Question

7 is added to a certain number; the sum is multiplied by 5, the product is divided by 9 and 3 is subtracted from the quotient. The remainder left is 12. The number is
A) 20
B) 30
C) 40
D) 60

Answer 1

Hint:
Here, we will assume the required number to be \[x\]. Then, we will follow the steps given in the question to get a linear expression. We will then equate this expression to the remainder and form a linear equation. We will solve the linear equation to find the required number.

Complete Step by step Solution:
Let the required number be \[x\].
According to the question, 7 is added to a certain number
Hence, we get the number as \[\left( {x + 7} \right)\]
Now, it is given that; the sum is multiplied by 5
So, after multiplying by 5, we have: \[5\left( {x + 7} \right)\]
Now, the product is divided by 9
Hence, dividing the product obtained by 9: \[\dfrac{{5\left( {x + 7} \right)}}{9}\]
In the next step, 3 is subtracted from the quotient
Therefore, we get: \[\dfrac{{5\left( {x + 7} \right)}}{9} - 3\]
Now, it is given that the remainder left is 12.
Therefore, equating the remainder obtained by 12, we get the equation:
\[\dfrac{{5\left( {x + 7} \right)}}{9} - 3 = 12\]
Taking LCM on the LHS, we get
\[ \Rightarrow \dfrac{{5\left( {x + 7} \right) - 27}}{9} = 12\]
Multiplying 9 on both the sides, we get
\[ \Rightarrow 5\left( {x + 7} \right) - 27 = 12 \times 9\]
Multiplying the terms, we get
\[ \Rightarrow 5x + 35 - 27 = 108\]
Subtracting the like terms, we get
\[ \Rightarrow 5x + 8 = 108\]
Subtracting 8 from both sides, we get
\[ \Rightarrow 5x = 108 - 8 = 100\]
Dividing both sides by 5, we get
\[ \Rightarrow x = 20\]
Therefore, the number is 20.

Hence, option A is the correct answer.

Note:
An alternate way of finding the required number is that we should go in the reverse order.
The given remainder left is 12.
It is given that 3 is subtracted from the quotient to get the remainder 12.
So, we will add 3 to the remainder, \[12 + 3 = 15\]
Before this step, the product was divided by 9. Hence, we will multiply our result by 9.
Therefore, we get,
\[15 \times 9 = 135\]
Before this, the sum was multiplied by 5,
So, we will divide by 5 and get
 \[\dfrac{{135}}{5} = 27\]
And finally, 7 was added to a certain number, hence we will subtract 7 to get the required number.
Therefore, we get
\[27 - 7 = 20\]
Therefore, the number is 20
Hence, option A is the correct answer.