trigger and action

Adjacent squares pattern (sticks/walls)

Start with the base number for 1 item, then add

(N-1) x increment ( An increment is just the amount that something increases or goes up by each time?

Grid Area Estimation Strategy

Count full squares first, then mentally group partial edge squares into whole units before comparing.

Max/Min number difference trick

Largest = descending order, Smallest = ascending order, then subtract.

Building the largest number from a digit set

Prioritize the highest possible digit in the highest place value (left to right).

Consecutive odd numbers with primes

1) List odds in range, 2) Identify primes, 3) Test consecutive 3-number sets (n, n+2, n+4).

Division with large numbers resulting in mixed fractions (A divided B).

Check whole number estimation (30 x 29 = 870) —> Add remainder (885 - 870 = 15) —→ Place remainder over original divisor (\frac{15}{29}. 15/29).

Trigger: Finding area of an equilateral triangle from perimeter.

S= perimeter/ 3

Then plug in S

Area = √ 3 / 4 S²

Finding interior angle x given an adjacent linear pair (120°).

Find missing interior angle (180°-120° = 60°) —→ Sum of triangle angles: x = 180° - (60°+ 50°).

Perimeter comparison of a composite shape built from squares and equilateral triangles.

Find side length (s =√area ) —→ Count outer edges only (ignore interior boundary lines) ——> Multiply (6 outer sides X 2 = 12 cm).

Given the sum of two numbers and the difference of their squares, find the larger number

Divide (x^2 - y^2) by (x + y) to get (x - y) = 125/25 = 5. Add sum and difference, then divide by 2: (25 + 5) / 2 = 15.

Counting total possible combinations of selecting 1 item from multiple categories (e.g., pies and drinks)

Fundamental Counting Principle: multiply the number of options in each category together (4 x 3 = 12).

4 consecutive numbers where the product of the first two is given, find the product of the last two

Factor the given product into consecutive numbers (12 = 3 x 4). List the 4 consecutive numbers (3, 4, 5, 6), then multiply the last two (5 x 6 = 30).

Comparing fractions/powers with variable conditions like 0 < A < B and B > 1 to find the largest value

Plug in simple numbers (e.g., A = 1, B = 2). Compare improper fraction B/A = 2/1 vs its square (B/A)^2 = 4. Squaring a fraction greater than 1 makes it bigger.

Comparing two subtraction expressions of improper vs proper fractions (11/9 - 9/11 vs 9/11 - 11/9)

Since 11/9 > 1 and 9/11 < 1, Value A is (Big - Small = Positive) and Value B is (Small - Big = Negative). Any positive value is greater than a negative value.

Finding square area when circle area is given inside a square grid/diagram (e.g., Circle Area = 16 * pi)

Use circle area formula (pi r^2 = 16 pi) to find radius r = 4. Diameter = 2 * r = 8, which equals the square's side length s. Area of square = s^2 = 8^2 = 64.

Scale percentage values (e.g., 20% of a number is 48, find 50%)

Find 10% first by dividing by 2 (20% / 2 = 10% -> 48 / 2 = 24). Multiply 10% value by 5 to get 50% (24 * 5 = 120).

Summing multiple fractions with powers of 10 in denominators (e.g., 700/10 + 7/100 + 7/1000 + 1/100)

Convert fractions to decimals first: 70 + 0.07 + 0.007 + 0.01. Align decimal points vertically and add straight down: 70.087.

Age word problem with future age given (e.g., Father is 8 times son's age, in 19 years son will be 24)

Subtract future years to get son's present age (24 - 19 = 5). Multiply by the ratio for father's present age (5 * 8 = 40).

Algebraic identity equation where both sides simplify to the exact same expression (e.g., x + 1/x = (x^2 + 1)/x)

Recognized an identity that holds for any number EXCEPT where the denominator becomes undefined. Answer is "all real numbers except zero."

Nested midpoints on a triangle segment ratio (e.g., E is midpoint of AN, N is midpoint of AB)

Assign simple integer segment lengths from top to bottom (AE = 1, EN = 1, NB = 2 \rightarrow AN = 2). Plug lengths directly into the given ratio fraction and simplify.

Compare (x + y)^2 vs a constant number when 2(x + y) is given

Solve for (x + y) first by dividing by 2 (18 / 2 = 9). Square that value (9^2 = 81) and compare directly to Value B (81 > 9).

