Question

Two scales of equal length are divided into 162 and 209 equal parts respectively; if their zero points are coincident. Show that the ${31}^{st}$ division of one nearly coincides with the ${40}^{th}$ division of the other.

Answer 1

Hint: To show that ${31}^{st}$ division nearly coincides with ${40}^{th}$ division of the other, show that the ratio of ${31}^{st}$ division of one scale and ${40}^{th}$ division of second scale is approximately 1.

Complete step-by-step answer:
Let, the ${1}^{st}$ scale be x and ${2}^{nd}$ scale be y. As, given in question ${1}^{st}$ scale is divided into 162 parts, that is, length of each part is \[\left( \dfrac{x}{162} \right)\] and it is also given that ${2}^{nd}$ scale is divided into 209 parts, that is, length of each part of second scale is \[\left( \dfrac{y}{209} \right)\].
Now, if we consider the length ${31}^{st}$ part of ${1}^{st}$ scale, then it is equal to \[\left( \dfrac{31x}{162} \right)\] and the length of ${40}^{th}$ part of ${2}^{nd}$ scale is \[\left( \dfrac{40y}{209} \right)\].
We will approach this question by contradiction. So, let us assume that the ratio of ${31}^{st}$ part of ${1}^{st}$ scale and ${40}^{th}$ part of ${2}^{nd}$ scale do not coincide.
Therefore, we can write \[\left( \dfrac{31x}{162} \right)\ne \left( \dfrac{40y}{209} \right)\]
\[\Rightarrow \left( \dfrac{x}{y} \right)\ne \left( \dfrac{40\times 162}{31\times 209} \right)\]
\[\Rightarrow \left( \dfrac{x}{y} \right)\ne \left( \dfrac{6480}{6479} \right)\]
\[\Rightarrow \left( \dfrac{x}{y} \right)\ne \left( 1.0001 \right)\]
But the ratio of x and y is approximately 1, which contradicts the assumption. Therefore, the ${31}^{st}$ part of the ${1}^{st}$ scale nearly coincides with the ${40}^{th}$ part of the ${2}^{nd}$ scale.
Hence proved.

Note: Sometimes one might get confused with the\[\left( \dfrac{x}{y} \right)\] part. The \[\left( \dfrac{x}{y} \right)\] do not depicts the general ratio of length of the scales, because we have considered ${31}^{st}$ and ${40}^{th}$ ratio and after simplification we have obtained \[\left( \dfrac{x}{y} \right)=\left( 1.0001 \right)\].