Mathematical Equations — Strategy & Tricks

📐 What the item actually asks

Every item hands you a tiny system: four equations, four unknowns called A, B, C and D, and six candidate assignments underneath. Exactly one satisfies all four. Nothing says you have to solve the system to find it.

The numbers are friendly on purpose: whole numbers, mostly 1 to 20, joined by +, -, × and ÷. No calculator, and nothing here is harder than adding two-digit numbers.

C = 2
B + C = 8
A = B - -1
D + A = 15

Two consequences. The candidates are printed, so checking one often beats solving for it. And a wrong answer costs nothing, so never leave a blank. This is not an algebra test — it is a test of reading a minus sign under pressure.

👀 Read the shape before the numbers

Spend five seconds on the shape of the four lines, not their values. Each variant has a best first move, and naming it turns a 90-second slog into a 40-second routine.

Early items may use only two or three unknowns. Same game, fewer pieces — pin, propagate, done. Bank the seconds; the four-letter items will want them.

🧠 Two routes: pin-and-propagate, or test-and-kill

Two ways to finish an item, and good solvers switch mid-item. Solve it: pin one value and propagate. Test it: push each printed candidate through an equation and see which survive. The exam scores the letter, not the route.

Route one — the anchor method. For a direct value or a short chain.

1. Find the anchor: the line with fewest unknowns. A direct value (B = 5) is best, a one-link line (C = 2 × B) next.
2. Pin it and say it: B is five.
3. Substitute: find the line where only one letter is now unknown; solve that one-liner.
4. Repeat until all four are pinned. Each step is one addition, subtraction or small multiplication.
5. Verify with the line you have not used — usually the long one — then find your assignment.

D = 4
A - D = 3
C = A - -2
B + C = 16

No anchor? Eliminate a letter instead: combine two lines so it cancels. The pair-sum trick is exactly that — A + B = 13, B + C = 11, A + C = 10 add to 34, twice the total, so A + B + C = 17 and each letter is 17 minus the pair that excludes it: C = 4, A = 6, B = 7.

Route two — option elimination. Six options and four equations is not 24 checks. Run all six through the easiest equation, cross out the failures, then run only the survivors through the next. Most items settle after two.

1. Pick the easiest equation: a direct value first (C = 4 kills every option whose C is not 4 at a glance), then a one-link line, then a two-letter sum.
2. Run all six down it, reading only the letters involved. If all six agree on a letter, it is free — never spend an equation on it.
3. Take the next-easiest equation; run only the survivors through it.
4. One option left means done. Do not check the rest — that is the time saving.
5. Two still standing? The line you skipped separates them.

✂️ Reading signs and doing sums in your head

Double negatives. The writers love A = B - -1. Read it as B minus negative one — that is B plus one. D - -3 is D + 3; C - -5 = 12 means C = 7. Train your eye until - - reads as + with no conscious step in between.

Leading minus. - A + B = 3 is B - A = 3; B is bigger. Tails. 2 × C - 1 = B means double C then drop one; 13 - C = A means A and C add to 13. Division. B ÷ 4 = A says B is four times A — the letter beside the ÷ is the big one. Never flip it.

• Parity. A + B = 13 is odd, so A and B differ in parity — two odds or two evens die without adding.
• Odd by construction. 2 × C - 1 = B forces B odd. An even B is dead on sight.
• Multiples. 3 × B = C makes C one of 3, 6, 9, 12, 15, 18; B ÷ 4 = A makes B a multiple of 4.
• Group the signs. In A - B + C - D = 6, add the pluses (A + C) and the minuses (B + D); the difference must be 6.
• Count up. To take 7 from 15, think 7 and what makes 15? Counting up slips less.

⚡ The 75-second routine

Seventy-five seconds is generous until you spend twenty staring at the wrong line. The routine below doubles as your scanning order: shortest line first, long line last.

🪤 Where the points leak

The arithmetic is easy, so the errors are not arithmetic errors. They are reading errors and stopping errors — and the options reward every one.

• Misreading a minus. A = B - -1 read as A = B - 1 lands two out — and that value is printed in the options. Plug: rewrite - - as + before computing.
• The option that fits three equations. A decoy is the right answer with one letter changed, so it passes every line that skips that letter. Plug: if two survive, test the skipped line on both. An equation passes options; only the last survivor is the answer.
• The redundant line. A + C = 15, when you already have A + B = 12 and C - B = 3, cannot fail — so it separates nothing. Plug: move to the letter the options disagree on.
• Swapped letters. A=6, B=7 and A=7, B=6 both pass A + B = 13. Plug: after a sum, test a difference or a link.
• Division direction. B ÷ 3 = A makes B the multiple of 3, not A. Plug: say B is three A.

