BSEB Matric Adv Math (Opt) cheat sheet

Euclid's division lemma: for integers a and positive b there exist unique q, r with a = bq + r and 0 ≤ r < b.

Euclid's algorithm for HCF: repeatedly replace (a, b) with (b, r) until r = 0; the last non-zero divisor is HCF(a, b).

Fundamental Theorem of Arithmetic: every integer > 1 is a unique product of primes up to order.

HCF(a, b) × LCM(a, b) = a × b holds for exactly two positive integers; extend to three numbers only via prime factorisation.

√n is irrational for every prime n; more generally √n is irrational unless n is a perfect square.

A rational number p/q (in lowest terms) has a terminating decimal iff q = 2^m · 5^n; otherwise the decimal is non-terminating repeating.

Sum or product of a rational and an irrational is irrational; sum of two irrationals may be rational (√2 + (−√2) = 0).

For positive reals a, b: AM ≥ GM ≥ HM, with equality only when a = b.

Remainder Theorem: the remainder on dividing p(x) by (x − a) is p(a); p(a) = 0 iff (x − a) is a factor.

For ax² + bx + c: sum of roots = −b/a, product of roots = c/a.

For ax³ + bx² + cx + d: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a.

Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a; discriminant D = b² − 4ac decides the nature of roots.

D > 0 gives two distinct real roots, D = 0 gives equal real roots, D < 0 gives no real roots (complex conjugates).

An equation has equal roots iff D = 0; use this to find unknown coefficients k.

Sum-to-product: to build a quadratic from roots α, β use x² − (α + β)x + αβ = 0.

Graph of y = ax² + bx + c is a parabola opening upward for a > 0, downward for a < 0; vertex at x = −b/2a.

nth term of an AP: aₙ = a + (n − 1)d, where a is the first term and d the common difference.

Sum of n terms: Sₙ = (n/2)[2a + (n − 1)d] = (n/2)(a + l) when the last term l is known.

d = aₙ − aₙ₋₁ is constant for an AP; three terms in AP are best written a − d, a, a + d.

If a, b, c are in AP then 2b = a + c; the AP mean of two numbers is their average.

Sum of first n natural numbers: n(n + 1)/2; sum of first n odd numbers is n².

Sₙ = n² type formulas imply aₙ = Sₙ − Sₙ₋₁; use this to recover the nth term from a sum formula.

For a GP: aₙ = ar^(n−1) and Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1; S∞ = a/(1 − r) when |r| < 1.

aₙ = Sₙ − Sₙ₋₁ with S₀ = 0 gives the term sequence; verify by checking n = 1 separately.

Distance between (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²].

Section formula: point dividing AB in ratio m : n internally is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)); midpoint is m = n = 1.

Slope m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes, perpendicular lines have m₁m₂ = −1.

Line forms: y = mx + c (slope–intercept), y − y₁ = m(x − x₁) (point–slope), x/a + y/b = 1 (intercept form).

General line Ax + By + C = 0 has slope −A/B; distance from (x₁, y₁) is |Ax₁ + By₁ + C| / √(A² + B²).

Area of triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|; zero area means collinear.

Three points are collinear iff the slope between any two pairs is equal.

x-axis equation is y = 0, y-axis is x = 0; a line through origin has c = 0.

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

Standard values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2; sin 60° = √3/2, tan 60° = √3.

sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, sec(90° − θ) = cosec θ.

Angle of elevation is measured upward from the horizontal to the line of sight; angle of depression is measured downward.

For a vertical tower of height h: horizontal distance d = h / tan(elevation); line-of-sight length = h / sin(elevation).

Two-observer problem: if angles of elevation from points a apart are α and β on the same side, h = a·tanα·tanβ / (tanβ − tanα).

Shadow problem: h = shadow length × tan(elevation angle).

tan θ = sin θ / cos θ; cot θ = 1/tan θ; write all ratios in sin and cos to simplify identities.

A tangent at a point is perpendicular to the radius drawn to that point.

Tangents from an external point are equal in length; they subtend equal angles at the centre.

Angle in a semicircle is 90°; angle in the same segment is equal.

The angle between a tangent and a chord equals the angle in the alternate segment.

Central angle = 2 × inscribed angle standing on the same arc.

Length of arc = (θ/360°) × 2πr; area of sector = (θ/360°) × πr².

Area of segment = area of sector − area of triangle formed by the two radii and the chord.

For two concentric circles with radii R and r, area of ring = π(R² − r²); tangent chord of outer circle to inner has length 2√(R² − r²).

Basic Proportionality Theorem (Thales): a line parallel to one side of a triangle divides the other two sides in the same ratio.

