If `a^(3)+b^(3):a^(3)-b^(3)=133:117,` Find a:b

Options

A. `2:3`
B. `5:4`
C. `5:2`
D. None of these

Answer

To solve the problem, we start with the given ratio:

\[ \frac{a^3 + b^3}{a^3 - b^3} = \frac{133}{117} \]

### Step 1: Cross-multiply the ratios

Cross-multiplying gives us:

\[ 117(a^3 + b^3) = 133(a^3 - b^3) \]

### Step 2: Expand both sides

Expanding both sides results in:

\[ 117a^3 + 117b^3 = 133a^3 - 133b^3 \]

### Step 3: Rearrange the equation

Now, we move all terms involving \(a^3\) to one side and all terms involving \(b^3\) to the other side:

\[ 117b^3 + 133b^3 = 133a^3 - 117a^3 \]

This simplifies to:

\[ 250b^3 = 16a^3 \]

### Step 4: Divide both sides by \(b^3\)

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

\[ \frac{a^3}{b^3} = \frac{250}{16} \]

### Step 5: Simplify the fraction

We simplify \(\frac{250}{16}\):

\[ \frac{250 \div 2}{16 \div 2} = \frac{125}{8} \]

### Step 6: Express \(a\) in terms of \(b\)

Taking the cube root of both sides gives us:

\[ \frac{a}{b} = \frac{\sqrt[3]{125}}{\sqrt[3]{8}} = \frac{5}{2} \]

### Step 7: Write the final ratio

Thus, the ratio \(a:b\) is:

\[ a:b = 5:2 \]