- Percent change formula
- Change ÷ original × 100. From 80 to 100 is 20 ÷ 80 = 25% increase.
- 'More than' in a word problem
- Means add (+). '8 more than 5 times a number' = 5x + 8.
- 'Less than' in a word problem
- Means subtract — and it REVERSES order. '3 less than x' = x − 3, not 3 − x.
- 'Product of' means
- Multiply (×). 'The product of a number and 5' = 5x.
- 'Quotient of' means
- Divide (÷). 'The quotient of a number and 4' = x ÷ 4.
- 'Of' in a percent problem means
- Multiply. '25% of 80' = 0.25 × 80 = 20.
- 'Is' / 'equals' / 'results in' means
- The equals sign (=) — it separates the two sides of the equation.
- Order of operations (PEMDAS)
- Parentheses, Exponents, Multiply/Divide (left→right), Add/Subtract (left→right).
- 2 + 3 × 4 = ?
- 14 — multiply first (3 × 4 = 12), then add 2. Not 20.
- Distance, rate, time formula
- Distance = rate × time. So rate = distance ÷ time and time = distance ÷ rate.
- How to solve a ratio/rate problem
- Set up a proportion (a/b = c/d) and cross-multiply to find the unknown.
- What is a proportion?
- An equation saying two ratios are equal (a/b = c/d). Cross-multiply: a × d = b × c.
- Increase a number by 25% in one step
- Multiply by 1.25. (To decrease by 25%, multiply by 0.75.)
- Decrease a number by 20% in one step
- Multiply by 0.80 (because 100% − 20% = 80%).
- Average (mean) of a set
- Add all values, then divide by how many there are.
- Find a missing value from an average
- Total = average × count. Subtract the known values to get the missing one.
- Adding fractions
- Find a common denominator, convert, then add the numerators. 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2.
- Multiplying fractions
- Multiply straight across: (a/b) × (c/d) = ac/bd, then simplify.
- Dividing fractions
- Multiply by the reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).
- Convert a percent to a decimal
- Divide by 100 (move the decimal two places left). 25% = 0.25.
- Convert a fraction to a percent
- Divide numerator by denominator, then × 100. 3/4 = 0.75 = 75%.
- Solve a one-variable linear equation
- Isolate the variable: undo addition/subtraction first, then multiplication/division.
- Solve 5x + 8 = 23
- Subtract 8: 5x = 15. Divide by 5: x = 3.
- Translate: 'twice a number increased by 7'
- 2x + 7.
- Translate: 'the sum of a number and 9 is 15'
- x + 9 = 15, so x = 6.
- Percent vs percentage points
- A change from 10% to 15% is +5 percentage points, but a 50% increase relative to 10%.
- Working backward from an answer
- Plug the answer choices into the problem and keep the one that fits — fast on a no-calculator test.
- Estimate to check arithmetic
- Round numbers, do the easy math, and confirm your exact answer is in the right ballpark.
- Commission / tip / tax problems
- Take the percent OF the base amount, then add (or it IS the amount). 8% tax on $50 = 0.08 × 50 = $4.
- Speed: 120 miles in 2 hours
- Rate = distance ÷ time = 120 ÷ 2 = 60 miles per hour.
- Unit rate
- A rate with a denominator of 1 (e.g. miles PER hour, cost PER item). Divide to find it.
- Consecutive integers
- x, x+1, x+2, … For consecutive EVEN or ODD integers, step by 2: x, x+2, x+4.
- No calculator — keep numbers simple
- Reduce fractions early, factor out common amounts, and use round numbers when estimating.
- Area of a rectangle
- Length × width.
- Perimeter of a rectangle
- 2 × (length + width).
- What is a verbal analogy?
- A question 'A is to B as C is to ?' that asks you to keep the SAME relationship in the second pair.
- The bridge-sentence method
- State how the first pair relates in a short sentence, then apply that exact sentence to the second pair.
- Analogy: CAR is to GARAGE as PLANE is to ?
- HANGAR — the place where each is kept/parked.
- Analogy: HAPPY is to SAD as HOT is to ?
- COLD — an antonym (opposite) relationship.
- Analogy: PETAL is to FLOWER as ? is to BOOK
- PAGE — part to whole.
