Question

The ratio of length of Rope A to the length of rope B is $ 3:4 $ . The ratio of length of rope C to length of rope B is $ 7:6 $ . If the length of longest rope is 84cm, find the total length of all three ropes.
A.210cm
B.235cm
C.185cm
D.200cm

Answer 1

Hint: We are given the ratio of lengths and the length of longest rope. So, first of all we are going to find the longest rope and then suppose the ratio as x, y and solve the equations.

Complete step by step solution:
We are given three ropes A, B, C and the longest one of them has length 84cm.
The ratio of rope A to rope B $ = 3:4 $ .
Let the ratio be x.
Therefore, length of rope A $ = 3x $
And length of rope B $ = 4x $
Out of these two ropes, rope B is longer as x is multiplied with 4 and in rope A, x is multiplied with 3.
Now, the ratio of rope C to rope B $ = 7:6 $
Let the ratio be y.
Therefore, length of rope C $ = 7y $
And length of rope B $ = 6y $
Out of these two ropes, rope C is longer as y is multiplied with 7 and in rope B, y is multiplied with 6.
Therefore, rope C is the longest out of all three ropes.
Hence, length of rope C $ = 84cm $
Now, $ \dfrac{{length\;of\;rope\;C}}{{length\;of\;rope\;B}} = \dfrac{7}{6} $
\[
   \Rightarrow \dfrac{{84}}{B} = \dfrac{7}{6} \\
   \Rightarrow B = \dfrac{{84 \times 6}}{7} \\
   \Rightarrow B = 72 \\
 \]
Therefore, length of rope B is 72cm.
Now, $ \dfrac{{length\;of\;rope\;A}}{{length\;of\;rope\;B}} = \dfrac{3}{4} $
 $
   \Rightarrow \dfrac{A}{{72}} = \dfrac{3}{4} \\
   \Rightarrow A = \dfrac{{72 \times 3}}{4} \\
   \Rightarrow A = 54 \;
  $
Therefore, the length of rope A is 54cm.
Therefore, total length of all three ropes will be,
 $ A + B + C = 54 + 72 + 84 $
 $ = 210 $
Therefore the total length of all three ropes is 210 cm.
Hence, the answer is option A.
So, the correct answer is “Option A”.

Note: Whenever you are given a ratio of lengths in a question, you can always suppose the ratio as some variable and find out the individual length in terms of that variable. Doing so will give you some equations and solving them you can get the value of the individual term.