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A man bought a certain number of oranges, out of which 13 percent were rotten. He gave \[75\% \] of the remaining in charity and still has 522 oranges left. Find how many had he bought?
(a) 2,000
(b) 2,400
(c) 2,500
 (d) 2,250

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Last updated date: 25th Jun 2024
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Answer
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Hint: Here, we have to find the number of oranges the man bought. We will assume the number of oranges bought be \[x\]. We will use the given information to form a linear equation in terms of \[x\]. Then, we will solve the obtained equation to get the value of \[x\], and hence, the number of oranges bought by the man.

Complete step-by-step answer:
Let the number of oranges bought be \[x\].
We will apply the stated operations on \[x\] to form an equation in terms of \[x\].
First, we know that \[13\% \] of the oranges bought were rotten.
Therefore, we get
Number of rotten oranges \[ = 13\% {\rm{ of }}x = \dfrac{{13}}{{100}} \times x = \dfrac{{13x}}{{100}}\]
We know that the number of remaining oranges is the difference in the number of oranges bought, and the number of rotten oranges.
Therefore, we get
Number of remaining oranges \[ = x - \dfrac{{13x}}{{100}}\]
Taking the L.C.M., we get
Number of remaining oranges \[ = \dfrac{{100x - 13x}}{{100}}\]
Subtracting the like terms, we get
Number of remaining oranges \[ = \dfrac{{87x}}{{100}}\]
Now, the man gave away \[75\% \] of the remaining oranges as charity.
Therefore, we get
Number of oranges given away as charity \[ = 75\% \]of Remaining Oranges \[ = \dfrac{{75}}{{100}} \times \dfrac{{87x}}{{100}}\]
Multiplying the terms of the expression, we get
Number of oranges given away as charity \[ = \dfrac{{6525x}}{{10000}} = \dfrac{{261x}}{{400}}\]
We know that the number of oranges left with the man is the difference in the number of oranges remaining after removing the rotten oranges, and the number of oranges given away as charity.
Therefore, we get
Number of oranges left with the man \[ = \dfrac{{87x}}{{100}} - \dfrac{{261x}}{{400}}\]
Taking the L.C.M., we get
Number of oranges left with the man \[ = \dfrac{{348x - 261x}}{{400}}\]
Subtracting the like terms, we get
Number of oranges left with the man \[ = \dfrac{{87x}}{{400}}\]
Finally, we know that the number of oranges left with the man is 522.
Therefore, we get
\[ \Rightarrow \dfrac{{87x}}{{400}} = 522\]
Multiplying both sides by 400, we get
\[\begin{array}{l} \Rightarrow \dfrac{{87x}}{{400}} \times 400 = 522 \times 400\\ \Rightarrow 87x = 208800\end{array}\]
Dividing both sides of the equation by 87, we get
\[\begin{array}{l} \Rightarrow \dfrac{{87x}}{{87}} = \dfrac{{208800}}{{87}}\\ \Rightarrow x = 2400\end{array}\]
Therefore, the number of oranges the man bought is 2,400.
The correct option is option (b).

Note: In the question it is given that \[75\% \] of remaining oranges was given to the charity. Here, remaining means the good oranges that remained after separating the rotten oranges. And \[75\% \] is sent to charity from the good oranges and not the total oranges. We can make a mistake by using the number of oranges remaining with the man as \[\dfrac{{261x}}{{400}}\] instead of \[\dfrac{{87x}}{{100}} - \dfrac{{261x}}{{400}}\].