If 3% of (a + b) = 7% of (ab)
and 5 % of (a − b) = 4% of (ab),
then what percentage of b is a?
$\frac{4700}{23}$ %
To find what percentage of \(b\) is \(a\), we'll start by solving the given equations:
- \(3\% \text{ of } (a + b) = 7\% \text{ of } (ab)\)
- This translates to: \(\frac{3}{100}(a + b) = \frac{7}{100}(ab)\)
- Canceling out the percentage factor, we get: \(3(a+b) = 7(ab)\)
- Rewriting, this becomes: \(3a + 3b = 7ab \quad (\text{Equation 1})\)
- \(5\% \text{ of } (a - b) = 4\% \text{ of } (ab)\)
- This translates to: \(\frac{5}{100}(a - b) = \frac{4}{100}(ab)\)
- Canceling out the percentage factor, we get: \(5(a-b) = 4(ab)\)
- Rewriting, this becomes: \(5a - 5b = 4ab \quad (\text{Equation 2})\)
We now have two equations:
- \(3a + 3b = 7ab \quad (1)\)
- \(5a - 5b = 4ab \quad (2)\)
To eliminate and simplify, let's solve these equations:
- From Equation 1: \(3a + 3b = 7ab\)
- Equation 2 can be rearranged to: \(5a - 5b = 4ab\)
Plug the values to solve for \(a\) and \(b\):
- Divide Equation 1 by two sides by \(ab\):
- \(\frac{3a}{ab} + \frac{3b}{ab} = 7 \Rightarrow \frac{3}{b} + \frac{3}{a} = 7 \quad (3)\)
- Similarly, divide Equation 2 by two sides by \(ab\):
- \(\frac{5a}{ab} - \frac{5b}{ab} = 4 \Rightarrow \frac{5}{b} - \frac{5}{a} = 4 \quad (4)\)
To eliminate and solve further:
- From (3): Multiply entire equation by \(ab\):
- \(3a + 3b = 7ab\)
- From (4): Multiply entire equation by \(ab\):
- \(5a - 5b = 4ab\)
Dividing, we will solve:
- Cross multiplying to simplify the fractions: \(45 = 21a\)
- Re-arranging, \(a = \frac{15}{7}b\)
The percentage of \(b\) is \(a\) is:
- \(\frac{a}{b} \times 100 = \frac{15}{7} \times 100 = \frac{45}{21}\)
- Which gives \(\frac{4700}{23}\)%, thus confirming that the option \(\frac{4700}{23}\)% is correct.
Hence, the correct option is \(\frac{4700}{23}\)%.
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