Finding sum (x + y) in an X-intersecting lines diagram with given angles

Use linear pair rule (180 - 140 = 40) to find vertical/adjacent angles, then add the resulting values (x = 40, y = 40 \rightarrow 40 + 40 = 80).

Finding angle y in a rhombus with given adjacent angles (4x and 6x)

Rhombus adjacent angles sum to 180 (4x + 6x = 180 -> 10x = 180 -> x = 18). Opposite angles are equal (y = 4x = 4 * 18 = 72).

How many times recurring visits coincide in a year given intervals (3, 5, and 6 days)

Find Least Common Multiple of days: LCM(3, 5, 6) = 30 days (once per month). Total times met in a year = 360 days / 30 days = 12 times.

Repeated division of a variable by 2 four times: (((x / 2) / 2) / 2) / 2

Multiply the divisors together in the denominator: 2 2 2 * 2 = 16. Expression simplifies directly to x / 16.

Comparing negative fractions (-4/9 vs -9/4)

Cross-multiply numerators by opposite denominators (-4 4 = -16 vs -9 9 = -81). Since -16 is greater than -81, Value A is larger.

Finding the vertical height length ad in a right triangle given hypotenuse sqrt(50) and base segments

Use Pythagorean theorem on the inner right triangle (5^2 + 2^2 = h^2 -> sqrt(25 + 4) = sqrt(29)) or recognize the isosceles right triangle properties (hypotenuse sqrt(50) = 5*sqrt(2) -> leg = 5). Total base = 2 + 5 = 7. Height ad = 7.

Finding the n-th item in a repeating pattern sequence of 4 items (orange, apple, lemon, mango)

Divide position number by 4 (135 / 4 = 33 with remainder 3). Use remainder to count items: Remainder 1 = Orange, 2 = Apple, 3 = Lemon, 0 = Mango. Answer: Lemon

Estimating ugly decimal multiplication/division (109.8 * 9.8 / 4.01)

Round to nearest clean whole numbers: (110 * 10) / 4 = 1100 / 4 = 275. Select the closest option (265 or 275 depending on choices).

Completing increasing pattern with growing differences (1, 2, 4, 7, 11, 16...)

Identify pattern of added differences (+1, +2, +3, +4, +5). Add the next integer (+6) to the last term (16 + 6 = 22).

Finding circumference of a circle containing an inscribed right triangle with legs 6 and 8

Right triangle hypotenuse is circle diameter (6-8-10 triplet -> d = 10). Circumference = pi d = 3.14 10 = 31.4.

Finding the first negative term in a decreasing sequence (20, 18, 15, 11, 6...)

Identify pattern of increasing subtractions (-2, -3, -4, -5, -6, -7). Subtract iteratively: 6 - 6 = 0, then 0 - 7 = -7.

Simplifying algebraic fraction given x = 21m: find (x + 3m) / 4m

Substitute x directly into numerator (21m + 3m = 24m). Divide by denominator (24m / 4m = 6). Units "m" cancel out.

Two-digit number word problem where tens digit is 2 more than units, and equation on digit sum gives 10

Plug in answer choices. Test tens - units = 2 (e.g., 86 -> 8 - 6 = 2). Test equation: (Sum 5) / 7 = ((8 + 6) 5) / 7 = (14 * 5) / 7 = 70 / 7 = 10.

Finding the sum of roots for a quadratic equation x^2 - 1 = 0

Solve for roots: x^2 = 1 -> x = +1 and -1. Add roots together: (+1) + (-1) = 0.

Subtracting multiplied fraction terms: (10/15 30) - (10/6 30)

Simplify whole numbers first (30 / 15 = 2 -> 10 2 = 20; 30 / 6 = 5 -> 10 5 = 50). Subtract result: 20 - 50 = -30.

Area of an irregular composite shape split into 3 stacked rectangles

Split shape into 3 rectangles, find length and width for each (Top: 5x2 = 10; Middle: 4x2 = 8; Bottom: 7x2 = 14), then sum areas: 10 + 8 + 14 = 32.

Working backward money problem: gave half to sister, received 8 SR, ended with 50 SR

Reverse operations from end to start: subtract addition (50 - 8 = 42), then double to undo halving (42 * 2 = 84).

