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Mathematical Operations

1. Outline & Learning Goals

In this chapter you will master:

• Symbol substitution — where `+` means `−`, `×` means `÷`, and so on.
• Interchanging signs and numbers — which swap makes an equation correct?
• Balancing equations — inserting the right signs to make a statement true.
• BODMAS-based evaluation after decoding the symbols.
• Coded operations — letters or words standing for operators.

By the end, you should be able to decode any symbol puzzle in under 45 seconds and spot the examiner's favourite traps before they catch you.

2. The Hook

Imagine the examiner hands you this:

> If `+` means `÷`, `÷` means `×`, `×` means `−` and `−` means `+`, what is the value of `8 + 4 ÷ 2 × 6 − 3`?

Most students freeze. But here is the secret: this is not a maths question at all. It is a *translation* question. The arithmetic itself is Class-5 level. The only skill being tested is whether you can calmly rewrite the sentence in a new language and then follow one sacred rule — BODMAS. Once you see it that way, this chapter becomes free marks.

3. The Common Struggle (You Are Not Alone)

Almost every aspirant makes the same three mistakes here:

1. Solving left to right instead of following BODMAS after substitution.
2. Substituting symbols halfway — changing the first two signs, then forgetting and using the original meaning for the third.
3. Panicking at the options when two options look "almost right."

The fix for all three is one habit: rewrite the entire expression in real symbols first, on paper, before touching any calculation. Never decode in your head.

🧠 Active Recall: Before reading on, close your eyes and answer — why should you never solve a substituted expression left to right?

4. Chunk 1 — Symbol Substitution (The Core Type)

The rule: Replace every symbol with its new meaning, *then* apply BODMAS.

Example: If `+` means `×`, `−` means `+`, `×` means `÷`, `÷` means `−`, find:
`16 ÷ 4 + 2 × 3 − 1`

Step 1 — Translate:
`16 − 4 × 2 ÷ 3 + 1` … wait, careful! Translate symbol by symbol:

• `÷` → `−`
• `+` → `×`
• `×` → `÷`
• `−` → `+`

So: `16 − 4 × 2 ÷ 3 + 1`

Step 2 — BODMAS: Division first: `2 ÷ 3 = 2/3`. Hmm — that gives a fraction, which is a red flag. In exam questions the numbers are usually chosen to divide cleanly, so if you get an ugly fraction, re-check your translation. That habit alone saves more marks than any shortcut.

🧠 Active Recall: What are the two steps of every substitution question, in order?

5. Chunk 2 — Interchanging Signs or Numbers

Here the question says: *"Which interchange of two signs (or two numbers) makes the equation correct?"*

The strategy — do NOT try all combinations. Use these filters:

• Division filter: If the equation contains `÷`, after interchange the division must still produce a whole number (usually). Test only interchanges that keep division clean.
• Magnitude filter: If the LHS is much smaller than the RHS, the interchange probably converts a `−` or `÷` into a `×` or `+` (something that *grows* the value).

Example: `5 + 3 × 8 − 12 ÷ 4 = 3`. Which interchange works?
Try `+` and `−`: `5 − 3 × 8 + 12 ÷ 4 = 5 − 24 + 3 = −16`. No.
Try `×` and `÷`: `5 + 3 ÷ 8 × 12 − 4` — messy fraction, skip.
Try `−` and `÷`: `5 + 3 × 8 ÷ 12 − 4 = 5 + 2 − 4 = 3`. ✓

Notice how the filters eliminated two of the three tries instantly.

🧠 Active Recall: Name the two filters that save you from testing every interchange.

6. Chunk 3 — Balancing Equations (Insert the Signs)

You are given something like: `7 ? 3 ? 4 = 25` and must pick the right pair of signs.

Golden method — work backwards from the answer:

• Answer 25 is *large* → at least one `×` is almost certain.
• `7 × 3 = 21`, and `21 + 4 = 25`. Done: `×` then `+`.

Backwards-thinking turns a guessing game into a 10-second check. Always ask: *"Which operation can produce a number this big (or this small)?"*

🧠 Active Recall: In balancing questions, why do we start from the result instead of the numbers?

7. Chunk 4 — Coded Operations (Letters & Words)

Sometimes operators hide behind letters: *"A means +, B means −, C means ×, D means ÷."* Or words: *"'goes' means +, 'into' means ×…"*

The method never changes:

1. Write a mini dictionary on your rough sheet: `A = +`, `B = −`…
2. Translate the full expression.
3. BODMAS.

The dictionary step feels slow but is actually the fastest path — it kills the "halfway substitution" error from Section 3.

8. Mnemonic Trick 🎯

Remember the workflow with "T-B-C" — *Translate, BODMAS, Check*:

> "The Brave Cat" — Translate first, BODMAS second, Check the division is clean.

And for BODMAS order itself: "Big Dragons Make Apple Sauce" — Brackets, Division, Multiplication, Addition, Subtraction.

🧠 Active Recall: Recite the T-B-C workflow and what each letter stands for.

9. Summary Table

📅 Review this chapter after 2 days. Redo one question of each type from memory — if the T-B-C workflow comes back to you without notes, you own this chapter.

⚠️ Caution: Examiner's Trap

Trap 1 — The "almost right" option. After substitution, the examiner places one option that you get if you solve *left to right* instead of BODMAS, and another you get if you forget to substitute one symbol. Defence: translate fully on paper, then strictly follow BODMAS.

Trap 2 — Confusing language to hide a simple operation. Phrases like "P is multiplied by the difference of Q and R" are designed to make you misplace brackets. It means `P × (Q − R)`, not `P × Q − R`. Defence: convert wordy sentences into brackets first — "difference of", "sum of", "product of" always create a bracket.

Trap 3 — Double-relationship questions. In interchange questions, sometimes *two* interchanges appear to work if you check carelessly (e.g., forgetting that division must yield a whole number, or checking only part of the equation). Defence: verify the complete equation after interchange, and apply the clean-division filter before committing.

Trap 3 — Double-relationship questions. In interchange questions, sometimes *two* interchanges appear to work if you check carelessly (e.g., forgetting that division must yield a whole number, or checking only part of the equation). Defence: verify the complete equation after interchange, and apply the clean-division filter before committing.