Which two numbers (not digits) should be interchanged to make the given equation correct? 19 + 125 ÷ 25 – 100 ÷ 5 + 4 × 3 = 52

Correct — B. Rule: swap the two numbers, then evaluate by BODMAS — divisions and multiplications first, then the additions and subtractions left to right.

Swapping 25 and 5 turns
19 + 125 ÷ 25 - 100 ÷ 5 + 4 × 3
into
19 + 125 ÷ 5 - 100 ÷ 25 + 4 × 3

125 ÷ 5 = 25
100 ÷ 25 = 4
4 × 3 = 12
19 + 25 - 4 + 12 = 52

The right-hand side is 52, so the swap works — option (b).

Why the others are wrong

• (a)
  125 and 100 — Swapping 125 and 100 gives 19 + 100 ÷ 25 - 125 ÷ 5 + 12 = 19 + 4 - 25 + 12 = 10, nowhere near 52.
• (c)
  19 and 4 — Swapping 19 and 4 gives 4 + 125 ÷ 25 - 100 ÷ 5 + 19 × 3 = 4 + 5 - 20 + 57 = 46, six short.
• (d)
  4 and 5 — Swapping 4 and 5 gives 19 + 125 ÷ 25 - 100 ÷ 4 + 5 × 3 = 19 + 5 - 25 + 15 = 14.

Left exactly as printed the line comes to 19 + 5 - 20 + 12 = 16, so the swap you are looking for has to add 36. Knowing the gap tells you the swap must act on a division, which is where the large jumps live.

The bracketed rider is doing real work: numbers, not digits. 125 moves as 125, and you may not lift the 1, 2 or 5 out of it.

Test an option by recomputing only the terms that changed. Re-evaluating the whole line four times is how this question eats two minutes.

Key facts

• As printed the line evaluates to 19 + 5 - 20 + 12 = 16, against a stated 52.
• After swapping 25 and 5 the two divisions give 25 and 4, and 19 + 25 - 4 + 12 = 52.
• The rider 'two numbers (not digits)' forbids moving a digit out of 125 or 100.

Common traps

• Swapping digits instead of whole numbers, which the rider bans
• Working strictly left to right and ignoring BODMAS
• Re-evaluating the entire line once per option instead of the changed terms

No directly related past PYQ was found.