Converse of BPT: if a line divides two sides in the same ratio, it is parallel to the third side.

Similar triangles: ratio of areas equals the square of the ratio of corresponding sides.

Pythagoras theorem: in a right triangle, hypotenuse² = sum of squares of the other two sides; converse also holds.

In a 30°–60°–90° triangle, sides are in ratio 1 : √3 : 2; in 45°–45°–90°, 1 : 1 : √2.

Angle bisector theorem: BD/DC = AB/AC when AD bisects angle A.

Median divides a triangle into two equal-area triangles; centroid divides each median in 2 : 1 from the vertex.

Construction steps must list: given data, drawn figure with measurements, construction steps, and justification.

Cylinder: curved surface 2πrh, total surface 2πr(r + h), volume πr²h.

Cone: slant l = √(r² + h²), curved surface πrl, total surface πr(l + r), volume (1/3)πr²h.

Sphere: surface 4πr², volume (4/3)πr³; hemisphere: curved 2πr², total 3πr², volume (2/3)πr³.

Frustum: volume (1/3)πh(R² + Rr + r²), slant l = √[h² + (R − r)²], curved surface πl(R + r).

Cube: total surface 6a², volume a³, diagonal a√3; cuboid: surface 2(lb + bh + hl), volume lbh.

When a solid is melted and recast, volume is conserved; equate initial and final volumes to find the new dimension.

Sphere inscribed in a cylinder of same radius: sphere volume is 2/3 of cylinder volume.

1 m³ = 1000 litres; 1 litre = 1000 cm³; convert units before applying formulas.

Mean of grouped data: x̄ = Σfᵢxᵢ / Σfᵢ; use assumed mean or step-deviation for large values.

Median of grouped data: l + [(N/2 − cf)/f] × h, where l is lower limit of median class, cf cumulative frequency before it, f its frequency, h class width.

Mode of grouped data: l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.

Empirical relation: Mode = 3 Median − 2 Mean.

Probability of event E: P(E) = favourable outcomes / total equally likely outcomes; 0 ≤ P(E) ≤ 1.

P(not E) = 1 − P(E); P(E) + P(not E) = 1.

For a sure event P = 1; for an impossible event P = 0; sum of probabilities of all elementary outcomes is 1.

Cumulative frequency curve (ogive) gives median at the x-value where cumulative frequency equals N/2.

Verify a result by substitution or by checking a limiting case (e.g. set a variable to 0 or 1).

For 'find the number' problems, translate each clause into an equation, then solve and check the answer satisfies the original wording.

Work-rate: if A finishes in a days and B in b days, together they finish in ab/(a + b) days.

Percentage change: successive changes of x% and y% give net change x + y + xy/100 percent.

Time–speed–distance: distance = speed × time; for equal distances, average speed = 2v₁v₂/(v₁ + v₂).

Ratio problems: express each quantity as a multiple of a common factor k, then use the given total or difference to find k.

For 'two-digit number' puzzles, write the number as 10a + b and the reversed number as 10b + a.

Eliminate options by checking units, parity, and magnitude before doing full algebra.

Euclid's division lemma: for integers a and positive b there exist unique q, r with a = bq + r and 0 ≤ r < b.

Euclid's algorithm for HCF: repeatedly replace (a, b) with (b, r) until r = 0; the last non-zero divisor is HCF(a, b).

Fundamental Theorem of Arithmetic: every integer > 1 is a unique product of primes up to order.

HCF(a, b) × LCM(a, b) = a × b holds for exactly two positive integers; extend to three numbers only via prime factorisation.

√n is irrational for every prime n; more generally √n is irrational unless n is a perfect square.

A rational number p/q (in lowest terms) has a terminating decimal iff q = 2^m · 5^n; otherwise the decimal is non-terminating repeating.

Sum or product of a rational and an irrational is irrational; sum of two irrationals may be rational (√2 + (−√2) = 0).

For positive reals a, b: AM ≥ GM ≥ HM, with equality only when a = b.

Remainder Theorem: the remainder on dividing p(x) by (x − a) is p(a); p(a) = 0 iff (x − a) is a factor.

For ax² + bx + c: sum of roots = −b/a, product of roots = c/a.

For ax³ + bx² + cx + d: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a.

Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a; discriminant D = b² − 4ac decides the nature of roots.

D > 0 gives two distinct real roots, D = 0 gives equal real roots, D < 0 gives no real roots (complex conjugates).

An equation has equal roots iff D = 0; use this to find unknown coefficients k.

Sum-to-product: to build a quadratic from roots α, β use x² − (α + β)x + αβ = 0.

