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: $${\displaystyle \frac{5\left(x+7\right)}{9}}$$
In the next step, 3 is subtracted from the quotient
Therefore, we get: $${\displaystyle \frac{5\left(x+7\right)}{9}}-3$$
Now, it is given that the remainder left is 12.
Therefore, equating the remainder obtained by 12, we get the equation:
$${\displaystyle \frac{5\left(x+7\right)}{9}}-3=12$$
Taking LCM on the LHS, we get
$$\Rightarrow {\displaystyle \frac{5\left(x+7\right)-27}{9}}=12$$
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
 $${\displaystyle \frac{135}{5}}=27$$
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.