# How to Simplify Algebraic Expressions Using Ratios

Simplifying algebraic expressions using ratios is a powerful technique that can help you solve complex problems more efficiently. When given a ratio involving algebraic terms, such as $\displaystyle \frac{a+b}{3} = \frac{b+c}{4} = \frac{c+a}{5}$, you can use this information to evaluate various expressions containing these terms.

## The Key Concept: Expressing Terms as Multiples of a Constant

The key to simplifying expressions using ratios is to express each term in the ratio as a multiple of a constant, typically denoted as $k$. By setting each part of the ratio equal to $k$, you can create a system of equations that allows you to solve for each variable in terms of $k$.

Once you have expressed each variable in terms of $k$, you can substitute these values into the given expressions and simplify them. This process helps you evaluate expressions such as ratios of the variables, ratios of their reciprocals, or more complex combinations of the variables.

## Developing Problem-Solving Skills

By mastering the technique of simplifying algebraic expressions using ratios, you can develop a deeper understanding of the relationships between variables and enhance your problem-solving skills. This method is particularly useful when dealing with expressions that involve multiple variables and complex relationships between them.

To become proficient in this technique, it is essential to practice with a variety of problems and ratios. As you work through more examples, you will develop a stronger intuition for recognizing patterns and applying the appropriate steps to simplify the expressions.

### Practicing with Various Problems

As you practice simplifying algebraic expressions using ratios, consider the following tips:

1. Identify the ratio and the variables involved in the problem.
2. Express each term in the ratio as a multiple of a constant $k$.
3. Solve for each variable in terms of k.
4. Substitute the values of the variables into the given expressions.
5. Simplify the expressions and evaluate the results.

In conclusion, simplifying algebraic expressions using ratios is a valuable skill that can help you tackle a wide range of mathematical problems. By expressing each term in the ratio as a multiple of a constant and substituting the values into the given expressions, you can efficiently evaluate and simplify complex algebraic statements.

Given $\displaystyle \frac{a+b}{3} = \frac{b+c}{4} = \frac{c+a}{5}$, evaluate the following.

(a)   $a : b : c$

\displaystyle \begin{align} \require{AMSsymbols} \text{Let }\frac{a+b}{3} &= \frac{b+c}{4} = \frac{c+a}{5} = k \\ a+b &= 3k \cdots (1) \\ b+c &= 4k \cdots (2) \\ c+a &= 5k \cdots (3) \\ (a+b)-(c+a) &= 3k-5k &\color{green}{(1)-(3)} \\ b-c &= -2k \cdots (4) \\ (b+c)+(b-c) &=4k-2k &\color{green}{(2)+(4)} \\ 2b &= 2k \\ b &= k \\ a+k &= 3k &\color{green}{\text{Substitute } b=k \text{ into } a+b=3k\cdots(1)} \\ a &= 2k \\ k+c &= 4k &\color{green}{\text{Substitute } b=k \text{ into } b+c=4k\cdots(2)} \\ c &= 3k \\ a : b : c &= 2k : k : 3k \\ \therefore a : b : c &= 2 : 1 : 3 \end{align}

(b)   $\displaystyle \frac{1}{ab} : \frac{1}{bc} : \frac{1}{ca}$

\displaystyle \begin{align} \frac{1}{ab} : \frac{1}{bc} : \frac{1}{ca} &= \frac{1}{2k \times k} : \frac{1}{k \times 3k} : \frac{1}{3k \times 2k} \\ &= \frac{1}{2k^2} : \frac{1}{3k^2} : \frac{1}{6k^2} \\ &= \frac{1}{2} : \frac{1}{3} : \frac{1}{6} \\ &= \frac{1}{2} \times 6 : \frac{1}{3} \times 6 : \frac{1}{6} \times 6 \\ \therefore \frac{1}{ab} : \frac{1}{bc} : \frac{1}{ca} &= 3 : 2 : 1 \end{align}

(c)   $\displaystyle \frac{ab+bc+ca}{a^2+b^2+c^2}$

\displaystyle \begin{align} \frac{ab+bc+ca}{a^2+b^2+c^2} &= \frac{2k \times k + k \times 3k + 3k \times 2k}{(2k)^2 + k^2 + (3k)^2} \\ &= \frac{2k^2+3k^2+6k^2}{4k^2+k^2+9k^2} \\ &= \frac{11k^2}{14k^2} \\ \therefore \frac{ab+bc+ca}{a^2+b^2+c^2} &= \frac{11}{14} \end{align}

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