- Relationship type: Synonym
- The two words mean the same thing. BIG : LARGE.
- Relationship type: Antonym
- The two words are opposites. UP : DOWN.
- Relationship type: Part to whole
- First word is a part of the second. WHEEL : CAR.
- Relationship type: Whole to part
- First word contains the second. CAR : WHEEL — note the reversed direction.
- Relationship type: Category (item to group)
- First is a member of the second. ROBIN : BIRD.
- Relationship type: Function / use
- The second is what the first does or is used for. KNIFE : CUT.
- Relationship type: Worker to tool
- The first uses the second. PAINTER : BRUSH.
- Relationship type: Worker to product
- The first makes the second. BAKER : BREAD.
- Relationship type: Degree / intensity
- Same idea, different strength. WARM : HOT, or COOL : COLD.
- Relationship type: Cause and effect
- The first leads to the second. FIRE : SMOKE.
- Relationship type: Object to material
- What the object is made of. SHIRT : COTTON.
- Why direction matters in analogies
- PART:WHOLE must stay PART:WHOLE in your answer. Keep the order consistent.
- Build the bridge BEFORE reading choices
- Decide the relationship first so a tempting but wrong-relationship choice doesn't fool you.
- Trap: same topic, wrong relationship
- A choice can be about the right subject but break the relationship — reject it.
- Analogy: DOCTOR is to HOSPITAL as TEACHER is to ?
- SCHOOL — the workplace of each.
- Analogy: BIRD is to FLY as FISH is to ?
- SWIM — how each moves.
- Analogy: COLD is to ICE as HEAT is to ?
- STEAM (or FIRE) — cause and effect / result.
- Analogy: PUPPY is to DOG as KITTEN is to ?
- CAT — young animal to adult animal.
- Analogy: HAND is to GLOVE as FOOT is to ?
- SOCK (or SHOE) — what covers it.
- Analogy: AUTHOR is to BOOK as COMPOSER is to ?
- SYMPHONY (or MUSIC) — creator to creation.
- Analogy: LIBRARY is to BOOKS as ARSENAL is to ?
- WEAPONS — place to what it stores.
- Analogy: SECOND is to MINUTE as MINUTE is to ?
- HOUR — smaller unit to next-larger unit of time.
- Analogy: THERMOMETER is to TEMPERATURE as CLOCK is to ?
- TIME — instrument to what it measures.
- Analogy: ISLAND is to OCEAN as OASIS is to ?
- DESERT — surrounded by.
- Strategy for unfamiliar words
- Use word roots, prefixes, and how the word sounds/relates; then match the relationship, not the exact meaning.
- Analogy: CHAPTER is to NOVEL as SCENE is to ?
- PLAY (or MOVIE) — a part of the larger work.
- Analogy: HUNGRY is to EAT as TIRED is to ?
- SLEEP — condition to the action that relieves it.
- Analogy: DOG is to BARK as CAT is to ?
- MEOW — animal to the sound it makes.
- Analogy: PEN is to WRITE as SCISSORS is to ?
- CUT — tool to its function.
- Next: 9, 13, 17, 21, ?
- 25 — add 4 each time (arithmetic sequence, common difference 4).
- What is an arithmetic sequence?
- A series with a constant DIFFERENCE between terms (add/subtract the same amount).
- What is the common difference?
- The fixed amount added between consecutive terms of an arithmetic sequence.
- What is a geometric sequence?
- A series with a constant RATIO between terms (multiply/divide by the same number).
- What is the common ratio?
- The fixed number each term is multiplied by to get the next term.
- Next: 3, 6, 12, 24, ?
- 48 — each term doubles (geometric, common ratio 2).
- Next: 60, 52, 44, 36, ?
- 28 — subtract 8 each time (common difference −8).
- Next: 14, 22, 30, 38, ?
- 46 — add 8 each time.
- First step on any number series
- Look at the GAPS between terms. Constant gap = add; constant ratio = multiply.
- Two-step (growing gap) pattern
- The difference itself grows: 2, 4, 8, 14 (gaps +2, +4, +6) → next is 22.
- Next: 1, 4, 9, 16, ?
