Question

Three ropes are $7\text{ m}$, $12\text{ m }95\text{ cm}$, $3\text{ m }85\text{ cm}$ long. What is the greatest possible length that can be used to measure these ropes?

Answer 1

Hint: In this problem we need to find the greatest possible length that can be measured by the given ropes that means we need to calculate the Highest Common Factor of the given lengths. We can observe that the given lengths are not in the same units, that means some they are given in meters and some of them are given in centimeters. So, we will first convert all the lengths into any one unit probably in centimeters. For this we will use the relation $1\text{ m}=100\text{ cm }$. By using this relation, we will convert all the given lengths in centimeters and also, we will write the lengths in prime factorization form. From the prime factorization form we will calculate the HCF by multiplying the common factors in all the numbers.

Complete step by step solution:
Given lengths are $7\text{ m}$, $12\text{ m }95\text{ cm}$, $3\text{ m }85\text{ cm}$.

Considering the length $7\text{ m}$. Converting this length by using the relation $1\text{ m}=100\text{ cm }$, then we will get
$\begin{align}
  & 7\text{ m}=7\times 100\text{ cm} \\
 & \Rightarrow 7\text{ m}=700\text{ cm} \\
\end{align}$

Considering the length $12\text{ m }95\text{ cm}$. Converting this length by using the relation $1\text{ m}=100\text{ cm }$, then we will get
$\begin{align}
  & 12\text{ m }95\text{ cm}=12\times 100\text{ cm}+95\text{ cm} \\
 & \Rightarrow 12\text{ m }95\text{ cm}=1200\text{ cm}+95\text{ cm} \\
 & \Rightarrow 12\text{ m }95\text{ cm}=1295\text{ cm} \\
\end{align}$

Considering the length $3\text{ m }85\text{ cm}$. Converting this length by using the relation $1\text{ m}=100\text{ cm }$, then we will get
$\begin{align}
  & 3\text{ m }85\text{ cm}=3\times 100\text{ cm}+85\text{ cm} \\
 & \Rightarrow 3\text{ m }85\text{ cm}=300\text{ cm}+85\text{ cm} \\
 & \Rightarrow 3\text{ m }85\text{ cm}=385\text{ cm} \\
\end{align}$

Hence the converted lengths are $700\ \text{cm}$, $1295\text{ cm}$, $385\text{ cm}$.

Considering the value $700$. The prime factorization of the number $700$ is given by
$\begin{align}
  & 2\left| \!{\underline {\,
  700 \,}} \right. \\
 & 2\left| \!{\underline {\,
  350 \,}} \right. \\
 & 5\left| \!{\underline {\,
  175 \,}} \right. \\
 & 5\left| \!{\underline {\,
  35 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
From this we can write the number $700$ as
$700=2\times 2\times 5\times 5\times 7$

Considering the value $1295$. The prime factorization of the number $1295$ is given by
$\begin{align}
  & 5\left| \!{\underline {\,
  1295 \,}} \right. \\
 & 7\left| \!{\underline {\,
  259 \,}} \right. \\
 & 37\left| \!{\underline {\,
  37 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
From this we can write the number $1295$ as
$1295=5\times 7\times 37$

Considering the value $385$. The prime factorization of the number $385$ is given by
$\begin{align}
  & 5\left| \!{\underline {\,
  385 \,}} \right. \\
 & 7\left| \!{\underline {\,
  77 \,}} \right. \\
 & 11\left| \!{\underline {\,
  11 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
From this we can write the number $385$ as
$385=5\times 7\times 11$

We can observe that the factors $5$, $7$ are common in all the given lengths. So, the HCF of the given lengths will become
$\begin{align}
  & \text{HCF}=5\times 7 \\
 & \Rightarrow \text{HCF}=35 \\
\end{align}$

Hence the greatest possible length that can be measured by using the given ropes is $35\text{ cm}$.

Note: In this problem we have asked to calculate the greatest possible length that can be measured by using the given ropes, so we have calculated the HCF of the given lengths. In some cases, they may ask to calculate the least possible length that can be measured by using the given ropes, then we need to calculate the LCM (Least Common Multiple) of the given lengths.