Finding the Unknown · Lesson 8 of 13
Number Tricks and Equation Chains
“Follow and reverse operations to explain tricks and create equations.”
• Translate a number trick into an equation. • Reverse a chain by undoing operations in reverse order. • Express two related quantities using one unknown. • Create equivalent equations with a chosen solution.
Write the hidden number as a letter
A number trick starts with a number you cannot see and performs known operations on it. Give the starting number a letter and carry it through each operation. Brackets matter because they record which amount is multiplied. If you subtract 3 and then multiply by 4, the result is 4(x − 3), not 4x − 3.
Problem
Riyaz asks someone to subtract 3 from a chosen number, multiply the result by 4, then add 8. The final answer is 24. Find the starting number.
- 1.Let x be the starting number. The stages are x, x − 3, 4(x − 3), and 4(x − 3) + 8.
- 2.Write 4(x − 3) + 8 = 24. Subtract 8: 4(x − 3) = 16.
- 3.Divide by 4: x − 3 = 4. Add 3: x = 7.
- 4.Check forwards: 7 − 3 = 4; 4 × 4 = 16; 16 + 8 = 24.
The same final expression expands to 4x − 12 + 8 = 4x − 4. So Riyaz can recover the starting number by adding 4 to the reported result and dividing by 4. For a result of 24 this gives (24 + 4)/4 = 7. Simplifying the expression explains the trick rather than leaving it mysterious.
Reverse the order as well as the operations
To walk backwards through a process, undo its last operation first. An addition at the end becomes a subtraction at the beginning of the reverse route. Then undo the earlier multiplication, and finally the first subtraction. Undoing the operations in the original order usually acts on the wrong intermediate quantity.
| Forward stage | Starting with 7 | Reverse stage from 24 |
|---|---|---|
| Subtract 3 | 4 | Subtract 8: 16 |
| Multiply by 4 | 16 | Divide by 4: 4 |
| Add 8 | 24 | Add 3: 7 |
Problem
A number is multiplied by 3 and then decreased by 5, giving 16. Find it.
- 1.The last operation was subtracting 5, so first add 5 to the output: 16 + 5 = 21.
- 2.The earlier operation was multiplying by 3, so divide: 21/3 = 7.
- 3.The equation is 3a − 5 = 16. Check: 3 × 7 − 5 = 16.
Two amounts can share one unknown
When two unknown amounts have a stated relationship, you can express both using one letter. If Ramesh has 30 more marbles than Suresh, let Suresh’s count be y and Ramesh’s count be y + 30. Writing two unrelated letters first is possible, but the relationship lets you replace one so that you have a single equation to solve.
If Ramesh’s count is first called x and Suresh’s y, the clues give x + y = 60 and x = y + 30. Replacing x in the total by its equal expression y + 30 gives (y + 30) + y = 60. This substitution explains how two linked unknown quantities become one equation in y; no separate method for solving two independent unknowns is needed.
Problem
Ramesh and Suresh have 60 marbles together; Ramesh has 30 more.
- 1.Let Suresh have y marbles. Ramesh then has y + 30, so y + (y + 30) = 60.
- 2.Combine the copies of y: 2y + 30 = 60. Subtract 30: 2y = 30.
- 3.Divide by 2: y = 15. Suresh has 15 and Ramesh has 45.
- 4.Check both conditions: 15 + 45 = 60 and 45 − 15 = 30.
Build and read equation chains
You can start with a chosen solution and apply the same operation to both sides to make a new equation. The resulting equation still has that solution if your operations are reversible. This helps you understand solving from another direction: creating an equation builds a chain, while solving it can unwind that chain.
Problem
Explain the chain y + 1 = 6, then −y − 1 = −6, then −1 = −6 + y, then 5 = y.
- 1.The first equation has solution 5 because 5 + 1 = 6.
- 2.Multiply both sides by −1 to obtain −y − 1 = −6.
- 3.Add y to both sides: −1 = −6 + y. Add 6 to both sides: 5 = y.
- 4.Every change preserved equality. Read backwards by subtracting 6, subtracting y, and multiplying by −1 in that order.
Problem
Explain 3y = 15 → 3y + 6 = 21 → y + 2 = 7 → y = 5.
- 1.Add 6 to both sides of 3y = 15 to get 3y + 6 = 21.
- 2.Divide the entire equation by 3: (3y + 6)/3 = 21/3, so y + 2 = 7.
- 3.Subtract 2: y = 5. Reversing uses add 2, multiply by 3, then subtract 6.
- 4.Test y = 5 in the starting equation: 3 × 5 = 15.
Problem
Create a situation for 100x + 75 = 250 and solve it.
- 1.One possible story is a product costing ₹100 per kilogram, with a fixed ₹75 delivery fee. Buying x kg costs ₹250 in total.
- 2.Subtract 75: 100x = 175. Divide by 100: x = 1.75 kg.
- 3.Check: 100 × 1.75 + 75 = 175 + 75 = ₹250. Here x is a measured mass, so a decimal answer makes sense.
When reversing a process, reverse the operation order too. When multiplying a bracket, include every term. Two letters in a story need not mean two independent unknowns if one amount is given in terms of the other.
Quiz
Which represents subtract 3, then multiply the result by 4?
Which operation is first when reversing “multiply by 3, then add 7”?
If Suresh has y marbles and Ramesh has 30 more, what is their total?
What gives y + 2 = 7 from 3y + 6 = 21?
Which equation has solution x = −2?
Practice Problems
- Write five different equations with solution x = −2. Check each by substitution.
- A trick adds 5, doubles the result, and subtracts 6. The output is 20. Find the starting number and write its forward stages.
- Start from u = 6 and create four equivalent equations using reversible operations.
- Two people have 84 cards together; one has 18 more. Use one unknown to find both counts.
- Read the chain 3y = 15 → 3y + 6 = 21 → y + 2 = 7 backwards, naming every inverse operation.
- Create a different sensible story for 100x + 75 = 250. Define x and its unit.
- Create a number trick that allows you to recover the starting number from its output, and explain the reverse route.
Key Takeaways
• Carry the unknown through every operation in a trick. • Brackets preserve the intended order. • Reverse both the operations and their order. • A relationship lets two quantities be expressed using one letter. • Reversible equal operations create equation chains with the same solution.