- 3, 6, 12, 24, 48, …
- Next term is 96 — a geometric series doubling each step (× 2).
- Arithmetic sequence
- A number series with a constant difference between terms, such as 5, 9, 13, 17 (adding 4 each time).
- Geometric sequence
- A number series with a constant multiplier between terms, such as 1, 3, 9, 27 (each term × 3).
- Number series
- A sequence that follows a hidden rule; you find the rule and give the next (or missing) term. The most common CCAT Math & Logic item.
- How to crack a number series
- Test for a constant difference (arithmetic) first, then a constant multiplier (geometric); if neither fits, look for two rules alternating or two sequences interleaved.
- 1, 4, 9, 16, 25, …
- Next term is 36 — these are perfect squares (1², 2², 3², 4², 5²), so 6² = 36.
- Interleaved series
- Two separate sequences woven together (odd positions one rule, even positions another). Split the odd- and even-position terms to see each pattern.
- part = percent × whole
- The core percent setup. 35% of 80 = 0.35 × 80 = 28. Turn the percent into a decimal first.
- Percent change formula
- change ÷ original × 100. A price rising from 20 to 25 is 5 ÷ 20 × 100 = 25% increase.
- Percentage markup
- (selling price − cost) ÷ cost × 100. A jacket bought for 40 and sold for 52: 12 ÷ 40 × 100 = 30% markup.
- Ratio / proportion
- Set equal fractions and cross-multiply: a/b = c/d. Cats:dogs = 3:5 with 24 cats → 24/3 = 8 groups, so dogs = 5 × 8 = 40.
- Distance = rate × time
- The motion relationship. 150 miles in 3 hours → rate 50 mph; in 5 hours → 50 × 5 = 250 miles.
- Average (mean)
- sum of the values ÷ how many values there are.
- ½ = ? as a percent
- 50%. Memorize the common fraction ↔ percent pairs to save seconds on a 15-minute test.
- ¼ = ? as a percent
- 25%.
- ⅓ = ? as a percent
- ≈ 33.3%.
- ⅕ = ? as a percent
- 20%.
- ⅛ = ? as a percent
- 12.5%.
- ¾ = ? as a percent
- 75%.
- Solve: 4x − 7 = 21
- x = 7. Add 7 to both sides (4x = 28), then divide by 4.
- Solve: 3(x + 2) = 18
- x = 4. Divide by 3 (x + 2 = 6), then subtract 2.
- Solve a basic linear equation
- Undo operations in reverse order: clear parentheses, move the constant, then divide by the coefficient to isolate x.
- Successive discounts
- Apply them one after another, not added. 300 discounted 20% → 240, then 10% off → 216. (Not a flat 30%.)
- A number increased by 25% equals 90 — find it
- 72. The original × 1.25 = 90, so original = 90 ÷ 1.25 = 72.
- ¼ + ⅔ = ?
- 11/12. Use a common denominator of 12: 3/12 + 8/12 = 11/12.
- Logical deduction (CCAT)
- Given statements assumed true, judge whether a conclusion must follow. Accept only what the premises prove — choose 'uncertain' for any gap.
- Syllogism
- A two-premise argument: 'All A are B; all B are C; therefore all A are C.' Valid when the conclusion must follow, regardless of real-world truth.
- Syllogism traps
- 'Some' never becomes 'all,' and a statement does not reverse — 'all A are B' does not mean 'all B are A.'
- Word problem strategy
- Translate words to a setup: name what's asked, label the numbers, pick the relationship (part = percent × whole, d = rt), then do clean mental math.
- Reading a chart/graph item
- Read the axis labels and units first, find the exact values the question asks for, then compute (often a percent change or a share of a total).
- Take 10% of a number fast
- Move the decimal one place left. Build other percents from it: 5% is half of 10%, 20% is double, 15% is 10% + 5%.
- ⅗ as a percentage
- 60%. Divide and multiply by 100: 3 ÷ 5 = 0.6 = 60%.