Graph of y = ax² + bx + c is a parabola opening upward for a > 0, downward for a < 0; vertex at x = −b/2a.

nth term of an AP: aₙ = a + (n − 1)d, where a is the first term and d the common difference.

Sum of n terms: Sₙ = (n/2)[2a + (n − 1)d] = (n/2)(a + l) when the last term l is known.

d = aₙ − aₙ₋₁ is constant for an AP; three terms in AP are best written a − d, a, a + d.

If a, b, c are in AP then 2b = a + c; the AP mean of two numbers is their average.

Sum of first n natural numbers: n(n + 1)/2; sum of first n odd numbers is n².

Sₙ = n² type formulas imply aₙ = Sₙ − Sₙ₋₁; use this to recover the nth term from a sum formula.

For a GP: aₙ = ar^(n−1) and Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1; S∞ = a/(1 − r) when |r| < 1.

aₙ = Sₙ − Sₙ₋₁ with S₀ = 0 gives the term sequence; verify by checking n = 1 separately.

Distance between (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²].

Section formula: point dividing AB in ratio m : n internally is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)); midpoint is m = n = 1.

Slope m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes, perpendicular lines have m₁m₂ = −1.

Line forms: y = mx + c (slope–intercept), y − y₁ = m(x − x₁) (point–slope), x/a + y/b = 1 (intercept form).

General line Ax + By + C = 0 has slope −A/B; distance from (x₁, y₁) is |Ax₁ + By₁ + C| / √(A² + B²).

Area of triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|; zero area means collinear.

Three points are collinear iff the slope between any two pairs is equal.

x-axis equation is y = 0, y-axis is x = 0; a line through origin has c = 0.

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

Standard values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2; sin 60° = √3/2, tan 60° = √3.

sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, sec(90° − θ) = cosec θ.

Angle of elevation is measured upward from the horizontal to the line of sight; angle of depression is measured downward.

For a vertical tower of height h: horizontal distance d = h / tan(elevation); line-of-sight length = h / sin(elevation).

Two-observer problem: if angles of elevation from points a apart are α and β on the same side, h = a·tanα·tanβ / (tanβ − tanα).

Shadow problem: h = shadow length × tan(elevation angle).

tan θ = sin θ / cos θ; cot θ = 1/tan θ; write all ratios in sin and cos to simplify identities.

A tangent at a point is perpendicular to the radius drawn to that point.

Tangents from an external point are equal in length; they subtend equal angles at the centre.

Angle in a semicircle is 90°; angle in the same segment is equal.

The angle between a tangent and a chord equals the angle in the alternate segment.

Central angle = 2 × inscribed angle standing on the same arc.

Length of arc = (θ/360°) × 2πr; area of sector = (θ/360°) × πr².

Area of segment = area of sector − area of triangle formed by the two radii and the chord.

For two concentric circles with radii R and r, area of ring = π(R² − r²); tangent chord of outer circle to inner has length 2√(R² − r²).

Basic Proportionality Theorem (Thales): a line parallel to one side of a triangle divides the other two sides in the same ratio.

Converse of BPT: if a line divides two sides in the same ratio, it is parallel to the third side.

Similar triangles: ratio of areas equals the square of the ratio of corresponding sides.

Pythagoras theorem: in a right triangle, hypotenuse² = sum of squares of the other two sides; converse also holds.

In a 30°–60°–90° triangle, sides are in ratio 1 : √3 : 2; in 45°–45°–90°, 1 : 1 : √2.

Angle bisector theorem: BD/DC = AB/AC when AD bisects angle A.

Median divides a triangle into two equal-area triangles; centroid divides each median in 2 : 1 from the vertex.

Construction steps must list: given data, drawn figure with measurements, construction steps, and justification.

Cylinder: curved surface 2πrh, total surface 2πr(r + h), volume πr²h.

Cone: slant l = √(r² + h²), curved surface πrl, total surface πr(l + r), volume (1/3)πr²h.

Sphere: surface 4πr², volume (4/3)πr³; hemisphere: curved 2πr², total 3πr², volume (2/3)πr³.

Frustum: volume (1/3)πh(R² + Rr + r²), slant l = √[h² + (R − r)²], curved surface πl(R + r).

Cube: total surface 6a², volume a³, diagonal a√3; cuboid: surface 2(lb + bh + hl), volume lbh.

When a solid is melted and recast, volume is conserved; equate initial and final volumes to find the new dimension.

Sphere inscribed in a cylinder of same radius: sphere volume is 2/3 of cylinder volume.

