If `2/3` of A = 75% of B = 0.6 of C, then `A : B : C` is

Options

A. `2 : 3 : 3 `
B. `3 : 4 : 5`
C. `4 : 5 : 6 `
D. `9 : 8 : 10`

Answer

To solve the problem, we need to find the ratio \( A : B : C \) given that \( \frac{2}{3} A = 75\% B = 0.6 C \).

### Step 1: Convert all expressions to a common form
We start by expressing everything in terms of fractions:
- \( 75\% \) can be written as \( \frac{3}{4} \).
- \( 0.6 \) can be expressed as \( \frac{6}{10} \) or simplified to \( \frac{3}{5} \).

So we have:
\[ \frac{2}{3} A = \frac{3}{4} B = \frac{3}{5} C \]

### Step 2: Set a common variable
Let \( k \) be the common value such that:
\[ \frac{2}{3} A = k, \quad \frac{3}{4} B = k, \quad \frac{3}{5} C = k \]

### Step 3: Express A, B, and C in terms of k
Now we can express \( A \), \( B \), and \( C \) in terms of \( k \):
1. From \( \frac{2}{3} A = k \):
\[ A = \frac{3}{2} k \]
2. From \( \frac{3}{4} B = k \):
\[ B = \frac{4}{3} k \]
3. From \( \frac{3}{5} C = k \):
\[ C = \frac{5}{3} k \]

### Step 4: Find the ratio A : B : C
Now we have:
\[ A = \frac{3}{2} k, \quad B = \frac{4}{3} k, \quad C = \frac{5}{3} k \]

To find the ratio \( A : B : C \), we can write:
\[ A : B : C = \frac{3}{2} k : \frac{4}{3} k : \frac{5}{3} k \]

### Step 5: Eliminate k and simplify the ratios
Since \( k \) is common in all terms, we can eliminate it:
\[ A : B : C = \frac{3}{2} : \frac{4}{3} : \frac{5}{3} \]

To simplify this ratio, we can find a common denominator. The least common multiple of the denominators (2, 3) is 6.

Now, we can convert each term:
1. \( \frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6} \)
2. \( \frac{4}{3} = \frac{4 \times 2}{3 \times 2} = \frac{8}{6} \)
3. \( \frac{5}{3} = \frac{5 \times 2}{3 \times 2} = \frac{10}{6} \)

Thus, the ratio becomes:
\[ A : B : C = 9 : 8 : 10 \]

### Final Answer
The ratio \( A : B : C \) is \( 9 : 8 : 10 \).

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