- 25 — the perfect squares (1², 2², 3², 4², 5²).
- Next: 1, 8, 27, 64, ?
- 125 — the perfect cubes (1³, 2³, 3³, 4³, 5³).
- What is an alternating series?
- Two interleaved patterns — odd positions follow one rule, even positions another.
- Next: 2, 20, 4, 18, 6, ?
- 16 — two series interleaved: 2,4,6 (+2) and 20,18,16 (−2).
- 'Multiply then add' pattern
- Each term = previous × a, then + b. E.g. ×2 then +1: 1, 3, 7, 15 → next 31.
- Next: 5, 11, 23, 47, ?
- 95 — each term is ×2 + 1 (5×2+1=11, 11×2+1=23 …).
- Fibonacci-style pattern
- Each term is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13 …
- Next: 1, 1, 2, 3, 5, 8, ?
- 13 — add the previous two terms (Fibonacci).
- What is a letter series?
- A pattern of letters; convert to alphabet positions (A=1 … Z=26) and treat as numbers.
- Next letter: A, C, E, G, ?
- I — positions 1, 3, 5, 7 (gap +2), so next is position 9 = I.
- Next letter: B, D, F, H, ?
- J — every other letter (skip one each time).
- Next letter: Z, X, V, T, ?
- R — going backward, skipping one letter each step.
- Tip: write the alphabet with positions
- At test start, jot A=1 … Z=26 so every letter pattern becomes a quick number problem.
- Decreasing geometric: 80, 40, 20, ?
- 10 — divide by 2 each time (common ratio 1/2).
- Next: 100, 90, 81, 73, ?
- 66 — the gap shrinks by 1 each step (−10, −9, −8, −7).
- Spotting the rule fast
- Check addition first, then multiplication, then growing/shrinking gaps, then alternating.
- Next: 7, 14, 28, 56, ?
- 112 — each term doubles.
- Next: 2, 5, 11, 23, ?
- 47 — each term is ×2 + 1.
- Next: 100, 95, 85, 70, ?
- 50 — subtract a growing amount (−5, −10, −15, −20).
- Next: 3, 9, 27, 81, ?
- 243 — multiply by 3 each time.
- Next: 64, 32, 16, 8, ?
- 4 — halve each time.
- Next: 11, 14, 18, 23, ?
- 29 — gaps grow by 1 (+3, +4, +5, +6).
- If no simple rule fits
- Try splitting into two interleaved series, or look for squares/cubes or a ×then+ rule.
- Next: 1, 2, 4, 7, 11, ?
- 16 — add 1, 2, 3, 4, 5 … (the gap grows by 1).
- What is a figure analogy?
- A non-verbal item: the first figure changes into the second by ONE transformation; apply that same change to the third.
- First step on a figure analogy
- Find the SINGLE transformation between the first two figures, then apply it to the third.
- Transformation: Rotation
- Turning a figure around a point (90° or 180°) without flipping it. An up-arrow becomes a right-arrow.
- Transformation: Reflection
- Flipping a figure across a line into its mirror image — left and right reverse.
- Rotation vs reflection — quick test
- If spinning the page would do it, it's a rotation; if you'd have to flip the page over, it's a reflection.
- Transformation: Add an element
- A dot, line, or smaller shape appears in the second figure.
- Transformation: Remove an element
- A feature present in the first figure is gone in the second.
- Transformation: Count change
- The number of items increases or decreases by a fixed amount.
- Transformation: Size change
- The shape gets larger or smaller while keeping its form.
- Transformation: Shading change
- An unshaded shape becomes shaded (or vice versa), or the pattern fill changes.
- 90° clockwise rotation of ↑
- → (up becomes right).
- 90° clockwise rotation of →
- ↓ (right becomes down).
- 180° rotation of ↑
- ↓ (up becomes down).
- Reflection of a left-facing shape
- It becomes right-facing — a mirror image.
- If several things seem to change
- Look for the simplest single rule that explains all of them — the test rewards the clean pattern.
- Up-arrow → right-arrow, so left-arrow → ?
- Up — apply the same 90° clockwise turn to the left-arrow.
- Why isolate ONE change
- Figure analogies almost always use a single transformation; finding it makes the answer obvious.
