Interchange Problems — Balancing the Equation

🎯 Learning Objective

Find which two signs (or numbers) must be interchanged to make a wrong equation correct.

💡 Concept

• The printed equation is FALSE; exactly one option's swap makes LHS = RHS
• Method: take an option, swap the two signs (or numbers) everywhere in the equation, evaluate with BODMAS, compare with RHS
• Start with options that swap × or ÷ — they change the value most and get confirmed or rejected fastest
• Sign swaps apply to every occurrence of both signs, not just one spot
• Number-interchange questions work the same way: swap the two numbers, then BODMAS
• Balancing variant: choose the sign set that makes both sides equal — test options the same way

✏️ Easy Example

Q. 2 + 6 − 4 = 0 is wrong. Which two signs should be interchanged to correct it?

1. Try swapping + and −: 2 − 6 + 4
2. 2 − 6 = −4, then −4 + 4 = 0 = RHS

Answer: + and −

🇮🇳 Real-Life Example

A shop bill totals wrong because two keys on the calculator were pressed in swapped order — the cashier re-checks by redoing the doubtful pair, exactly your option-testing method.

📝 Exam-Level Example

Q. 5 + 6 ÷ 3 − 12 × 2 = 17. Which two signs must be interchanged?

1. As printed: 5 + 2 − 24 = −17 ≠ 17, so test swaps
2. Swap ÷ and ×: 5 + 6 × 3 − 12 ÷ 2
3. BODMAS: 5 + 18 − 6 = 17 = RHS

Answer: ÷ and ×

📝 Exam-Level Example

Q. 20 ÷ 8 + 5 = 12. Which two NUMBERS should be interchanged to make it correct?

1. As printed: 2.5 + 5 = 7.5 ≠ 12
2. Swap 8 and 5: 20 ÷ 5 + 8
3. BODMAS: 4 + 8 = 12 = RHS

Answer: 8 and 5

🪄 Memory Trick

Before testing, estimate: is the printed LHS too big or too small versus RHS? Too small → the swap must inject multiplication or remove a division. This points to the right option in one glance.

⚠️ Common Mistakes

• ❌ Evaluating the swapped equation left to right instead of by BODMAS
• ❌ Swapping the signs at only one place when they occur more than once
• ❌ Stopping at an option that comes 'close' to RHS — the correct swap gives an EXACT match

🏆 Exam Tips

• ✅ Compute the printed LHS once at the start — it tells you how far you are and in which direction
• ✅ Divisions that produce fractions midway usually signal a wrong option; exam answers stay clean

📌 Summary

• Swap per the option, rewrite fully, then BODMAS
• Test ×/÷ swaps first — biggest impact, fastest decision
• Swaps apply to every occurrence of both signs
• Exact equality only; 'nearly equal' means wrong option