1 m³ = 1000 litres; 1 litre = 1000 cm³; convert units before applying formulas.

Mean of grouped data: x̄ = Σfᵢxᵢ / Σfᵢ; use assumed mean or step-deviation for large values.

Median of grouped data: l + [(N/2 − cf)/f] × h, where l is lower limit of median class, cf cumulative frequency before it, f its frequency, h class width.

Mode of grouped data: l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.

Empirical relation: Mode = 3 Median − 2 Mean.

Probability of event E: P(E) = favourable outcomes / total equally likely outcomes; 0 ≤ P(E) ≤ 1.

P(not E) = 1 − P(E); P(E) + P(not E) = 1.

For a sure event P = 1; for an impossible event P = 0; sum of probabilities of all elementary outcomes is 1.

Cumulative frequency curve (ogive) gives median at the x-value where cumulative frequency equals N/2.

Verify a result by substitution or by checking a limiting case (e.g. set a variable to 0 or 1).

For 'find the number' problems, translate each clause into an equation, then solve and check the answer satisfies the original wording.

Work-rate: if A finishes in a days and B in b days, together they finish in ab/(a + b) days.

Percentage change: successive changes of x% and y% give net change x + y + xy/100 percent.

Time–speed–distance: distance = speed × time; for equal distances, average speed = 2v₁v₂/(v₁ + v₂).

Ratio problems: express each quantity as a multiple of a common factor k, then use the given total or difference to find k.

For 'two-digit number' puzzles, write the number as 10a + b and the reversed number as 10b + a.

Eliminate options by checking units, parity, and magnitude before doing full algebra.

Euclid's division lemma: for integers a and positive b there exist unique q, r with a = bq + r and 0 ≤ r < b.

Euclid's algorithm for HCF: repeatedly replace (a, b) with (b, r) until r = 0; the last non-zero divisor is HCF(a, b).

Fundamental Theorem of Arithmetic: every integer > 1 is a unique product of primes up to order.

HCF(a, b) × LCM(a, b) = a × b holds for exactly two positive integers; extend to three numbers only via prime factorisation.

√n is irrational for every prime n; more generally √n is irrational unless n is a perfect square.

A rational number p/q (in lowest terms) has a terminating decimal iff q = 2^m · 5^n; otherwise the decimal is non-terminating repeating.

Sum or product of a rational and an irrational is irrational; sum of two irrationals may be rational (√2 + (−√2) = 0).

For positive reals a, b: AM ≥ GM ≥ HM, with equality only when a = b.

Remainder Theorem: the remainder on dividing p(x) by (x − a) is p(a); p(a) = 0 iff (x − a) is a factor.

For ax² + bx + c: sum of roots = −b/a, product of roots = c/a.

For ax³ + bx² + cx + d: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a.

Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a; discriminant D = b² − 4ac decides the nature of roots.

D > 0 gives two distinct real roots, D = 0 gives equal real roots, D < 0 gives no real roots (complex conjugates).

An equation has equal roots iff D = 0; use this to find unknown coefficients k.

Sum-to-product: to build a quadratic from roots α, β use x² − (α + β)x + αβ = 0.

Graph of y = ax² + bx + c is a parabola opening upward for a > 0, downward for a < 0; vertex at x = −b/2a.

nth term of an AP: aₙ = a + (n − 1)d, where a is the first term and d the common difference.

Sum of n terms: Sₙ = (n/2)[2a + (n − 1)d] = (n/2)(a + l) when the last term l is known.

d = aₙ − aₙ₋₁ is constant for an AP; three terms in AP are best written a − d, a, a + d.

If a, b, c are in AP then 2b = a + c; the AP mean of two numbers is their average.

Sum of first n natural numbers: n(n + 1)/2; sum of first n odd numbers is n².

Sₙ = n² type formulas imply aₙ = Sₙ − Sₙ₋₁; use this to recover the nth term from a sum formula.

For a GP: aₙ = ar^(n−1) and Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1; S∞ = a/(1 − r) when |r| < 1.

aₙ = Sₙ − Sₙ₋₁ with S₀ = 0 gives the term sequence; verify by checking n = 1 separately.

Distance between (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²].

Section formula: point dividing AB in ratio m : n internally is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)); midpoint is m = n = 1.

Slope m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes, perpendicular lines have m₁m₂ = −1.

Line forms: y = mx + c (slope–intercept), y − y₁ = m(x − x₁) (point–slope), x/a + y/b = 1 (intercept form).

General line Ax + By + C = 0 has slope −A/B; distance from (x₁, y₁) is |Ax₁ + By₁ + C| / √(A² + B²).

Area of triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|; zero area means collinear.

