Question

Today I received $25,000 from a bank as a loan. The loan was agreed upon at 8%, capitalized quarterly. I must make a payment after one year for $12,000 and another two years after (the first payment) to pay off the loan. How much is the second payment?

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Answer to a math question Today I received $25,000 from a bank as a loan. The loan was agreed upon at 8%, capitalized quarterly. I must make a payment after one year for $12,000 and another two years after (the first payment) to pay off the loan. How much is the second payment?

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Lurline
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104 Answers
Para resolver este problema, primero debemos calcular cuánto será la deuda luego de un año de capitalización con un interés del 8% capitalizable trimestralmente.

La fórmula para calcular el monto final de un préstamo con interés compuesto es:

A = P(1 + r/n)^{nt}

Donde:
- A es la cantidad final después de t años,
- P es la cantidad principal (préstamo) recibida,
- r es la tasa de interés anual,
- n es el número de veces que se capitaliza el interés por año, y
- t es el número de años.

En este caso, el préstamo es de $25,000, la tasa de interés anual es del 8% (o 0.08 en forma decimal), se capitaliza trimestralmente (entonces n = 4), y queremos calcular el monto después de un año, por lo que t = 1.

Sustituyendo los valores en la fórmula:

A = 25,000(1 + 0.08/4)^{4(1)}

A = 25,000(1 + 0.02)^{4}

A = 25,000(1.02)^{4}

A = 25,000(1.0824)

A = 27,060

Por lo tanto, al transcurrir un año, la deuda será de $27,060.

Ahora, el pago que debes hacer al transcurrir un año es de $12,000, por lo que la deuda restante después de este pago es de $27,060 - $12,000 = $15,060.

Ahora, queremos calcular cuánto deberás pagar dos años después del primer pago para liquidar la deuda restante de $15,060. De nuevo, utilizaremos la fórmula del monto final del préstamo con interés compuesto.

A = 15,060(1 + 0.08/4)^{4(2)}

A = 15,060(1.02)^{8}

A = 15,060(1.17166144)

A = 17,655.63

Por lo tanto, el segundo pago que debes hacer para liquidar completamente el préstamo será de $\$17,655.63$.

\boxed{17,655.63}

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