SECTION NOTES: MENTAL MATH QUICK RECALL
FRACTION TO DECIMAL CONVERSIONS ($\dfrac{1}{2}$ TO $\dfrac{1}{50}$)
$\dfrac{1}{2} = 0.5$
$\dfrac{1}{3} = 0.3333$
$\dfrac{1}{4} = 0.25$
$\dfrac{1}{5} = 0.2$
$\dfrac{1}{6} = 0.1667$
$\dfrac{1}{7} = 0.1429$
$\dfrac{1}{8} = 0.125$
$\dfrac{1}{9} = 0.1111$
$\dfrac{1}{10} = 0.1$
$\dfrac{1}{11} = 0.0909$
$\dfrac{1}{12} = 0.0833$
$\dfrac{1}{13} = 0.0769$
$\dfrac{1}{14} = 0.0714$
$\dfrac{1}{15} = 0.0667$
$\dfrac{1}{16} = 0.0625$
$\dfrac{1}{17} = 0.0588$
$\dfrac{1}{18} = 0.0556$
$\dfrac{1}{19} = 0.0526$
$\dfrac{1}{20} = 0.05$
$\dfrac{1}{21} = 0.0476$
$\dfrac{1}{22} = 0.0455$
$\dfrac{1}{23} = 0.0435$
$\dfrac{1}{24} = 0.0417$
$\dfrac{1}{25} = 0.04$
$\dfrac{1}{26} = 0.0385$
$\dfrac{1}{27} = 0.0370$
$\dfrac{1}{28} = 0.0357$
$\dfrac{1}{29} = 0.0345$
$\dfrac{1}{30} = 0.0333$
$\dfrac{1}{31} = 0.0323$
$\dfrac{1}{32} = 0.0312$
$\dfrac{1}{33} = 0.0303$
$\dfrac{1}{34} = 0.0294$
$\dfrac{1}{35} = 0.0286$
$\dfrac{1}{36} = 0.0278$
$\dfrac{1}{37} = 0.0270$
$\dfrac{1}{38} = 0.0263$
$\dfrac{1}{39} = 0.0256$
$\dfrac{1}{40} = 0.025$
$\dfrac{1}{41} = 0.0244$
$\dfrac{1}{42} = 0.0238$
$\dfrac{1}{43} = 0.0233$
$\dfrac{1}{44} = 0.0227$
$\dfrac{1}{45} = 0.0222$
$\dfrac{1}{46} = 0.0217$
$\dfrac{1}{47} = 0.0213$
$\dfrac{1}{48} = 0.0208$
$\dfrac{1}{49} = 0.0204$
$\dfrac{1}{50} = 0.02$
COMMON NON-UNIT RATIOS (CRITICAL FOR CALCULATIONS)
These extend the list above beyond 1/n to the other fraction/decimal conversions that show up constantly in physics problems.
HALVES
$\dfrac{3}{2} = 1.5$
$\dfrac{5}{2} = 2.5$
$\dfrac{7}{2} = 3.5$
$\dfrac{9}{2} = 4.5$
THIRDS
$\dfrac{2}{3} = 0.6667$
$\dfrac{4}{3} = 1.3333$
$\dfrac{5}{3} = 1.6667$
$\dfrac{7}{3} = 2.3333$
$\dfrac{8}{3} = 2.6667$
QUARTERS
$\dfrac{3}{4} = 0.75$
$\dfrac{5}{4} = 1.25$
$\dfrac{7}{4} = 1.75$
$\dfrac{9}{4} = 2.25$
$\dfrac{11}{4} = 2.75$
FIFTHS
$\dfrac{2}{5} = 0.4$
$\dfrac{3}{5} = 0.6$
$\dfrac{4}{5} = 0.8$
$\dfrac{6}{5} = 1.2$
$\dfrac{7}{5} = 1.4$
$\dfrac{8}{5} = 1.6$
$\dfrac{9}{5} = 1.8$
SIXTHS
$\dfrac{5}{6} = 0.8333$
$\dfrac{7}{6} = 1.1667$
$\dfrac{11}{6} = 1.8333$
SEVENTHS
$\dfrac{2}{7} = 0.2857$
$\dfrac{3}{7} = 0.4286$
$\dfrac{4}{7} = 0.5714$
$\dfrac{5}{7} = 0.7143$
$\dfrac{6}{7} = 0.8571$
EIGHTHS
$\dfrac{3}{8} = 0.375$
$\dfrac{5}{8} = 0.625$
$\dfrac{7}{8} = 0.875$
$\dfrac{9}{8} = 1.125$
$\dfrac{11}{8} = 1.375$
NINTHS
$\dfrac{2}{9} = 0.2222$
$\dfrac{4}{9} = 0.4444$
$\dfrac{5}{9} = 0.5556$
$\dfrac{7}{9} = 0.7778$
$\dfrac{8}{9} = 0.8889$
TENTHS
$\dfrac{3}{10} = 0.3$
$\dfrac{7}{10} = 0.7$
$\dfrac{9}{10} = 0.9$
TWELFTHS
$\dfrac{5}{12} = 0.4167$
$\dfrac{7}{12} = 0.5833$
$\dfrac{11}{12} = 0.9167$
PI APPROXIMATIONS
$\dfrac{22}{7} = 3.1429$
$\dfrac{355}{113} = 3.1416$ (very precise)
WHERE THESE SHOW UP
Specific heat ratio $\gamma$: monoatomic gas $=\dfrac{5}{3}=1.6667$; diatomic gas $=\dfrac{7}{5}=1.4$
Moment of inertia coefficients: solid sphere $=\dfrac{2}{5}=0.4$; hollow sphere $=\dfrac{2}{3}=0.6667$; disk/solid cylinder $=\dfrac{1}{2}=0.5$; rod about centre $=\dfrac{1}{12}=0.0833$; rod about end $=\dfrac{1}{3}=0.3333$
FROM SCANNING ACTUAL PAST CONTEST QUESTIONS (CHEMISTRY + PHYSICS)
Every fraction/decimal below was confirmed appearing directly inside a QUESTION (not just a solution) across the chemistry and physics past-contest database, and recurs at least 3 times.
