If (10a 3+ 4b 3) : (11a 3- 15b 3) = 7 : 5, then (3a + 5b) : (9a - 2b) =?
10 : 13
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
| Step | Description | Action Taken |
|---|---|---|
| 1 | Understand the given ratio relationship. | Interpreted \((10a^3 + 4b^3) : (11a^3 - 15b^3) = 7 : 5\). |
| 2 | Convert ratio to fractional form. | Wrote as \(\frac{10a^3 + 4b^3}{11a^3 - 15b^3} = \frac{7}{5}\). |
| 3 | Solve for the relationship between variables (e.g., \(a:b\)). | Used cross-multiplication and algebraic manipulation to find \(\frac{a}{b} = \frac{5}{3}\). |
| 4 | Identify the target ratio to evaluate. | Noted the target ratio is \((3a + 5b) : (9a - 2b)\). |
| 5 | Substitute the relationship found in Step 3 into the target ratio expression. | Substituted \(a=5k\) and \(b=3k\). |
| 6 | Simplify the expression to find the final ratio value. | Calculated \(30k : 39k\) and simplified to \(10 : 13\). |
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.
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