Question

Which term of the following series is 17.25?
\(-0.25, 0.25, 0.75, \dots\)

• A. \(34^{\text{th}}\)
• B. \(36^{\text{th}}\)
• C. \(31^{\text{st}}\)
• D. \(32^{\text{nd}}\)

Hint:

To quickly check divisibility by 0.5, simply double the numerator. \( \frac{x}{0.5} = 2x \).

Answer 1

The Correct Option is B

Strategy:

Ratio of shares \(A : B : C = \frac{4}{3} : \frac{7}{2} : \frac{6}{5}\). Multiply by LCM of denominators \((3, 2, 5) = 30\): \[ A : B : C = (4/3 \times 30) : (7/2 \times 30) : (6/5 \times 30) \] \[ A : B : C = 40 : 105 : 36 \] Let initial investments be \(40x\), \(105x\), and \(36x\).
A increases his share by \(108.75%\). Percentage increase = \(108.75% = \frac{10875}{10000} = \frac{435}{400} = \frac{87}{80}\). Increase in A's capital = \(40x \times \frac{87}{80} = \frac{87}{2}x = 43.5x\). New Capital for A = \(40x + 43.5x = 83.5x\).
Profit is distributed based on (Capital \(\times\) Time).

A: Invests \(40x\) for 4 months, then \(83.5x\) for 8 months. \[ \text{Total}_A = (40 \times 4) + (83.5 \times 8) = 160 + 668 = 828 \]
B: Invests \(105x\) for 12 months. \[ \text{Total}_B = 105 \times 12 = 1260 \]
C: Invests \(36x\) for 12 months. \[ \text{Total}_C = 36 \times 12 = 432 \]

Total Ratio Sum = \(828 + 1260 + 432 = 2520\). B's Share of Profit = \(\frac{1260}{2520} \times 17208\). Notice that \(\frac{1260}{2520} = \frac{1}{2}\). \[ \text{B's Share} = \frac{1}{2} \times 17208 = 8604 \]