`\color{#9400D3}{-k} \color{#800080}{o} \color{#4B0082}{r} \color{#00008B}{e} \color{#0000CD}{a} \color{#0000FF}{a} \color{#1E90FF}{a-}`

`(19/1 + 18/2 + ... + 2/18 + 1/19) / (1/2 + 1/3 + 1/4 + ... + 1/20)`

`= ((20 - 1)/1 + (20 - 2)/2 + (20 - 3)/3 + ... + (20 - 19)/19) / ((1/1 + 1/2 + 1/3 + ... + 1/20) - 1 )`

`=( 20/1 - 1 + 20/2 - 1 + 20/3 - 1 + ... + 20/19 - 1) / ((1/1 + 1/2 + 1/3 + ... + 1/20) - 1) `

`= ((20/1 + 20/2 + 20/3 + ... + 20/19) - 19) / ((1/1 + 1/2 + 1/3 + ... + 1/20) - 1) `

`= 20.(1/1 + 1/2 + 1/3 + ... + 1/19) - 19 / (1/1 + 1/2 + 1/3 + ... + 1/20) - 1 `

`= (20A - 19) / (A - 1)` với `A = 1 + 1/2 + ... + 1/20 `

`= (20A - 20) / (A - 1) `

`= 20`