Understanding the Ratio Problem

The question asks us to find the value of a specific algebraic ratio, $(3a + 5b) : (9a - 2b)$, given a relationship between two other algebraic expressions involving $a^3$ and $b^3$. The given relationship is $(10a^3 + 4b^3) : (11a^3 - 15b^3) = 7 : 5$. To solve this, we first need to use the given ratio to find the relationship between $a$ and $b$. Once we have the ratio $a:b$, we can substitute it into the second expression to find its value.

Solving the Given Algebraic Ratio

The given ratio can be written as a fraction:

\(\frac{10a^3 + 4b^3}{11a^3 - 15b^3} = \frac{7}{5}\)

To find the relationship between \(a\) and \(b\), we can cross-multiply:

\(5(10a^3 + 4b^3) = 7(11a^3 - 15b^3)\)

Now, expand both sides of the equation:

\(50a^3 + 20b^3 = 77a^3 - 105b^3\)

Rearrange the terms to group \(a^3\) on one side and \(b^3\) on the other side:

\(20b^3 + 105b^3 = 77a^3 - 50a^3\)

Combine the like terms:

\(125b^3 = 27a^3\)

Now, we can find the ratio of \(a^3\) to \(b^3\):

\(\frac{a^3}{b^3} = \frac{125}{27}\)

To find the ratio of \(a\) to \(b\), we take the cube root of both sides:

\(\frac{a}{b} = \sqrt[3]{\frac{125}{27}} = \frac{\sqrt[3]{125}}{\sqrt[3]{27}} = \frac{5}{3}\)

So, the ratio \(a:b\) is \(5:3\). This means we can write \(a = 5k\) and \(b = 3k\) for some constant \(k\) (where \(k \neq 0\)).

Calculating the Second Algebraic Ratio

We need to find the value of the ratio \((3a + 5b) : (9a - 2b)\). We substitute \(a = 5k\) and \(b = 3k\) into this expression:

First part of the ratio:

\(3a + 5b = 3(5k) + 5(3k)\)

\(= 15k + 15k\)

\(= 30k\)

Second part of the ratio:

\(9a - 2b = 9(5k) - 2(3k)\)

\(= 45k - 6k\)

\(= 39k\)

Now form the ratio using these results:

\((3a + 5b) : (9a - 2b) = 30k : 39k\)

Since \(k \neq 0\), we can cancel \(k\) from both parts of the ratio:

\(30 : 39\)

Simplify the ratio by dividing both numbers by their greatest common divisor, which is 3:

\(\frac{30}{3} : \frac{39}{3} = 10 : 13\)

Thus, the value of the ratio \((3a + 5b) : (9a - 2b)\) is \(10 : 13\).

Conclusion on the Ratio Problem

By first using the given ratio \((10a^3 + 4b^3) : (11a^3 - 15b^3) = 7 : 5\) to establish that \(a:b = 5:3\), we were able to substitute this relationship into the expression \((3a + 5b) : (9a - 2b)\) and simplify to find the final ratio \(10 : 13\).

Revision Table: Key Steps for Solving Ratio Problems

Additional Information on Ratios and Proportions

A ratio is a comparison of two quantities by division. It can be written as \(a:b\) or \(\frac{a}{b}\). A proportion is an equation stating that two ratios are equal, like \(\frac{a}{b} = \frac{c}{d}\). In this problem, we used the property of proportions that if \(\frac{a}{b} = \frac{c}{d}\), then \(ad = bc\) (cross-multiplication). This fundamental property is crucial for solving equations involving ratios.

When a ratio like \(a:b = 5:3\) is found, it means that \(a\) is \(\frac{5}{3}\) times \(b\). We can represent \(a\) and \(b\) using a common multiplier \(k\), such that \(a=5k\) and \(b=3k\). Using this representation simplifies the process of substituting the values into other algebraic expressions involving \(a\) and \(b\), as the \(k\) often cancels out in the final ratio.