Los 2/7 de un capital se imponen al 20% y el resto al 40% si luego de 9 meses el monto es S/. 7040. ¿Cuál es el capital?a.3500b.5600c.6300d.4200e.4000

1. Break Down the Problem

• Let $C$ be the total capital.
• The portion of the capital that is invested at 20% is $\frac{2}{7}C$.
• The remaining portion, which is invested at 40%, is $C - \frac{2}{7}C = \frac{5}{7}C$.
• The time period is 9 months (which is $\frac{9}{12} = \frac{3}{4}$ years).

2. Relevant Concepts

• The formula for simple interest is given by:

  $I = P \cdot r \cdot t$

  where:

  • $I$ = interest earned
  • $P$ = principal amount (initial investment)
  • $r$ = rate of interest (as a decimal)
  • $t$ = time (in years)

• Total amount after interest is:

  $A = P + I = P + P \cdot r \cdot t = P(1 + r \cdot t)$

3. Analysis and Detail

• Calculate the amount from the first portion at 20%:

  $A_1 = \left(\frac{2}{7}C\right) \left(1 + 0.20 \cdot \frac{3}{4}\right) = \left(\frac{2}{7}C\right) \left(1 + 0.15\right) = \left(\frac{2}{7}C\right) \left(1.15\right) = \frac{2.3}{7}C$

• Calculate the amount from the second portion at 40%:

  $A_2 = \left(\frac{5}{7}C\right) \left(1 + 0.40 \cdot \frac{3}{4}\right) = \left(\frac{5}{7}C\right) \left(1 + 0.30\right) = \left(\frac{5}{7}C\right) \left(1.30\right) = \frac{6.5}{7}C$

• Total amount after 9 months:

  $A = A_1 + A_2 = \frac{2.3}{7}C + \frac{6.5}{7}C = \frac{8.8}{7}C$

• Set this equal to the total amount given, $S/. 7040$:

  $\frac{8.8}{7}C = 7040$

4. Verify and Summarize

• Solve for $C$:

  $C = 7040 \cdot \frac{7}{8.8} = 7040 \cdot \frac{70}{88} = 7040 \cdot \frac{35}{44}$

• Performing the multiplication and division:

  $C = 7040 \cdot \frac{35}{44} = 1760 \cdot 35 = 61600 \div 44 = 1400$

• Correct the calculations:

  $C = 7040 \div \frac{8.8}{7} = 7040 \cdot \frac{7}{8.8} = \frac{7040 \cdot 7}{8.8} = \frac{49280}{8.8} = 5600$

Final Answer

The total capital $C$ is S/. 5600. Thus, the correct option is b. 5600.