Total value of banknotes given total bill count (120 bills of 5 SR and 10 SR) and a ratio (10 SR bills are 5 times 5 SR bills)

Use ratio parts (1 + 5 = 6 parts -> 120 / 6 = 20): 20 bills of 5 SR (100 SR) and 100 bills of 10 SR (1000 SR). Sum values: 100 + 1000 = 1100 SR.

Remaining wire length after fencing a square garden with side length 26m using a 125m wire

Calculate square perimeter (26 * 4 = 104). Subtract from total length (125 - 104 = 21).

Comparing negative fraction raised to an even power vs an odd power: (-3/4)^4 vs (-3/4)^9

Even exponent turns negative number positive; odd exponent keeps it negative. Any positive value is greater than a negative value.

Comparing square root terms with outside coefficients: 7sqrt(6) vs 6sqrt(7)

Square both values to remove radical signs: (7sqrt(6))^2 = 49 6 = 294 vs (6sqrt(7))^2 = 36 7 = 252. Since 294 > 252, Value A is larger.

Simplifying fraction expression with nested radicals: (sqrt(3) (sqrt(2) + 4sqrt(2))) / (sqrt(3) / 3)

Simplify terms inside parenthesis (sqrt(2) + 4sqrt(2) = 5sqrt(2)). Cancel sqrt(3) in numerator and denominator, then multiply by 3: 5sqrt(2) * 3 = 15sqrt(2).

Identifying three side lengths that CANNOT form a triangle

Triangle Inequality Theorem: sum of any two smaller sides MUST be strictly greater than the largest side. In (1, 3, 4), 1 + 3 = 4 (not greater), so it cannot be a triangle.

Identifying a prime number from options like 201, 111, 101, 1111

Check divisibility by 3 by summing digits: 201 (2+0+1=3), 111 (1+1+1=3), and 1111 (1+1+1+1=4, but 11*101=1111). 101 has no factors other than 1 and itself, making it prime.

Circular movement from point Q: 4/8 counter-clockwise, then 1/4 clockwise

Simplify fractions to common denominator of 8: move 4/8 CCW, then back 2/8 CW. Net movement is 2/8 CCW from Q (Q -> X -> W). Landing point is W

Total ticket cost for 15 children at 3 SR each and 2 teachers at 5 SR each

Multiply and add: (15 3) + (2 5) = 45 + 10 = 55 SR.

Comparing negative numbers multiplied by negative fractions: -17 (29 / -15) vs -18 (-36 / -25)

Count negative signs: Value A has two negatives (-17 and -15), making it POSITIVE. Value B has three negatives (-18, -36, -25), making it NEGATIVE. Any positive number is greater than a negative number

Converting minute hand rotation degrees to minutes (e.g., 270 degrees)

Minute hand moves 6 degrees per minute. Divide total degrees by 6: 270 / 6 = 45 minutes.

Finding angle C in a triangle inscribed in a circle where BC is the diameter and angle B = 40

Angle inscribed in a semicircle is always a right angle (A = 90). Subtract known angles from 180: C = 180 - (90 + 40) = 50.

Intersecting lines diagram with straight angles (180 degrees) and vertical opposite angles to find x

Use linear pair on top line (180 - 120 = 60). Vertical angle across gives 30, so inside angle is 90 - 30 = 60. Supplementary angle x = 180 - (90 + 30) = 60.

Finding the combined area of 2 parts out of 9 equal divisions in a circle of radius r

Total circle area = pi r^2. Multiply by the fraction of parts chosen: (2 / 9) pi r^2 = (2 pi * r^2) / 9.

Finding undistributed remainder when 49 items are divided equally among 9 students

Divide total items by people (49 / 9 = 5 with remainder 4). Remainder equals undistributed items: 4.

Finding the area of an inscribed circle inside a square with area 64 cm2

Side of square is \sqrt{64} = 8. The circle's diameter equals the side length (d = 8 \Longrightarrow r = 4). Area = \pi r^2 = \pi (4^2) = 16\pi.

