50% of a% of b is 75% of b% of c. Which of the following is c?

Options

A. (a) 1.5a
B. (b) 0.667a
C. (c) 0.5a
D. (d) 1.25a

Answer

To solve the equation "50% of a% of b is 75% of b% of c", we can follow these steps:

**Step 1: Write the equation based on the problem statement.**
We can express the given statement mathematically:
\[ 50\% \text{ of } (a\% \text{ of } b) = 75\% \text{ of } (b\% \text{ of } c) \]

**Step 2: Convert percentages to fractions.**
We know that:
- \( 50\% = \frac{50}{100} = \frac{1}{2} \)
- \( a\% = \frac{a}{100} \)
- \( b\% = \frac{b}{100} \)
- \( 75\% = \frac{75}{100} = \frac{3}{4} \)

Now we can rewrite the equation:
\[ \frac{1}{2} \left(\frac{a}{100} b\right) = \frac{3}{4} \left(\frac{b}{100} c\right) \]

**Step 3: Simplify both sides.**
Multiplying both sides by 100 to eliminate the denominators:
\[ \frac{1}{2} (a \cdot b) = \frac{3}{4} (b \cdot c) \]

**Step 4: Cancel out common terms.**
We can divide both sides by \( b \) (assuming \( b \neq 0 \)):
\[ \frac{1}{2} a = \frac{3}{4} c \]

**Step 5: Solve for c.**
To isolate \( c \), we can multiply both sides by \( \frac{4}{3} \):
\[ c = \frac{4}{3} \left(\frac{1}{2} a\right) \]
\[ c = \frac{4a}{6} \]
\[ c = \frac{2a}{3} \]

**Step 6: Final answer.**
Thus, \( c \) is:
\[ c = \frac{2}{3} a \]