A + B = 13
B + C = 11
C + D = 10
A + D = 12

🧩 Worked examples, easy to hard

Six items in the exam's own style. Cover the options, try each, then read on.

B = 6
A - B = 3
C = A + B
D + B = 10

C = 4
- C + D = 5
A = D - -2
B + C = A

A + B = 13
B + C = 11
A + C = 10
D - C = 5

B ÷ 4 = A
C - B = A
D = 2 × C - 1
A + B = 10

A - B + C - D = 6
3 × B = C
D - 3 = 2 × B
A = 4 × B - -1

A + B = 12
C - B = 3
A + C = 15
D = 2 × A - C

🎯 Recognition drills

Two drills that train the scan, not the sums. Answer out loud before reading the solution.

🧭 Guided practice: talk yourself through one

Cover the walkthrough and try this one first. Then read the self-talk of a well-drilled candidate, and copy the rhythm.

D = 5
A - D = 6
C = A - -3
B + C = 20

1. 0–5 s. Line one is a direct value, D is five. Line three has a - -, so that is a plus. Direct-value chain — solve, do not test.
2. 5–15 s. D = 5, so (5) is dead. A minus five is six, A is eleven — (3) is dead.
3. 15–30 s. C is A plus three, fourteen. (2) and (4) both have C = 8, the double-negative misread.
4. 30–40 s. B plus fourteen is twenty, B is six. That is (1); (6) has B = 4. Nothing checked twice, and the only line said out loud was the one with the - - — which is exactly where the two dead options came from.

📈 Build up: easy, medium, hard

Work the three tiers in order, untimed. The goal is not a score — it is that the variant name arrives before you compute. Move up at 7 of 8, and only when you can say why each wrong option was wrong.

Easy set — direct values and short chains → 8 real easy questions · free, no login

Medium set — pair sums and × ÷ links → 8 real medium questions · free, no login

Hard set — long lines, hubs, redundant equations → 8 real hard questions · free, no login

Between tiers, re-read the traps: the decoys one tier up are built from those misreads.

⏱️ Against the clock

Now the clock. Speed comes from the scan being automatic, not from adding faster. Time each item and note where the seconds went: scan, anchor, propagate or pick. Seconds saved on an easy item are for the hard item at the end.

Medium set — 60 seconds each → 8 real medium questions, against the clock · free, no login

Hard set — 75 seconds each → 8 real hard questions, against the clock · free, no login

🏆 Exam day

The set opens with lighter shapes and thickens: move fast early, reach the four-letter items with time in hand, never let one item eat three.

• Scan all four lines for shape before touching a number.
• Pin the direct value or shortest line; strike every option that disagrees.
• Rewrite every - - as + the moment you see it.
• Say each equation you use as a sentence, once.
• Easiest equation on all six; next-easiest on survivors only.
• Stop when one option remains. Do not confirm for fun.
• Two survivors? Test the skipped line on those two.
• A line that passes everything is redundant — move on.
• Answer not in the options? Re-read the line with the sign.
• Ninety seconds is the ceiling: take the best survivor and move.
• Never leave an item blank; there is no negative marking.

Clock checks: at 75 seconds an item, be through item 7 by 9 minutes and item 14 by 18. Behind? Switch fully to elimination.

Exam simulation — 8 items at exam pace → 8 real hard questions, against the clock · free, no login

📋 Cheat sheet

• Four equations, four letters, six options, one fits all. ~75 s each, no negative marking.
• Scan shape first: direct value → one-step link → two-letter sum → long line last.
• Anchor: pin, propagate to the line with one unknown, repeat.
• No anchor? Combine two lines to cancel a letter.
• Elimination: easiest line on all six, next on survivors, stop at one.
• - - is +. A = B - -1 means A = B + 1.
• - A + B = 3 means B - A = 3; B is bigger.
• B ÷ 4 = A: B is the big one, a multiple of 4.
• 2 × C - 1 = B: B is odd; even-B options die.
• Odd sum, mixed parity. Even sum, same parity.
• Three pair sums: add, halve, subtract each pair.
• Long line: (pluses) - (minuses). Survivors only.
• Options differ in one letter? Test the line with that letter.
• A line that passes everything is redundant — move on.
• Never blank. Past 90 s, take the best survivor and go.