AMC 8 topics and formulas

• Percent change: new value $=$ old value $\times \left(1\pm \frac{p}{100}\right)$.

• Average: total $÷$ count, so total $=$ average $\times$ count. Many questions are solved by working with totals.

• Rates: distance $=$ speed $\times$ time, and work done $=$ rate $\times$ time.

• Sum of the first $n$ whole numbers: $\frac{n(n+1)}{2}$.

• Ratios: if $a:b=2:3$, write $a=2k$ and $b=3k$ and find $k$.

• Multiply the choices when they are made one after another.

• Arrangements of $n$ different things: $n!=n\times (n-1)\times \cdots \times 1$.

• Probability $=$ favourable outcomes $÷$ all equally likely outcomes.

• Count what you do not want and subtract, when that is shorter.

• Angles in a triangle add to $180^{\circ }$; the angles of an $n$-sided polygon add to $(n-2)\times 180^{\circ }$.

• Pythagorean theorem: $a^2+b^2=c^2$, with 3-4-5 and 5-12-13 worth recognising.

• Areas: triangle $\frac{1}{2}bh$, trapezium $\frac{1}{2}(a+b)h$, circle $\pi r^2$; circumference $2\pi r$.

• Volume of a box $lwh$ and of a cylinder $\pi r^{2}h$.

• A composite shape is a big shape minus the pieces that are not there.

• Divisibility: by 3 or 9 through the digit sum, by 4 through the last two digits, by 5 and 10 through the last digit.

• Every whole number is a product of primes in one way; factor first, then count or compare.

• $\gcd (a,b)\times \mathrm{lcm}(a,b)=ab$.

• Remainders repeat in cycles: to find the last digit of a large power, find the cycle.

Original practice tests written by Math Webb in the style of the contest, free to read, each question with its answer and a full worked solution.