3 mice in 3 minutes, how many cats catch 100 mice in 100 minutes?' The student answers '100 cats.' Explain the flaw and state the correct answer.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
3 cats
Step-by-Step Explanation:
Rate of 3 cats is 1 mouse per minute. In 100 minutes, those same 3 cats catch 100 mice. 3 cats are needed.
22
Fill in the Blanks
Medium
1 Mark
Work backwards to find X: ((X + 6) × 3) - 9 = 45. X = [ ? ].
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
12
Step-by-Step Explanation:
45 + 9 = 54; 54 ÷ 3 = 18; 18 - 6 = 12.
23
Fill in the Blanks
Medium
1 Mark
Solve the logic puzzle: [ ? ] + [ ? ] = 18 and [ ? ] × [ ? ] = 77. The two numbers are 7 and [ ? ].
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
11
Step-by-Step Explanation:
7 + 11 = 18 and 7 × 11 = 77.
24
Short Answer
Medium
1 Mark
A water tank empties at a rate of 18 liters per minute while a pipe fills it at 25 liters per minute. What is the net increase in water volume after 20 minutes?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
140 liters
Step-by-Step Explanation:
Net fill rate = 25 - 18 = 7 L/min. In 20 minutes: 7 × 20 = 140 liters.
25
Short Answer
Medium
1 Mark
Together, Ethan and his sister have 64 marbles. Ethan has 12 more marbles than his sister. How many marbles does his sister have?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
26 marbles
Step-by-Step Explanation:
Subtract the difference: 64 - 12 = 52. Divide equally: 52 ÷ 2 = 26 marbles for the sister (Ethan has 26 + 12 = 38).
26
Short Answer
Medium
1 Mark
At the Eco-Friendly Solar Park, an event organizer arranges 144 chairs into rows. The number of chairs in each row is 2 more than the number of rows. How many rows of chairs are there?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
10 rows or 12 chairs
Step-by-Step Explanation:
10 rows × 14 = 140; 12 rows × 12 = 144. If square: 12 rows of 12 chairs.
27
Multiple Choice
Easy
1 Mark
Which two consecutive whole numbers have a product of 156?
A
12 and 13
B
11 and 12
C
13 and 14
D
10 and 15
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
12 and 13
Step-by-Step Explanation:
12 × 13 = 156.
28
Multiple Choice
Easy
1 Mark
A mystery number is multiplied by 4, and then 12 is added. The result is 52. What is the mystery number?
A
10
B
12
C
8
D
11
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
10
Step-by-Step Explanation:
Work backwards: 52 - 12 = 40. 40 ÷ 4 = 10.
29
Multiple Choice
Easy
1 Mark
If 5 identical notebooks cost $15, how much will 12 of the same notebooks cost?
A
$36
B
$30
C
$40
D
$48
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
$36
Step-by-Step Explanation:
Unit price = $15 ÷ 5 = $3. For 12 notebooks = 12 × $3 = $36.
30
Multiple Choice
Easy
1 Mark
A farmer has 15 cows and 25 chickens. How many total legs are there on the farm?
A
110 legs
B
80 legs
C
100 legs
D
120 legs
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
110 legs
Step-by-Step Explanation:
(15 cows × 4 legs = 60) + (25 chickens × 2 legs = 50) = 110 legs.
31
Short Answer
Hard
2 Marks
A grandfather's age is 6 times his grandson's age. In 12 years, the grandfather will be only 3 times as old as his grandson. What are their current ages?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
Grandfather is 48, grandson is 8
Step-by-Step Explanation:
Let grandson = g, grandfather = 6g. In 12 years: 6g + 12 = 3(g + 12) -> 6g + 12 = 3g + 36 -> 3g = 24 -> g = 8. Grandfather = 6 × 8 = 48.
32
Short Answer
Hard
2 Marks
In a barn, there are horses and ducks. Counting heads gives 35 heads; counting legs gives 110 legs. How many horses and how many ducks are in the barn?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
20 horses and 15 ducks
Step-by-Step Explanation:
If all 35 were ducks: 35 × 2 = 70 legs. Extra legs = 110 - 70 = 40 legs. Each horse contributes 2 extra legs: 40 ÷ 2 = 20 horses. Ducks = 35 - 20 = 15 ducks.
33
Short Answer
Hard
2 Marks
If the perimeter of a rectangle is fixed at 24 cm, which dimensions (length and width) yield the greatest possible area? Explain why.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A square of 6 cm × 6 cm gives the greatest area (36 cm²). As the rectangle becomes more square-like, the product of length and width is maximized for a fixed sum of sides.
Step-by-Step Explanation:
Equal side distribution maximizes area for any given perimeter boundary.
34
Short Answer
Hard
2 Marks
Explain why working backwards from the final result is an effective strategy for solving multi-step arithmetic mystery problems.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
Working backwards inverts each consecutive operation (addition becomes subtraction, multiplication becomes division), systematically unwrapping the mystery quantity without guessing.
Step-by-Step Explanation:
Inverse operations systematically undo transformations step-by-step to recover the starting value.
35
Short Answer
Medium
1 Mark
To divide $120 between two friends in a 2:1 ratio, a student calculates $120 ÷ 2 = $60 each. What is the correct division?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
$80 and $40
Step-by-Step Explanation:
A 2:1 ratio has 2 + 1 = 3 equal shares. $120 ÷ 3 = $40 per share. The shares are $80 (2 shares) and $40 (1 share).
