Question

A boy was asked to multiply a given number by \[\dfrac{8}{{17}}\]. Instead, he divided the given number by \[\dfrac{8}{{17}}\] and got the result 225 more than what he should have got if he multiplied the number by \[\dfrac{8}{{17}}\]. The given number was
A) 8
B) 17
C) 64
D) 136

Answer 1

Hint:
Here, we will assume the required number to be some variable. Then we will form an equation based on the given information. We will solve the equation further to get the required number.

Complete Step by step Solution:
Let \[x\] be the required number.
It is given that the boy was supposed to multiply number \[x\]with \[\dfrac{8}{{17}}\].
So, correct result that the boy should get \[ = x \times \dfrac{8}{{17}}\]\[ = \dfrac{{8x}}{{17}}\]\[\] ………………………\[\left( 1 \right)\]
He divided the number \[x\] by \[\dfrac{8}{{17}}\] instead of multiplying.
So, wrong result that the boy got \[ = \dfrac{x}{{\dfrac{8}{{17}}}} = \dfrac{{17x}}{8}\] ………………………\[\left( 2 \right)\]
It’s given that he got result 225 more than what he should have gotten if he had multiplied \[\dfrac{8}{{17}}\] with the number \[x\].
Therefore, we can write the equation as
Equation \[\left( 2 \right)\]\[ = \] Equation \[\left( 1 \right)\] \[ + 225\]
On substituting the value of Equation \[\left( 2 \right)\] and Equation \[\left( 1 \right)\], we get
\[ \Rightarrow \dfrac{{17x}}{8} = 225 + \dfrac{{8x}}{{17}}\]
Subtracting the like terms, we get
\[ \Rightarrow \dfrac{{17x}}{8} - \dfrac{{8x}}{{17}} = 225\]
Now taking L.C.M on left hand side of the equation, we get
\[\begin{array}{l} \Rightarrow \dfrac{{{{(17)}^2}x - {{(8)}^2}x}}{{17 \times 8}} = 225\\ \Rightarrow x \times (\dfrac{{{{(17)}^2} - {{(8)}^2}}}{{17 \times 8}}) = 225\end{array}\]
Using the formula \[\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)\], we get
\[\begin{array}{l} \Rightarrow x \times \dfrac{{(17 + 8)(17 - 8)}}{{17 \times 8}} = 225\\ \Rightarrow x \times \dfrac{{25 \times 9}}{{17 \times 8}} = 225\end{array}\]
On cross multiplication, we get
\[ \Rightarrow x = \dfrac{{225 \times 17 \times 8}}{{25 \times 9}}\]
Simplifying the expression, we get
\[ \Rightarrow x = 136\]
Therefore, the given number was 136.

Hence, option (d) is correct.

Note:
Here, we have formed a linear equation based on the information given in the question. A linear equation is an equation which has the highest degree of variable as 1 and has only one solution. As we had one linear equation, so we got only one solution. We need to be careful while forming the equation because instead of subtracting the correct answer from the wrong answer of the boy we might add it. This will give us the wrong answer.