Formulas & Methods

Study Sheet

Formulas & Methods

Every formula the AMC 8 uses, with the methods that apply them

Arithmetic, Ratios & Percents

Tip
Percents and ratios

Percent change =new−oldold= \tfrac{\text{new} - \text{old}}{\text{old}}; a p%p\% increase then p%p\% decrease leaves (1−p21002)(1 - \tfrac{p^2}{100^2}) of the start. Ratio a:b:ca : b : c with total TT gives parts aTa+b+c\tfrac{aT}{a+b+c}. "a%a\% of bb" equals "b%b\% of aa".

Convert every percent to a multiplier (30%30\% off is ×0.7\times0.7) and chain multipliers for successive changes — never add percents of different bases. Ratio problems: assign kk to one part and solve for kk from the given total or difference.

Tip
Rates, work, and averages

Distance =rt= rt. Average speed =total distancetotal time= \tfrac{\text{total distance}}{\text{total time}} — the harmonic mean for equal distances (2v1v2v1+v2\tfrac{2v_1v_2}{v_1 + v_2}). Work rates ADD: 1t1+1t2=1t\tfrac{1}{t_1} + \tfrac{1}{t_2} = \tfrac{1}{t}. Mean =sumcount= \tfrac{\text{sum}}{\text{count}}; changing the mean by removing one item: removed =old total−new total= \text{old total} - \text{new total}.

Set up a table with columns rate, time, amount. For "how many of each" mixture problems, use one variable and the total. For averages, always work with TOTALS, not averages.

Number Theory & Counting

Tip
Divisibility, primes, LCM/GCD

Divisible by 33/99: digit sum; by 44/88: last two/three digits; by 1111: alternating digit sum. gcd⁡⋅lcm=ab\gcd\cdot\mathrm{lcm} = ab. Number of divisors of paqbp^aq^b: (a+1)(b+1)(a+1)(b+1). Remainder cycles: last digits of powers repeat with period ≤4\le 4.

Factor completely before anything else. Units-digit questions: find the cycle length, reduce the exponent. "Smallest number leaving remainders…": list one sequence and filter by the other condition.

Tip
Counting and probability

Multiplication principle for independent choices; permutations n!n!, with repeats n!a!b!\tfrac{n!}{a!b!}; combinations (nk)\binom nk. Complement: P(at least one)=1−P(none)P(\text{at least one}) = 1 - P(\text{none}). Probability =favorabletotal= \tfrac{\text{favorable}}{\text{total}} with equally likely outcomes.

Make the sample space explicit (a table for two dice, a tree for coins). Casework by a stated criterion. When "at least" appears, use the complement.

Geometry

Tip
Angles, triangles, circles

Triangle angles sum to 180∘180^\circ; polygon interior sum (n−2)180∘(n-2)180^\circ; exterior angles sum to 360∘360^\circ. Pythagorean triples 33-44-55, 55-1212-1313, 88-1515-1717, 77-2424-2525. Area: triangle 12bh\tfrac12bh, trapezoid (b1+b2)h2\tfrac{(b_1 + b_2)h}{2}, circle πr2\pi r^2, circumference 2πr2\pi r. Similar figures: lengths scale by kk, areas by k2k^2, volumes by k3k^3.

Draw to scale and label everything known. Look for right triangles to apply Pythagoras; for overlapping shapes, compute (whole) −- (parts). Coordinates: distance formula Δx2+Δy2\sqrt{\Delta x^2 + \Delta y^2}, midpoint (x1+x22,y1+y22)\left(\tfrac{x_1+x_2}{2}, \tfrac{y_1+y_2}{2}\right).

Tip
Clocks, calendars, patterns

Minute hand 6∘6^\circ/min, hour hand 0.5∘0.5^\circ/min. Days cycle mod 77. Arithmetic sequence: an=a1+(n−1)da_n = a_1 + (n-1)d, sum n(a1+an)2\tfrac{n(a_1 + a_n)}{2}; 1+2+⋯+n=n(n+1)21 + 2 + \cdots + n = \tfrac{n(n+1)}{2}.

For patterns, compute the first four terms and identify the rule before extrapolating; for cyclic patterns, find the period and reduce.

Concept
The universal problem-solving loop

Read twice. Restate the goal in your own words. List the givens. Pick a representation. Try small cases. Look for symmetry, an invariant, or an extremal object. Compute, then CHECK against a second method or a sanity bound.

On the AMC 8 and MathCounts the check is the whole strategy: answer choices can be tested, units can be verified, and a quick estimate ("about 3030, so 2828 not 280280") catches most slips. Draw every geometry problem to scale. Make a table for every counting problem. Write the equation before solving it.