Understanding the Ratio and Expression

The problem gives us a ratio relating two variables, \(a\) and \(b\), as \(\frac{a}{b} = \frac{7}{6}\). We need to find the value of a specific algebraic expression involving \(a\) and \(b\), which is \(\frac{6a+13b}{6a-13b}\).

To solve this, we can use the given ratio to simplify the expression. The expression \(\frac{6a+13b}{6a-13b}\) contains terms involving both \(a\) and \(b\). Our goal is to transform this expression so that we can substitute the value of the ratio \(\frac{a}{b}\).

Using the Given Ratio \(\frac{a}{b} = \frac{7}{6}\)

We can manipulate the given expression by dividing both the numerator and the denominator by \(b\) (assuming \(b \neq 0\)). This is a common technique used when dealing with expressions involving ratios like \(\frac{a}{b}\).

Let's divide the numerator and the denominator of the expression \(\frac{6a+13b}{6a-13b}\) by \(b\):

Numerator: \(\frac{6a+13b}{b} = \frac{6a}{b} + \frac{13b}{b} = 6\left(\frac{a}{b}\right) + 13\)

Denominator: \(\frac{6a-13b}{b} = \frac{6a}{b} - \frac{13b}{b} = 6\left(\frac{a}{b}\right) - 13\)

So, the expression becomes: \(\frac{6\left(\frac{a}{b}\right) + 13}{6\left(\frac{a}{b}\right) - 13}\)

Step-by-Step Calculation

Now we can substitute the given value of the ratio, \(\frac{a}{b} = \frac{7}{6}\), into the simplified expression.

1. Substitute \(\frac{a}{b} = \frac{7}{6}\) into the numerator:
2. Numerator = \(6 \times \left(\frac{7}{6}\right) + 13\)
3. Numerator = \(7 + 13\)
4. Numerator = \(20\)

1. Substitute \(\frac{a}{b} = \frac{7}{6}\) into the denominator:
2. Denominator = \(6 \times \left(\frac{7}{6}\right) - 13\)
3. Denominator = \(7 - 13\)
4. Denominator = \(-6\)

Now, combine the results for the numerator and the denominator to find the value of the expression:

Value of expression = \(\frac{\text{Numerator}}{\text{Denominator}} = \frac{20}{-6}\)

Simplify the fraction \(\frac{20}{-6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

Value of expression = \(\frac{20 \div 2}{-6 \div 2} = \frac{10}{-3} = -\frac{10}{3}\)

Result

The value of the expression \(\frac{6a+13b}{6a-13b}\) when \(\frac{a}{b} = \frac{7}{6}\) is \(-\frac{10}{3}\).

Revision Table: Key Concepts

Additional Information: Solving Ratio Problems

When you are given a ratio \(\frac{a}{b} = k\) and asked to evaluate an expression that is homogeneous in \(a\) and \(b\) (meaning all terms have the same total degree in \(a\) and \(b\), like \(6a\) (degree 1), \(13b\) (degree 1)), you can often divide the numerator and denominator by a suitable power of \(b\) (or \(a\)) to get the ratio \(\frac{a}{b}\) (or \(\frac{b}{a}\)).

In this case, all terms in the numerator (\(6a\) and \(13b\)) and the denominator (\(6a\) and \(-13b\)) are of degree 1. Dividing by \(b\) allows us to replace all occurrences of \(\frac{a}{b}\) with its given value.

Alternatively, one could assume \(a = 7k\) and \(b = 6k\) for some constant \(k\) (where \(k \neq 0\)). Substituting these into the expression:

\(\frac{6(7k)+13(6k)}{6(7k)-13(6k)} = \frac{42k+78k}{42k-78k} = \frac{120k}{-36k}\)

Since \(k \neq 0\), we can cancel \(k\):

\(\frac{120}{-36}\)

Simplifying this fraction:

\(\frac{120}{-36} = \frac{120 \div 12}{-36 \div 12} = \frac{10}{-3} = -\frac{10}{3}\)

Both methods yield the same result, confirming the answer.