`(2)/(3)a=(4)/(5)b=(1)/(2)c`, find `a:b:c`

Options

A. A) 6:5:8
B. B) 2:3:4
C. C) 1:1:1
D. D) 2:3:5

Answer

To solve the equation \(\frac{2}{3}a = \frac{4}{5}b = \frac{1}{2}c\) and find the ratio \(a:b:c\), we can follow these steps:

### Step 1: Set the common value
Let \(\frac{2}{3}a = \frac{4}{5}b = \frac{1}{2}c = k\).

### Step 2: Express \(a\), \(b\), and \(c\) in terms of \(k\)
From the equation \(\frac{2}{3}a = k\):
\[ a = \frac{3}{2}k \]

From the equation \(\frac{4}{5}b = k\):
\[ b = \frac{5}{4}k \]

From the equation \(\frac{1}{2}c = k\):
\[ c = 2k \]

### Step 3: Write the expressions for \(a\), \(b\), and \(c\)
Now we have:
- \(a = \frac{3}{2}k\)
- \(b = \frac{5}{4}k\)
- \(c = 2k\)

### Step 4: Find a common denominator to express the ratios
To compare \(a\), \(b\), and \(c\), we can express them with a common denominator. The least common multiple of the denominators (2, 4, and 1) is 4.

Rewriting each expression:
- \(a = \frac{3}{2}k = \frac{6}{4}k\)
- \(b = \frac{5}{4}k\)
- \(c = 2k = \frac{8}{4}k\)

### Step 5: Form the ratio \(a:b:c\)
Now, we can write the ratio:
\[ a:b:c = \frac{6}{4}k : \frac{5}{4}k : \frac{8}{4}k \]
This simplifies to:
\[ a:b:c = 6 : 5 : 8 \]

### Final Answer
Thus, the ratio \(a:b:c\) is \(6:5:8\).

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