Question

There are \[3\] strings of different length which together is $35cm$. The smallest is $\dfrac{2}{3}$ of the middle one which is $\dfrac{4}{5}$ of the longest string. What is the length of the longest string?
A.$12cm$
B.$22cm$
C.$15cm$
D.$18cm$

Answer 1

Hint: Here, in the given question, we are given two statements and we need to solve these to find the length of the longest string. We can let the required numbers equal to variables (A variable is a symbol usually a letter) standing for unknown numerical values in an equation), then make an equation according to the statement. By solving this equation, we will get the value of the variable and thus the required answer.

Complete answer:
We are given that there are \[3\] strings of different lengths which together is $35cm$. The smallest is $\dfrac{2}{3}$ of the middle one which is $\dfrac{4}{5}$ of the longest string.
We are required to find the number that satisfies the given statement. Let the length of smallest string be $xcm$, the length of the middle string be $ycm$ and the length of the longest string be $zcm$.
Length of three strings be,
$ \Rightarrow x + y + z = 35cm.......\left( i \right)$
We have, the smallest is $\dfrac{2}{3}$ of the middle.
$ \Rightarrow x = \dfrac{2}{3} \times y$
We have, the middle string is $\dfrac{4}{5}$ of the longest string.
$ \Rightarrow y = \dfrac{4}{5} \times z$
Now we have, $x = \dfrac{{2y}}{3}$ and $y = \dfrac{{4z}}{5}$.
We can also write $x$ in terms of $z$ by substituting the value of $y$ in $x$.
$ \Rightarrow x = \dfrac{2}{3} \times \dfrac{{4z}}{5}$
$ \Rightarrow x = \dfrac{{8z}}{{15}}$
On equating value of $x$ and $y$ in equation $\left( i \right)$, we get
$ \Rightarrow \dfrac{{8z}}{{15}} + \dfrac{{4z}}{5} + z = 35$
Now we will take LCM on the left hand side of the equation
$ \Rightarrow \dfrac{{8z + 12z + 15z}}{{15}} = 35$
$ \Rightarrow \dfrac{{35z}}{{15}} = 35$
On cross multiplication, we get
$ \Rightarrow 35z = 35 \times 15$
Divide both sides by $35$
$ \Rightarrow \dfrac{{35z}}{{35}} = \dfrac{{35 \times 15}}{{35}}$
$ \Rightarrow z = 15cm$
Hence, the length of the longest string is $15cm$.

Therefore, the correct option is C.

Note:
Remember that OF in algebra means to multiply and BY in algebra means to divide. Remember that if we add the same number to both sides, or subtract the same number from both sides, or multiply both sides by the same non-zero number, or divide both sides by the same non-zero number, the balance remains undisturbed. Transporting means moving from one side to the other. When a term is transposed from one side of the equation to the other side, its sign gets changed. We should take care of the calculations so as to be sure of our final answer.