- Inductive reasoning
- Reasoning from specific cases to a likely general rule — how you spot the pattern in a number or figure series.
- Deductive reasoning
- Reasoning from general rules to a guaranteed specific conclusion — the basis of syllogism and true/false/uncertain items.
- Share of a total (pie chart)
- Multiply the whole by the slice's percent. 40% food of a 4,000 budget = 0.40 × 4,000 = 1,600.
- Why no calculator hurts less than you think
- CCAT numbers are usually clean on purpose. Fast setup and memorized conversions beat heavy computation; a calculator just eats your 15 minutes.
- Alternating series
- Two rules applied in turn — e.g. +5, then ÷ 2, repeat. Identify the two operations before predicting the next term.
- Growing-gap series
- The differences themselves follow a pattern: 1, 2, 4, 7, 11 grows by +1, +2, +3, +4 — so the next gap is +5.
- Analogy
- A verbal item in the form A is to B as C is to ? — find the relationship in the first pair and apply the exact same relationship to the second.
- PAINTER : BRUSH :: WRITER : ?
- Pen. The relationship is 'uses as a tool to do their work.'
- How to solve an analogy
- Name the relationship in the first pair as a short sentence, then find the only choice that makes the same sentence true for the second pair.
- Antonym
- A word with the opposite meaning of another (transparent / opaque; rigid / flexible). CCAT verbal items ask for the most nearly opposite word.
- Synonym
- A word with nearly the same meaning as another (candid / frank; abate / lessen).
- Sentence completion
- Choose the word that best fits the blank using context clues — especially signal words like 'although,' 'because,' or 'despite' that flag contrast or cause.
- Opposite of TRANSPARENT
- Opaque — not able to be seen through.
- Opposite of RIGID
- Flexible — able to bend or adapt.
- Contrast signal words
- 'Although,' 'despite,' 'however,' 'yet,' 'whereas' signal that the missing word opposes the rest of the sentence — pick the contrasting choice.
- Cause/effect signal words
- 'Because,' 'since,' 'therefore,' 'so' signal that the blank continues or results from the rest — pick the consistent choice.
- Roots, prefixes, suffixes
- When a word is unfamiliar, decode its parts: 'mal-' (bad), 'bene-' (good), '-phobia' (fear) often reveal meaning or polarity.
- FRAGILE : SHATTER :: FLAMMABLE : ?
- Burn. The relationship is 'has the property → the event that property leads to.'
- Common analogy relationship: worker → workplace
- doctor : hospital, teacher : school, judge : courtroom.
- Common analogy relationship: tool → user
- scalpel : surgeon, brush : painter, wrench : mechanic.
- Common analogy relationship: part → whole
- page : book, petal : flower, room : house.
- Common analogy relationship: young → adult
- puppy : dog, kitten : cat, foal : horse.
- Common analogy relationship: object → function
- knife : cut, thermometer : temperature, scale : weight.
- Common analogy relationship: container → contents
- library : books, armory : weapons, wallet : money.
- Analogy trap to avoid
- A topically related word that breaks your relationship sentence. If 'a doctor works in a hospital,' the answer must fit 'a __ works in a __,' not just be hospital-related.
- Attention to detail (matching)
- Compare two codes, account numbers, or words character by character and decide whether they are exactly the same or different. The change is usually one digit or letter.
- Spotting a misspelling
- Look for swapped letters (receive vs. recieve) or a missing/extra letter. Scan left to right one character at a time.
- Speeded test
- A test limited mainly by time, not difficulty. The CCAT is heavily speeded — very few candidates finish all 50 questions in 15 minutes.
- Verbal reasoning (CCAT)
- Working with language under time pressure — analogies, antonyms, and sentence completion that test vocabulary and how quickly you process meaning.
- Opposite of ABUNDANT
- Scarce — in short supply.
