Approximations, Euclid's Algorithm & Continued Fractions

${43\over 30} = 1+ {1\over\displaystyle 2+ { 1 \over \displaystyle 3+ { 1\over \displaystyle 4}}}.$

${43\over 20} = 2+ {1\over\displaystyle 6+ { 1 \over \displaystyle 1+ { 1\over \displaystyle 2}}}.$

${43\over 10} = 4+ {1\over\displaystyle 3+ { 1 \over \displaystyle 3}}.$