Three points are collinear iff the slope between any two pairs is equal.

x-axis equation is y = 0, y-axis is x = 0; a line through origin has c = 0.

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

Standard values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2; sin 60° = √3/2, tan 60° = √3.

sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, sec(90° − θ) = cosec θ.

Angle of elevation is measured upward from the horizontal to the line of sight; angle of depression is measured downward.

For a vertical tower of height h: horizontal distance d = h / tan(elevation); line-of-sight length = h / sin(elevation).

Two-observer problem: if angles of elevation from points a apart are α and β on the same side, h = a·tanα·tanβ / (tanβ − tanα).

Shadow problem: h = shadow length × tan(elevation angle).

tan θ = sin θ / cos θ; cot θ = 1/tan θ; write all ratios in sin and cos to simplify identities.

A tangent at a point is perpendicular to the radius drawn to that point.

Tangents from an external point are equal in length; they subtend equal angles at the centre.

Angle in a semicircle is 90°; angle in the same segment is equal.

The angle between a tangent and a chord equals the angle in the alternate segment.

Central angle = 2 × inscribed angle standing on the same arc.

Length of arc = (θ/360°) × 2πr; area of sector = (θ/360°) × πr².

Area of segment = area of sector − area of triangle formed by the two radii and the chord.

For two concentric circles with radii R and r, area of ring = π(R² − r²); tangent chord of outer circle to inner has length 2√(R² − r²).

Basic Proportionality Theorem (Thales): a line parallel to one side of a triangle divides the other two sides in the same ratio.

Converse of BPT: if a line divides two sides in the same ratio, it is parallel to the third side.

Similar triangles: ratio of areas equals the square of the ratio of corresponding sides.

Pythagoras theorem: in a right triangle, hypotenuse² = sum of squares of the other two sides; converse also holds.

In a 30°–60°–90° triangle, sides are in ratio 1 : √3 : 2; in 45°–45°–90°, 1 : 1 : √2.

Angle bisector theorem: BD/DC = AB/AC when AD bisects angle A.

Median divides a triangle into two equal-area triangles; centroid divides each median in 2 : 1 from the vertex.

Construction steps must list: given data, drawn figure with measurements, construction steps, and justification.

Cylinder: curved surface 2πrh, total surface 2πr(r + h), volume πr²h.

Cone: slant l = √(r² + h²), curved surface πrl, total surface πr(l + r), volume (1/3)πr²h.

Sphere: surface 4πr², volume (4/3)πr³; hemisphere: curved 2πr², total 3πr², volume (2/3)πr³.

Frustum: volume (1/3)πh(R² + Rr + r²), slant l = √[h² + (R − r)²], curved surface πl(R + r).

Cube: total surface 6a², volume a³, diagonal a√3; cuboid: surface 2(lb + bh + hl), volume lbh.

When a solid is melted and recast, volume is conserved; equate initial and final volumes to find the new dimension.

Sphere inscribed in a cylinder of same radius: sphere volume is 2/3 of cylinder volume.

1 m³ = 1000 litres; 1 litre = 1000 cm³; convert units before applying formulas.

Mean of grouped data: x̄ = Σfᵢxᵢ / Σfᵢ; use assumed mean or step-deviation for large values.

Median of grouped data: l + [(N/2 − cf)/f] × h, where l is lower limit of median class, cf cumulative frequency before it, f its frequency, h class width.

Mode of grouped data: l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.

Empirical relation: Mode = 3 Median − 2 Mean.

Probability of event E: P(E) = favourable outcomes / total equally likely outcomes; 0 ≤ P(E) ≤ 1.

P(not E) = 1 − P(E); P(E) + P(not E) = 1.

For a sure event P = 1; for an impossible event P = 0; sum of probabilities of all elementary outcomes is 1.

Cumulative frequency curve (ogive) gives median at the x-value where cumulative frequency equals N/2.

Verify a result by substitution or by checking a limiting case (e.g. set a variable to 0 or 1).

For 'find the number' problems, translate each clause into an equation, then solve and check the answer satisfies the original wording.

Work-rate: if A finishes in a days and B in b days, together they finish in ab/(a + b) days.

Percentage change: successive changes of x% and y% give net change x + y + xy/100 percent.

Time–speed–distance: distance = speed × time; for equal distances, average speed = 2v₁v₂/(v₁ + v₂).

Ratio problems: express each quantity as a multiple of a common factor k, then use the given total or difference to find k.

For 'two-digit number' puzzles, write the number as 10a + b and the reversed number as 10b + a.

Eliminate options by checking units, parity, and magnitude before doing full algebra.