- Reflection reverses…
- Left and right (handedness). Rotation does NOT reverse handedness.
- Clockwise vs counterclockwise
- Confirm the DIRECTION of a rotation from the first pair before applying it to the third.
- Combined transformations
- Occasionally two changes happen (e.g. rotate AND shade) — apply both, in the same way, to the third figure.
- Abstract reasoning — what it is
- Spotting patterns and relationships in shapes/symbols rather than words or numbers.
- Pattern recognition — what it is
- Finding the underlying rule in a series of numbers, letters, or figures to predict what comes next.
- Checking your figure answer
- Re-apply your rule to the first pair to confirm it produces the second figure exactly.
- Symmetry as a clue
- Note lines of symmetry — a reflection often flips a shape across one of them.
- Position change
- An element may move (e.g. a dot shifts from one corner to another) by a fixed step.
- Orientation matters
- A rotated shape keeps the same parts in the same order; a reflected one mirrors them.
- Eliminate impossible choices first
- Cross off any option that doesn't match your transformation, then choose among the rest.
- Quarter-turn = 90°
- A single quarter-turn. Two quarter-turns (180°) point a shape the opposite way.
- Figure analogy vs number series
- Both are pattern problems — figure analogies use shapes, number series use numbers; the strategy (find the rule) is the same.
- Watch for a constant element
- Part of the figure may stay the same while another part transforms — change only what changes.
- Subtracting fractions
- Common denominator, then subtract numerators. 3/4 − 1/4 = 2/4 = 1/2.
- Reduce a fraction
- Divide top and bottom by their greatest common factor. 6/8 = 3/4.
- Convert decimal to fraction
- Write over a power of 10 and reduce. 0.6 = 6/10 = 3/5.
- Find a percent of a number
- Turn the percent into a decimal and multiply. 30% of 50 = 0.30 × 50 = 15.
- What number is 20% of 90?
- 0.20 × 90 = 18.
- 15 is what percent of 60?
- 15 ÷ 60 = 0.25 = 25%.
- Translate: 'the difference between a number and 4'
- x − 4.
- Translate: 'a number divided by 3 is 7'
- x ÷ 3 = 7, so x = 21.
- Translate: 'half of a number'
- x ÷ 2 (or 0.5x).
- Translate: 'a number squared'
- x².
- Solve 3x − 5 = 16
- Add 5: 3x = 21. Divide by 3: x = 7.
- Solve 2(x + 3) = 14
- Distribute: 2x + 6 = 14, so 2x = 8 and x = 4.
- Combine like terms: 4x + 3x
- 7x.
- Distribute: 5(2x − 1)
- 10x − 5.
- Mixed number to improper fraction
- Multiply whole × denominator, add numerator, keep denominator. 2 1/3 = 7/3.
- Average speed for a round trip
- Total distance ÷ total time — NOT the average of the two speeds.
- Work-rate idea
- If a job takes 4 hours, the rate is 1/4 of the job per hour.
- Profit
- Selling price − cost. Markup is the amount added to the cost.
- Discount of 30% off $40
- 0.30 × 40 = $12 off, so the price is $28.
- Total with 8% tax on $25
- Tax = 0.08 × 25 = $2; total = $27.
- Ratio 3:2 of 30 items
- Parts total 5; each part = 30 ÷ 5 = 6, so 18 and 12.
- Scale a recipe (proportion)
- Set up known/known = new/unknown and cross-multiply.
- Negative times negative
- A positive. (−3) × (−4) = 12.
- Negative plus positive
- Subtract and keep the sign of the larger absolute value. −7 + 3 = −4.
- Absolute value
- Distance from zero, always non-negative. |−5| = 5.
- Squaring a number
- Multiply it by itself. 6² = 36.
- Square root
- The number that multiplies by itself to give it. √49 = 7.
- Round 0.376 to the nearest hundredth
- 0.38 (the thousandths digit 6 rounds up).
- Convert hours to minutes
- Multiply by 60. 2.5 hours = 150 minutes.
- Median of a data set
- The middle value when the numbers are in order.
- Analogy: GLOVE is to HAND as SHOE is to ?
- FOOT — what each is worn on.