- Opposite of GENEROUS
- Stingy — unwilling to give or share.
- BENEVOLENT means…
- Kind, well-meaning, generous. Its opposite is malevolent (cruel).
- FRUGAL means…
- Thrifty, careful with money. Its opposite is wasteful or extravagant.
- TENACIOUS means…
- Persistent, determined, not giving up.
- LETHARGIC means…
- Sluggish, lacking energy. Its opposite is energetic.
- AMBIGUOUS means…
- Open to more than one interpretation; unclear. Its opposite is clear or explicit.
- Eliminate in sentence completion
- Cross out choices that clash with the sentence's tone or logic first; CCAT options usually include one clear opposite you can drop immediately.
- BEE : HIVE :: BIRD : ?
- Nest. The relationship is 'animal → the structure it builds and lives in.'
- Spatial reasoning (CCAT)
- Mentally rotating, reflecting, folding, or completing shapes — picturing how a figure changes or how parts fit together, without words.
- Next-in-series (figures)
- A row of figures changes by a rule — rotation, a growing count, a size change, or shading. Identify what changes from one to the next, then continue it.
- Figure-matrix item
- A 3×3 grid with one cell missing; each row and/or column follows a rule. Read across a complete row and down a complete column to deduce the missing figure.
- Odd-one-out
- Several figures share a feature and one breaks it (orientation, count of sides, shading). Find the shared rule, then pick the figure that violates it.
- How to attack a rotation item
- Rotate the figure in your mind one step at a time (90° at a time), and rule out mirror images — the most common wrong answer.
- Mirror image vs. rotation
- A rotation turns a figure; a reflection flips it. A mirror image of an asymmetric shape can never be reached by rotation — eliminate it.
- Paper-folding item
- A sheet is folded, a hole or mark is made, then unfolded. The mark is duplicated symmetrically across each fold line — count the folds to count the resulting marks.
- Growing-count series
- The number of elements increases by a fixed amount each step (one square, two squares, three squares…) — continue the count.
- Side-count series
- Shapes gain a side each step: triangle (3) → square (4) → pentagon (5) → hexagon (6).
- Alternating-shape series
- Figures cycle through a fixed order (circle, triangle, circle, triangle…); follow the cycle to name what comes next.
- Shading-pattern series
- A figure alternates black/white or shifts which part is shaded each step — track the shading rule separately from the shape.
- Shrinking/growing series
- A shape changes size at a steady step (large → medium → small → even smaller) — continue the size trend.
- Rotation in a matrix
- Each cell rotates its element by a fixed angle (e.g. 90° clockwise) across a row — apply the same turn to find the missing cell.
- 'b' rotated 180°
- Turning 'b' completely upside down gives 'q' (the bowl moves to the top-right). Picture the flip carefully — letter-rotation items are about orientation, not the letter itself.
- Mirror a capital letter
- Flipping a letter left-to-right swaps its left and right. 'E' mirrored has its prongs pointing left instead of right.
- Spatial strategy checklist
- 1) Decide the operation (rotate, reflect, complete, find odd one). 2) Transform one step at a time. 3) Rule out mirror images. 4) Drop any choice with an extra or missing part.
- Count the parts
- Before choosing, count the elements in your transformed figure and in each option — eliminate any choice with one too many or one too few.
- Pattern identification
- Spotting the rule that governs a series of numbers or figures so you can predict what comes next — the skill behind both number-series and figure-series items.
- Read the matrix both ways
- In a 3×3 figure matrix, check the rule along the rows AND down the columns — the missing cell must satisfy both.
- Why spatial items reward practice
- They test reasoning, not memorized facts, so repeated exposure to rotation, folding, and matrix formats builds the biggest, fastest gains.
- 5, 9, 13, 17, 21, …
- Next term is 25 — an arithmetic series adding 4 each step.
- 100, 90, 80, 70, 60, …
- Next term is 50 — a decreasing arithmetic series subtracting 10 each step.
