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# [Smart Math] Arithmetic Problem 3

Here’s and example of a SMART MATH problem for ARITHMETIC.

### Problem

Reduce the Expression:
$\frac{4.5x^{3}\times 15}{\sqrt{y}(x+y)^{2}}$ = $\frac{3^{3}\times 5\sqrt{4y}}{2x^{2}y+4xy^{2}+2y^{3}}$

1. $x^{3}\sqrt{y}$ = 2
2. $x^{3}$ = 2
3. $xy^{2}$ = 9
4. $\sqrt{xy}$ = 4
5. $x^{2}y$ = 10

### The Usual Method

$(4.5x^{3}\times 15)\times (2x^{2}y+4xy^{2}+2y^{3})$ = $(3^{3}\times 5\sqrt{4y})\times (\sqrt{y}(x+y)^{2})$

i.e. $135x^{5}y+270x^{4}y^{2}+135x^{3}y^{3}$ = $270y^{3}+270x^{2}y+540xy^{2}$

i.e. $135x^{3}y(x^{2}+2xy+y^{2})$ = $270y(x^{2}+2xy+y^{2})$

Canceling the common term
$(x^{2}+2xy+y^{2})$, we get
$135x^{3}y$ = $270y$

$\therefore x^{3}=2$

(Ans: 2)

Estimated Time to arrive at the answer = 100 seconds

### Using Technique

Just rewrite the expression as:

$(4.5x^{3}\times 15)\times (2x^{2}y+4xy^{2}+2y^{3})$ = $(3^{3}\times 5\sqrt{4y})\times (\sqrt{y}(x+y)^{2})$

Now observe the variable ‘y’ and its powers
LHS => $y,y^{2}\And y^{3}$

RHS => $y,y^{2}\And y^{3}$
(after opening the brackets)
This means that the variable ‘y’ cancels out and so the answer should have only the variable ‘x’. The only option that satisfies this condition is option ‘2’.

(Ans: 2)

Estimated Time to arrive at the answer = 10 seconds
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