Question

The digits of a three-digit number from a G.P. If 400 is subtracted from it, then we get another three-digit number whose digits form an arithmetic series. What is the sum of these two numbers?
A. 1356
B. 1648
C. 1462
D. 1000

Answer 1

Hint: We need to form the G.P. form for three assumed numbers x, y, z using G.P. formula of $xz={{y}^{2}}$. Then we use trial and error methods to find the possible choices for y and try to find the values for x and z. We subtract 400 from the number and find if the new number forms an A.P. we find the sum of the two numbers.

Complete step-by-step answer:
It’s given that the digits of a three-digit number from a G.P. Now when we are subtracting 400 from it, we are actually keeping the unit and tenth place digit intact. We are just subtracting 4 from the hundredth placed digit as $400=4\times 100+0\times 10+0\times 1$.
Let the digits of a three-digit number be x, y and z for unit, tenth and hundredth placed digit respectively and the number be $100z+10y+x$ where x, y and z are in G.P. we know for three numbers to be in G.P. we can say that $xz={{y}^{2}}$.
Now if the right hand side is square then the value of xz also has to be square.
Here we are going to use a trial and error method to find the value of y. We first take $y=1$.
Then we get $xz={{1}^{2}}=1$. The only options will be both be 1. The number will be 111. We can’t subtract 400 from 111. So, this number is a wrong assumption.
We now take $y=2$.Then we get $xz={{2}^{2}}=4$. The options will be $1,4$ and $2,2$. The numbers will be 421 or 222. We can’t subtract 400 from 222. And if we subtract 400 from 421, then we get $421-400=21$ which is not a three-digit number. So, this assumption is wrong.
We now take $y=3$.Then we get $xz={{3}^{2}}=9$. The options will be $1,9$ and $3,3$. The numbers will be 931 or 333. We can’t subtract 400 from 333. And if we subtract 400 from 931, then we get $931-400=531$ which is a three-digit number and the digits 1, 3, 5 are in A.P. with a common difference $5-3=3-1=2$ So, this assumption is right.
So, the number is 931. After subtracting 400 we get $931-400=531$.
Their sum is $931+531=1462$.
The correct option is C.

So, the correct answer is “Option C”.

Note: We can’t form the whole problem assuming the common ratio and common difference as there is very little information given about them to form equations to solve the unknowns. We need to try and solve using assumption as the possible range for the digits are only 0 to 9.