Concept 7 of 7
Operator Substitution: When the Signs Lie
Today's big idea
Rewrite the sum with the real signs first, then work the new sum out in the usual order.
A printed sign can lie, so restore the real signs before doing any arithmetic.
How it works
👀 What's going on?
A question can change what a sign means before it asks you anything. A printed plus might really be a minus, so the sum in front of you is a disguise.
💡 Here's the trick
Look at the two rows above. The top row is the sum exactly as printed, and the bottom row is that same sum with the real signs put back. Work only on the bottom row. Rewriting costs a few seconds and saves the whole question.
🪄 How to do it
- 1.Copy the sum out with every printed sign replaced, as the rows above show.
- 2.Check the order on your new sum, because × and ÷ still go before + and −.
- 3.Work it out, and never glance back at the printed version.
✍️ Try it
If '+' means '−' and '−' means '+', what is 20 + 6 − 4? Then, with the same rule, what is 12 − 4 + 3?
The first rewrites to 20 − 6 + 4, which is 14 + 4, so the answer is 18. The second rewrites to 12 + 4 − 3, which is 16 − 3, giving 13.
⚠️ Watch out
You might be thinking you can work left to right once the signs are swapped. The rewritten sum still obeys the usual order, so 6 + 2 × 3 is 12, not 24.
🌟 Why this matters
Signs that mean something else are all around you, from a map key to a button that does two jobs. Reading the key before you act is the habit.
Exam tip: Write the rewritten sum down. Almost nobody does these reliably in their head.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“If ‘×’ means ‘+’ and ‘+’ means ‘×’, what is 5 + 2 × 4?”
- 1
Rewrite with the real signs
5 + 2 × 4 becomes 5 × 2 + 4. 🔀
- 2
Now check the order
The rewritten sum has a × in it, and × goes before +.
- 3
Compute
5 × 2 = 10; 10 + 4 = 14.
- 4
Answer ✓
14
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Step 1: rewrite the sum with the real signs. Step 2: work it out.
SWAP RULE · ROUND 1 OF 3
'+' really means '−'
Step 1 — which is the sum rewritten with the REAL signs?
Quick check
✅ VERIFICATION
If '−' means '×', what is 5 − 3?
⚡ Finish the activity above first to answer.
Keep going!