TWENTIETHS (denominator 20)
$\dfrac{1}{20} = 0.05$
$\dfrac{3}{20} = 0.15$
$\dfrac{7}{20} = 0.35$
$\dfrac{9}{20} = 0.45$
$\dfrac{21}{20} = 1.05$
$\dfrac{27}{20} = 1.35$
$\dfrac{29}{20} = 1.45$
TWENTY-FIFTHS (denominator 25)
$\dfrac{1}{25} = 0.04$
$\dfrac{2}{25} = 0.08$
$\dfrac{3}{25} = 0.12$
$\dfrac{4}{25} = 0.16$
$\dfrac{6}{25} = 0.24$
$\dfrac{9}{25} = 0.36$
$\dfrac{11}{25} = 0.44$
$\dfrac{12}{25} = 0.48$
$\dfrac{19}{25} = 0.76$
$\dfrac{23}{25} = 0.92$
$\dfrac{34}{25} = 1.36$
$\dfrac{36}{25} = 1.44$
$\dfrac{76}{25} = 3.04$
(7/25 and 24/25 are covered above as the 7-24-25 triple.)
TENTHS — EXTENDED
$\dfrac{11}{10} = 1.1$
$\dfrac{13}{10} = 1.3$
$\dfrac{17}{10} = 1.7$
$\dfrac{19}{10} = 1.9$
$\dfrac{21}{10} = 2.1$
$\dfrac{23}{10} = 2.3$
$\dfrac{27}{10} = 2.7$
$\dfrac{31}{10} = 3.1$
$\dfrac{49}{10} = 4.9$
$\dfrac{81}{10} = 8.1$
FIFTHS — EXTENDED
$\dfrac{11}{5} = 2.2$
$\dfrac{12}{5} = 2.4$
$\dfrac{14}{5} = 2.8$
$\dfrac{16}{5} = 3.2$
$\dfrac{18}{5} = 3.6$
$\dfrac{21}{5} = 4.2$
$\dfrac{24}{5} = 4.8$
$\dfrac{27}{5} = 5.4$
$\dfrac{28}{5} = 5.6$
$\dfrac{29}{5} = 5.8$
$\dfrac{32}{5} = 6.4$
$\dfrac{36}{5} = 7.2$
$\dfrac{42}{5} = 8.4$
$\dfrac{48}{5} = 9.6$
$\dfrac{64}{5} = 12.8$
$\dfrac{112}{5} = 22.4$
HALVES — EXTENDED
$\dfrac{11}{2} = 5.5$
$\dfrac{13}{2} = 6.5$
$\dfrac{15}{2} = 7.5$
$\dfrac{25}{2} = 12.5$
$\dfrac{127}{2} = 63.5$
PHYSICS & CHEMISTRY CONSTANTS AS FRACTIONS
$g = \dfrac{49}{5} = 9.8 \text{ m/s}^2$ (and $\dfrac{g}{2}=\dfrac{49}{10}=4.9$, common in $s=\frac{1}{2}gt^2$-type questions)
Molar volume of a gas at STP $= \dfrac{112}{5} = 22.4 \text{ dm}^3$
Permittivity of free space $\varepsilon_0 \approx \dfrac{177}{20}\times10^{-12} = 8.85\times10^{-12} \text{ F/m}$
Relative atomic mass of copper $\approx \dfrac{127}{2} = 63.5$
$2\pi \approx 6.28$