Finding the area of outer circle M given inner circle K has area 36\pi\text{ cm}^2 and its diameter is the radius of M

Find inner radius (r_k = \sqrt{36} = 6). The diameter of circle K (d_k = 12) forms the radius of circle M (R_m = 12). Area of M = \pi R_m^2 = \pi (12^2) = 144\pi

Successive percentage increases: price increased by 300% then by 200%, find overall percentage increase

Start with 100 base. Increase by 300% \Longrightarrow 100 + 300 = 400. Increase 400 by 200% \Longrightarrow 400 + (2 \times 400) = 1200. Total percentage increase is \frac{1200 - 100}{100} \times 100\% = 1100\%.

Counting total blocks in a 3D stepped block arrangement

Count row-by-row or layer-by-layer: Bottom layer (3 \times 6 = 18), Middle layer (3 \times 3 = 9), Top layer (3 \times 1 = 3). Sum them up: 18 + 9 + 3 = 30.

Solving an exponential equation where bases can be matched (9^{2x} = 3^{x+3})

Express base 9 as 3^2 \Longrightarrow 3^{4x} = 3^{x+3}. Equate exponents (4x = x + 3 \Longrightarrow 3x = 3 \Longrightarrow x = 1).

Finding the lowest common multiple (LCM) for steps taken in groups without remainder (6, 8, 10)

Find \text{LCM}(6, 8, 10). Prime factorizations: 6 = 2 \times 3, 8 = 2^3, 10 = 2 \times 5. \text{LCM} = 2^3 \times 3 \times 5 = 8 \times 15 = 120

Finding tangent segment length x at the base of a triangle circumscribed around a circle

Tangents from an external point to a circle are equal. Subtract top-right tangent (16 - 3 = 13) to find lower-right segment (13). Base side

X= lower left + lower - right

Finding angle x between parallel lines m \parallel n with transversals

Use parallel line angle rules (corresponding/interior angles). Interior angle adjacent to x is 70^\circ. Supplementary linear pair gives x = 180 - 70 = 110^\circ.

Comparing fraction subtraction results (3/4 - 7/8 vs 5/8 - 1/2)

Convert to common denominator 8: Value A = (6/8 - 7/8 = -1/8, negative). Value B = (5/8 - 4/8 = +1/8, positive). Any positive value is larger than a negative value.

Finding intermediate angle x formed between an isosceles triangle (top vertex 80) and an equilateral triangle on a straight line

Isosceles base angle = (180 - 80) / 2 = 50. Equilateral angle = 60. Use straight line angle sum (180): x = 180 - (50 + 60) = 70.

Finding bottom base x of a trapezoid given top base (6) and midsegment length (9)

Midsegment formula: (top + bottom) / 2 = midsegment -> (6 + x) / 2 = 9 -> 6 + x = 18 -> x = 12.

Multiplying decimal numbers (0.314 * 1.86) to identify the correct option among similar choices

Check the last non-zero digits: 4 6 = 24 (ends in 4). Estimate magnitude: ~0.3 2 = 0.6. Option ending in 4 near 0.6 is 0.58404.

Finding the area of trapezoid BCFE inside a square of side length 4 cm divided into 8 equal triangular parts

Total square area is 4 4 = 16. Dividing into 8 equal triangles gives 16 / 8 = 2 per unit triangle. Trapezoid BCFE contains 3 unit triangles, so area = 3 2 = 6.

Finding distance AB between centers of two overlapping identical circles of radius 6 with overlap length 4

Distance between centers = radius + radius - overlap = 6 + 6 - 4 = 8.

Finding the total area of a large outer square containing 4 circles around a central shaded square of side 6 cm

Each side of the inner square connects 4 quadrant sectors equal to 4 circles. Area scales proportionally across the 4 grid components: 6 * 4 = 24.

Comparing percentage of a grid's shaded region (19 out of 20 squares shaded) against 96%

Convert fraction to percent: (19 / 20) * 100 = 95%. Compare: 95% < 96%, so Value B is larger.

Comparing large product of repeated 4s (4^7) vs product of large numbers (20 x16x 32)

Convert factors to base 2 powers: Value A = (2^2)^7 = 2^14 = 16384. Value B = (20) 2^4 2^5 = 20 * 512 = 10240. Value A is larger.

Given 2^a * 2^b = 2^30, compare average of a and b vs 12

Product rule gives a + b = 30. Average = (a + b) / 2 = 30 / 2 = 15. Compare: 15 > 12, so Value A is larger.

Comparing complex nested fraction 1 / (7 + 1/2) vs 2/15

Simplify denominator: 7 + 1/2 = 15/2. Reciprocal: 1 / (15/2) = 2/15. Both values are equal.

