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?

Expert avatar
Lurline
4.6
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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