Study Guide & Educational Overview
### Powerful Mental Calculation Heuristics
Speed arithmetic relies on algebraic identities and number decomposition rather than mechanical column calculation:
- **Squaring Numbers Ending in 5:** To calculate $(10n + 5)^2$, compute \times (n + 1)$ and append $25$. For example, $65^2 = (6 \times 7)\text{ followed by }25 = 4225$.
- **Reversible Percentages:** \%\text{ of }y = y\%\text{ of }x$. Computing $16\%\text{ of }75$ is identical to $75\%\text{ of }16 = \frac{3}{4} \times 16 = 12$.
- **Difference of Two Squares:** Multiplying symmetrical pairs: $48 \times 52 = (50 - 2)(50 + 2) = 50^2 - 2^2 = 2500 - 4 = 2496$.
- **Divisibility Tests:** An integer is divisible by 4 if its last two digits form a multiple of 4; divisible by 3 (or 9) if the sum of its digits is divisible by 3 (or 9); and divisible by 11 if the alternating sum of its digits is a multiple of 11.
Speed arithmetic relies on algebraic identities and number decomposition rather than mechanical column calculation:
- **Squaring Numbers Ending in 5:** To calculate $(10n + 5)^2$, compute \times (n + 1)$ and append $25$. For example, $65^2 = (6 \times 7)\text{ followed by }25 = 4225$.
- **Reversible Percentages:** \%\text{ of }y = y\%\text{ of }x$. Computing $16\%\text{ of }75$ is identical to $75\%\text{ of }16 = \frac{3}{4} \times 16 = 12$.
- **Difference of Two Squares:** Multiplying symmetrical pairs: $48 \times 52 = (50 - 2)(50 + 2) = 50^2 - 2^2 = 2500 - 4 = 2496$.
- **Divisibility Tests:** An integer is divisible by 4 if its last two digits form a multiple of 4; divisible by 3 (or 9) if the sum of its digits is divisible by 3 (or 9); and divisible by 11 if the alternating sum of its digits is a multiple of 11.
Frequently Asked Questions
Try to solve each challenge purely mentally first to build mental processing speed, but scratch paper is always welcome.
Questions range from fast 15% tips to multi-step mental algebraic shortcuts and squaring tricks.