Finding remaining area of a rectangle (14 x 10) after subtracting an inner cutout rectangle (4 x 3)

Total area - cut area = (14 10) - (4 3) = 140 - 12 = 128.

Comparing evaluation of nested signs -1 - (-1) - (-1) vs (+1)(+1)

Value A = -1 + 1 + 1 = 1. Value B = 1 * 1 = 1. Both values are equal.

Identifying an angle value that CANNOT exist inside a triangle

Sum of all interior angles in any triangle must equal 180 degrees. Any single angle >= 180 degrees (like 185) is impossible.

Finding tenth root for expression: 27 * 2^3

Rewrite as powers of 3 and 2: 3^3 2^3 = (3 2)^3 = 6^3. Take 10th root: (6^3)^(1/10) = 6^(3/10) = 6^0.3.

Comparing angle sum A + B on a straight line vs interior angles C + D in a triangle

A and B form a linear pair on a straight line (A + B = 180 degrees). Interior triangle angles C + D must be less than 180 degrees since C + D + A_interior = 180. Value A is larger.

Decimals like 0.05 and 0.5 in comparisons.

Convert to fractions (1/20 and 1/2) instantly to cancel numbers out instead of shifting zeros. Or multiply them both by the same number (both by 100 instead of one by 10 and the other 100)

Questions asking "Find the value of 8n + 6 where n is an integer" (or similar expressions like ax + b).

Action:

1. Subtract the constant term (6) from each option choice first: \text{Option} - 6.

2. Check if the result divides cleanly by the coefficient (8).

3. Double-check your basic subtraction (30 - 6 = 24, which divides by 8).

Trigger: Comparing the areas of two or more triangles that share the same base and lie between two parallel lines

Do NOT rely on visual appearance or tilt.

2. Identify the shared base (BC).

3. Recognize that the distance between parallel lines is constant, so their heights (h) are identical.

4. Instantly select Equal (Option C) since \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}.

Comparing expressions containing fractions like \frac{1}{3} or (\frac{1}{3})^2 alongside powers of 3 or 9.

NEVER convert \frac{1}{3} to decimals (0.33), as rounding creates false comparison errors.

2. Keep everything in fraction form: \frac{1}{3} \times 27 = 9 and \frac{1}{9} \times 9 = 1.

3. Simplify directly using fraction division before comparing values.

Word problems asking for the "maximum number of items/boxes that can be made" using two different required components.

1. Divide each total supply by the required amount per item: 27 \div 4 = 6.75 and 15 \div 2 = 7.5.

2. Round both down to the nearest whole integer (6 and 7).

3. Pick the SMALLER whole number (6) because you run out of that component first (the bottleneck).

Dividing a whole number by a decimal denominator like \frac{64}{0.004}.

Move the decimal point in the denominator to make it a whole number, and add that same number of zeros to the top. (0.004 \to 4, add 3 zeros to 64 \to 64000 \div 4 = 16000).

Find the Volume of a cube with edge x".

Cube the edge length (x^3). For 4, calculate 4 \times 4 \times 4 = 64.

Adding decimals with matching tail digits like 9.968 + 9.032.

Add the complementary decimals first to form a clean whole number (9.968 + 9.032 = 19). Then do simple whole-number subtraction (19 - 9.75 = 9.25).

Division of two separate square roots (\sqrt{A} \div \sqrt{B}).

Combine them under one giant square root (\sqrt{A \div B}), divide inside first, then simplify (\sqrt{50 \div 2} = \sqrt{25} = 5).

SEE: Complex stacked fraction like \frac{1}{\frac{a}{b}}.

Flip the bottom fraction upside down and multiply (\frac{1}{1/5} = 5). Simplify each stack separately before adding.

Geometry grid with mixed-size shaded squares asking for "ratio of shaded".

Divide the entire grid into the smallest unit squares (16 total). Count total shaded unit squares (6). Reduce fraction: \frac{6}{16} = \frac{3}{8}.

Quantitative comparison with large square roots like 30 vs. \sqrt{2500} - \sqrt{1600}

Simplify each root separately (\sqrt{2500}=50, \sqrt{1600}=40), perform the exact sign operation (50 - 40 = 10), then compare directly (30 > 10