36
Short Answer
Medium
1 Mark
A student solves: 'If 3 cats catch 3 mice in 3 minutes, how many cats catch 100 mice in 100 minutes?' The student answers '100 cats.' Explain the flaw and state the correct answer.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
3 cats
Step-by-Step Explanation:
Rate of 3 cats is 1 mouse per minute. In 100 minutes, those same 3 cats catch 100 mice. 3 cats are needed.
37
Fill in the Blanks
Medium
1 Mark
Work backwards to find X: ((X + 6) × 3) - 9 = 45. X = [ ? ].
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
12
Step-by-Step Explanation:
45 + 9 = 54; 54 ÷ 3 = 18; 18 - 6 = 12.
38
Fill in the Blanks
Medium
1 Mark
Solve the logic puzzle: [ ? ] + [ ? ] = 18 and [ ? ] × [ ? ] = 77. The two numbers are 7 and [ ? ].
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
11
Step-by-Step Explanation:
7 + 11 = 18 and 7 × 11 = 77.
39
Short Answer
Medium
1 Mark
A water tank empties at a rate of 18 liters per minute while a pipe fills it at 25 liters per minute. What is the net increase in water volume after 20 minutes?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
140 liters
Step-by-Step Explanation:
Net fill rate = 25 - 18 = 7 L/min. In 20 minutes: 7 × 20 = 140 liters.
40
Short Answer
Medium
1 Mark
Together, Ethan and his sister have 64 marbles. Ethan has 12 more marbles than his sister. How many marbles does his sister have?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
26 marbles
Step-by-Step Explanation:
Subtract the difference: 64 - 12 = 52. Divide equally: 52 ÷ 2 = 26 marbles for the sister (Ethan has 26 + 12 = 38).
41
Short Answer
Medium
1 Mark
At the Harbor Marine Terminal, an event organizer arranges 144 chairs into rows. The number of chairs in each row is 2 more than the number of rows. How many rows of chairs are there?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
10 rows or 12 chairs
Step-by-Step Explanation:
10 rows × 14 = 140; 12 rows × 12 = 144. If square: 12 rows of 12 chairs.
42
Multiple Choice
Easy
1 Mark
Which two consecutive whole numbers have a product of 156?
A
12 and 13
B
11 and 12
C
13 and 14
D
10 and 15
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
12 and 13
Step-by-Step Explanation:
12 × 13 = 156.
43
Multiple Choice
Easy
1 Mark
A mystery number is multiplied by 4, and then 12 is added. The result is 52. What is the mystery number?
A
10
B
12
C
8
D
11
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
10
Step-by-Step Explanation:
Work backwards: 52 - 12 = 40. 40 ÷ 4 = 10.
44
Multiple Choice
Easy
1 Mark
If 5 identical notebooks cost $15, how much will 12 of the same notebooks cost?
A
$36
B
$30
C
$40
D
$48
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
$36
Step-by-Step Explanation:
Unit price = $15 ÷ 5 = $3. For 12 notebooks = 12 × $3 = $36.
45
Multiple Choice
Easy
1 Mark
A farmer has 15 cows and 25 chickens. How many total legs are there on the farm?
A
110 legs
B
80 legs
C
100 legs
D
120 legs
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
110 legs
Step-by-Step Explanation:
(15 cows × 4 legs = 60) + (25 chickens × 2 legs = 50) = 110 legs.
46
Short Answer
Hard
2 Marks
A grandfather's age is 6 times his grandson's age. In 12 years, the grandfather will be only 3 times as old as his grandson. What are their current ages?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
Grandfather is 48, grandson is 8
Step-by-Step Explanation:
Let grandson = g, grandfather = 6g. In 12 years: 6g + 12 = 3(g + 12) -> 6g + 12 = 3g + 36 -> 3g = 24 -> g = 8. Grandfather = 6 × 8 = 48.
47
Short Answer
Hard
2 Marks
In a barn, there are horses and ducks. Counting heads gives 35 heads; counting legs gives 110 legs. How many horses and how many ducks are in the barn?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
20 horses and 15 ducks
Step-by-Step Explanation:
If all 35 were ducks: 35 × 2 = 70 legs. Extra legs = 110 - 70 = 40 legs. Each horse contributes 2 extra legs: 40 ÷ 2 = 20 horses. Ducks = 35 - 20 = 15 ducks.
48
Short Answer
Hard
2 Marks
If the perimeter of a rectangle is fixed at 24 cm, which dimensions (length and width) yield the greatest possible area? Explain why.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A square of 6 cm × 6 cm gives the greatest area (36 cm²). As the rectangle becomes more square-like, the product of length and width is maximized for a fixed sum of sides.
Step-by-Step Explanation:
Equal side distribution maximizes area for any given perimeter boundary.
49
Short Answer
Hard
2 Marks
Explain why working backwards from the final result is an effective strategy for solving multi-step arithmetic mystery problems.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
Working backwards inverts each consecutive operation (addition becomes subtraction, multiplication becomes division), systematically unwrapping the mystery quantity without guessing.
Step-by-Step Explanation:
Inverse operations systematically undo transformations step-by-step to recover the starting value.
50
Short Answer
Medium
1 Mark
To divide $120 between two friends in a 2:1 ratio, a student calculates $120 ÷ 2 = $60 each. What is the correct division?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
$80 and $40
Step-by-Step Explanation:
A 2:1 ratio has 2 + 1 = 3 equal shares. $120 ÷ 3 = $40 per share. The shares are $80 (2 shares) and $40 (1 share).