- Analogy: SHARP is to DULL as FAST is to ?
- SLOW — antonyms.
- Analogy: KITTEN is to CAT as CALF is to ?
- COW — young to adult animal.
- Analogy: BEE is to HIVE as BIRD is to ?
- NEST — animal to its home.
- Analogy: WATER is to THIRST as FOOD is to ?
- HUNGER — what it relieves.
- Analogy: KEY is to LOCK as PASSWORD is to ?
- ACCOUNT (or COMPUTER) — what it unlocks.
- Analogy: SUN is to DAY as MOON is to ?
- NIGHT — associated time.
- Analogy: TEACHER is to STUDENT as DOCTOR is to ?
- PATIENT — who they serve.
- Analogy: ENGINE is to CAR as HEART is to ?
- BODY — the powering part of the whole.
- Analogy: FROWN is to SADNESS as SMILE is to ?
- HAPPINESS — expression to emotion.
- Analogy: SECONDS is to CLOCK as MILES is to ?
- ODOMETER — what measures it.
- Analogy: PAINTER is to CANVAS as WRITER is to ?
- PAPER (or PAGE) — surface they work on.
- Analogy: WHALE is to MAMMAL as SHARK is to ?
- FISH — member to its class.
- Analogy: TALL is to SHORT as WIDE is to ?
- NARROW — antonyms.
- Analogy: SEED is to PLANT as EGG is to ?
- CHICKEN (or BIRD) — early form to mature form.
- Analogy: FLOCK is to SHEEP as POD is to ?
- WHALES (or DOLPHINS) — group to its animal.
- Analogy: SCALE is to WEIGHT as RULER is to ?
- LENGTH — instrument to what it measures.
- Analogy: ACTOR is to STAGE as ATHLETE is to ?
- FIELD (or COURT) — where they perform.
- Analogy: WORD is to SENTENCE as SENTENCE is to ?
- PARAGRAPH — part to next-larger whole.
- Analogy: LIGHT is to DARK as LOUD is to ?
- QUIET — antonyms.
- Analogy: KNIFE is to FORK as PEN is to ?
- PENCIL — items in the same set/pair.
- Analogy: WET is to DAMP as HOT is to ?
- WARM — degree (less intense).
- Analogy: COW is to MILK as BEE is to ?
- HONEY — animal to product.
- Analogy: CAPTAIN is to SHIP as PILOT is to ?
- PLANE — who leads/operates it.
- Analogy: SPIDER is to WEB as BEAVER is to ?
- DAM — animal to what it builds.
- Analogy: SMOKE is to FIRE as STEAM is to ?
- BOILING WATER — effect to cause.
- Analogy: COWARD is to BRAVE as HONEST is to ?
- DISHONEST — antonyms (trait).
- Analogy: VERSE is to POEM as ACT is to ?
- PLAY — a section of the whole work.
- Analogy: GERM is to DISEASE as SPARK is to ?
- FIRE — cause to effect.
- Analogy: SCISSORS is to CUT as BROOM is to ?
- SWEEP — tool to its function.
- Next: 4, 8, 12, 16, ?
- 20 — add 4 each time.
- Next: 81, 27, 9, 3, ?
- 1 — divide by 3 each time.
- Next: 2, 6, 18, 54, ?
- 162 — multiply by 3 each time.
- Next: 50, 45, 40, 35, ?
- 30 — subtract 5 each time.
- Next: 1, 3, 6, 10, ?
- 15 — add 2, 3, 4, 5 … (triangular numbers).
- Next: 2, 4, 6, 10, 16, ?
- 26 — each term is the sum of the previous two (Fibonacci-style).
- Next: 6, 12, 24, 48, ?
- 96 — double each time.
- Next: 90, 80, 70, 60, ?
- 50 — subtract 10 each time.
- Next: 1, 4, 16, 64, ?
- 256 — multiply by 4 each time.
- Next: 3, 4, 6, 9, 13, ?
- 18 — gaps grow by 1 (+1, +2, +3, +4, +5).
- Next: 100, 50, 25, ?
- 12.5 — halve each time.
- Next: 5, 10, 20, 40, ?
- 80 — double each time.
- Next: 12, 10, 8, 6, ?
