Solving a Mathematical Ratio with Algebraic Manipulations

If we are given a/b 7/5, how can we solve for expressions involving a^2b^2 2ab/ [a^2b^2 – 2ab]? This detailed step-by-step guide explains the algebraic manipulations and simplifications needed to solve such expressions, providing clarity on complex mathematical concepts.

Simplification Using Basic Algebra

Given: a/b 7/5

We can interpret this as a 7x#39;#39; and b 5x#39;#39;, where x#39;#39; is some number.

We need to solve for a^2b^2 2ab/ [a^2b^2 – 2ab].

Note we can use the identity a^2b^2 2ab ab^2. Applying this, we have:

a^2b^2 2ab a^2b^2 – 2ab ab^2

So, the expression becomes:

(frac{var{2ab}}{var{ab^2}}) ( frac{2ab}{ab^2})

Substituting ab 35x#39;#39;' (combination of previous steps) into the expression, we get:

(frac{2(35x#39;#39;)}{(35x#39;#39;)^2}) ( frac{7#39;#39;}{1225x#39;#39;^2}) ( frac{70}{1225x#39;#39;^1}) ( frac{14#39;#39;^1}{245#39;#39;^2}) ( frac{140}{2450}) ( frac{140 ÷ 70}{2450 ÷ 70}) ( frac{2}{35}) ( 1/35 Note that x#39;#39; gets cancelled out.

Therefore, the simplified answer is 36.

Using Componendo and Dividendo

a/b 7/5 can be used in the identity:

( frac{var{ab}}{var{a-b}} frac{7n}{5n} ) (implies frac{ab}{a-b} frac{7}{5})

Squaring both sides:

( left( frac{ab}{a-b} right)^2 left( frac{7}{5} right)^2 ) (implies left( frac{ab}{a-b} right)^2 frac{49}{25}) (implies ab/a-b^2 frac{49}{25} implies 36)

Alternative Solution

Given: a/b 7/5

Expressing in terms of a common term:

( frac{var{a}var{b}^2var{a}var{b}}{[var{a}var{b}^2-2var{a}var{b}]} ) ( frac{var{a}var{b}var{a}var{b}}{var{a}var{b}(var{a}var{b}-2)} ) ( frac{var{ab}}{var{ab}-2} ) ( frac{7/5 cdot 5/7(7/5 cdot 5/7-1)}{1} )

Simplifying further:

( frac{7/5 (7/5 - 1)}{1} ) ( frac{7/5 cdot 2/5}{1} ) ( frac{14/25}{1} ) ( frac{14/25}{1} ) ( 36 )

Thus, the solution is 36.