Doing addition and multiplication between the numbers of the set $S$ to find $100$ and $75$. Consider the set $S=\{0.7,1.4,3.5,7,14,17.5,35,52.5,70\}.$
Now play with these numbers by doing addition and multiplication and we have to get  $100$ and $75$. Subtraction and division are not legal. 
Like: $14 \times 7 =98$, but I cannot find $2$ to add with $98$ to get $100$. Similarly I cannot get $75$ also only by multiplication between these numbers and addition between them.
Please help me to solve this. 
 A: I don't believe this is possible. Rewrite the question, multiplying everything by $10$, and we get $S = \{7, 14, 35, 70, 140, 175, 350, 525, 700\}$. Now we're trying to add and multiply to get $1000$ and $750$. Every integer in $S$ is divisible by $7$. So adding or multiplying any two of them results in another multiple of $7$. Neither $1000$ nor $750$ are divisible by $7$, so they cannot be obtained in this way.
EDIT: As fleablood mentions below, my answer is only valid once we show that $7$ divides $m$ if and only if $7$ divides $10^k\cdot m$, and that if $7$ divides both $10m$ and $10n$, then $7$ divides $10^j\cdot m + 10^k\cdot n$. Then powers of $10$ can be safely ignored when they arise in combining multiplications and additions.
The first statement is immediate from the fact that $10^k$ factorises into primes $(2\cdot 5)^k$, meaning that $10^k\cdot m$ is divisible by $7$ when and only when $m$ is.
Now we prove the second statement. Suppose $7$ divides both $10m$ and $10n$. Then
$$10^j\cdot m + 10^k\cdot n = 10^{j-1}(7r) + 10^{k-1}(7s)$$
for some integers $r$ and $s$. Rewriting this as $7\cdot(10^{j-1}r + 10^{k-1}s)$, it becomes clear that the sum was divisible by $7$.
But fleablood's full answer is quite a bit better than mine. Working with all the elements multiplied by $10$ introduces unneeded complexity.
A: All these numbers are an integer multiples of $0.7$.  So any sum or product will be an integer multiple $0.7^k$ for some positive integer $k$.
$75$ and $100$ are not such numbers.
If $75= m*(.7)^k$ then $75*10^k = m*7^k$ but $7$ is prime and divides neither $75$ or $10$ so it can not divide $75*10^k$.
Similarly if $100 = n*(.7)^k$ then we'd have $100*10^k=n*7^k$ but $7|100*10^k$ is impossible.
