One main scale division of a vernier callipers is $a$ $cm$ and $n ^{\text {th }}$ division of the vernier scale coincide with $( n -1)^{\text {th }}$ division of the main scale. The least count of the callipers in $mm$ is
A$\frac{10 na }{( n -1)}$
B$\frac{10 a }{( n -1)}$
C$\left(\frac{ n -1}{10 n }\right) a$
D$\frac{10 a }{ n }$
JEE MAIN 2021, Medium
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D$\frac{10 a }{ n }$
d $(n-1) a=n\left(a^{\prime}\right)$
$a^{\prime}=\frac{(n-1) a}{n}$
$\therefore L \cdot C \cdot=1 M S D-1 VSD$
$=\left(a-a^{\prime}\right) c m$
$=a-\frac{(n-1) a}{n}$
$=\frac{n a-n a+a}{n}=\frac{a}{n} c m$
$=\left(\frac{10 a}{n}\right) m m$
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