- 81, 27, 9, ?, 1
- Missing term is 3 — each term is divided by 3 (81 → 27 → 9 → 3 → 1).
- Cross-multiplying a proportion
- If a/b = c/d, then a × d = b × c. Use it to solve for any single unknown in a ratio problem.
- Mean vs. the question's wording
- Reread what's asked before committing — the trap on word problems is solving for the rate when the question wants the distance (or vice versa).
- Convert a fraction to a percent
- Divide the numerator by the denominator, then multiply by 100: 3 ÷ 4 = 0.75 = 75%.
- Solve: 5x = 3x + 14
- x = 7. Subtract 3x from both sides (2x = 14), then divide by 2.
- Attention-to-detail in Math & Logic
- Some CCAT logic items ask you to follow precise instructions or compare values exactly — read every word; a single misread changes the answer.
- PUPPY : DOG :: KITTEN : ?
- Cat. The relationship is 'young animal → its adult.'
- LIBRARY : BOOKS :: ARMORY : ?
- Weapons. The relationship is 'building → what is stored inside it.'
- Opposite of EXPAND
- Contract — to shrink or reduce in size.
- Opposite of PRAISE
- Criticize (or condemn) — to express disapproval.
- CANDID means…
- Honest and direct; frank. A synonym is forthright.
- Verbal speed tip
- Don't camp on one hard vocabulary word — make your best read of its polarity and move on; the clock is the real opponent on a speeded test.
- DOCTOR : PATIENT :: TEACHER : ?
- Student. The relationship is 'professional → the person they serve.'
- Two folds = how many marks?
- Each fold doubles a punched hole when unfolded — one hole through two folds typically opens to four holes, placed symmetrically about the fold lines.
- Touching vs. not touching
- In an odd-one-out where inner shapes touch the box edge, the exception is the one where the inner shape does NOT touch — judge the relationship, not the shapes.
- Clockwise vs. counterclockwise
- Fix a reference point on the figure (an arrow tip, a dot) and track which way it moves between steps to lock the rotation direction.
- Series with two changes at once
- A figure can rotate AND change shading each step. Track each rule separately, then combine them to predict the next figure.
- Eliminate, don't search
- On figure items it's faster to rule out wrong options (wrong orientation, mirror flip, wrong count) than to construct the answer from scratch.
- What CCAT 'Math & Logic' covers
- Number/letter series, arithmetic word problems, ratios, percentages, basic algebra, reading charts/graphs, and short logical-deduction items — solved without a calculator.
- What CCAT 'Verbal' covers
- Analogies, antonyms (opposite words), and sentence completion — vocabulary and language-processing items answered quickly.
- What CCAT 'Spatial Reasoning' covers
- Next-in-series figures, figure matrices, shape rotation and reflection, paper folding, and odd-one-out items — all visual, no reading.
- Raw score (CCAT)
- The number of questions you answer correctly out of 50 — one point each, with no penalty for wrong answers, so never leave a blank if time allows.
- Percentile rank (CCAT)
- How your raw score compares to a norm group: a 90th-percentile score beat 90% of test-takers. Employers usually judge candidates on percentile, not raw score.
- No 'passing score' on the CCAT
- There is no universal pass mark. Employers set their own target score ranges by role, so a competitive score depends entirely on the job you're applying for.
- Pacing the CCAT
- 50 questions in 15 minutes is about 18 seconds each. Bank the questions you find easy, guess and move on when stuck, and answer everything you can before time runs out.
- Guess when unsure
- Because there's no penalty for a wrong answer, eliminate what you can and guess — a blank scores the same as a wrong answer, but a guess might be right.
- Letter series
- Like number series but with the alphabet — find the step between letters (e.g. +2: A, C, E, G) and continue it; convert letters to their position numbers if it helps.
- Difficulty ramps up
- CCAT questions tend to get harder as you progress, so move efficiently early to leave time for the tougher later items.