- 4 — subtract 2 each time.
- Next: 1, 2, 4, 8, 16, ?
- 32 — double each time (powers of 2).
- Next: 0, 1, 4, 9, 16, ?
- 25 — the squares of 0, 1, 2, 3, 4, 5.
- Next letter: A, B, D, G, ?
- K — gaps grow (+1, +2, +3, +4): positions 1,2,4,7,11.
- Next letter: C, F, I, L, ?
- O — skip two letters each time (+3 positions).
- Next letter: Y, W, U, S, ?
- Q — backward, skip one each time (−2 positions).
- Next letter: A, D, G, J, ?
- M — +3 positions each step.
- Next: 7, 12, 17, 22, ?
- 27 — add 5 each time.
- Next: 4, 9, 19, 39, ?
- 79 — each term is ×2 + 1.
- Next: 1, 10, 100, ?
- 1000 — multiply by 10 (powers of 10).
- Next: 15, 13, 11, 9, ?
- 7 — subtract 2 each time.
- Next: 3, 6, 11, 18, ?
- 27 — gaps grow by 2 (+3, +5, +7, +9).
- Next: 2, 3, 5, 8, 12, ?
- 17 — gaps grow by 1 (+1, +2, +3, +4, +5).
- Next: 144, 121, 100, 81, ?
- 64 — descending perfect squares (12², 11², 10², 9², 8²).
- Next: 5, 8, 14, 26, ?
- 50 — each term is ×2 − 2.
- Next: 1, 1, 2, 6, 24, ?
- 120 — multiply by 1, 2, 3, 4, 5 (factorials).
- Spot a geometric ratio fast
- Divide a term by the one before it; if the result is constant, that's the common ratio.
- Spot an arithmetic difference fast
- Subtract a term from the next; if the result is constant, that's the common difference.
- Rotate ↓ by 90° clockwise
- ← (down becomes left).
- Rotate ← by 90° clockwise
- ↑ (left becomes up).
- 180° rotation of →
- ← (right becomes left).
- 180° rotation of ←
- → (left becomes right).
- Reflection of ↑ across a vertical line
- ↑ stays ↑ (an up-arrow is symmetric left-right).
- Reflection of a 'b' shape (horizontal flip)
- A 'd' shape — its mirror image.
- Reflection of a 'p' shape (vertical flip)
- A 'b' shape — flipped top to bottom.
- Count change: 1 dot → 2 dots, so 3 dots → ?
- 4 dots — add one each step.
- Size change: small → large, so large → ?
- Apply the same change consistently (e.g. larger, or back to small if it cycles).
- Shading: white → black, so striped → ?
- Apply the same fill change the first pair showed.
- Which is NOT a rotation
- A mirror image (reflection) — it reverses left and right.
- Number of sides changes
- A square (4 sides) → pentagon (5 sides): add one side each step.
- Element moves clockwise
- A dot in the top-left moves to the top-right, then bottom-right, … by a fixed step.
- Two changes at once
- If the figure both rotates and gains a dot, apply BOTH to the third figure.
- 90° vs 180° — how to tell
- 90° is a quarter-turn (orientation perpendicular); 180° points the shape the opposite way.
- Confirm the rule on the first pair
- Re-apply your transformation to figure 1 and check it produces figure 2 exactly.
- Reflection across a diagonal
- Swaps the shape's position relative to that diagonal line — still a mirror image.
- Constant element stays put
- Only transform the part that changed between the first two figures.
- Rotation preserves size
- A pure rotation doesn't make the shape bigger or smaller — only its orientation changes.
- Symmetric shapes can hide rotations
- A square looks the same after 90° turns — watch a marked corner or dot to see the rotation.
- Add-then-rotate vs rotate-then-add
- Order rarely matters for the answer, but apply the SAME set of changes to the third figure.
- Mirror-image clue
- If left/right features swap but top/bottom don't, it's a horizontal reflection.
- Eliminate by orientation
- Cross off any choice whose orientation can't come from your transformation.
- Count both shapes and lines
- A 'remove an element' rule may take away a line, not a whole shape.
- Cycle patterns
- Some figure series cycle through a fixed set of orientations — find the cycle length.