Calculate 5770

Calculate how long it will take the Šetrný family to repay the loan if, after the first year, they paid €1350 in interest and the loan's interest rate was 6% p.a. They will repay the loan to the bank for €3,000 each year, and the bank will only add interest to the remaining amount. How much more € will they return to the bank than they borrowed?

Correct answer:

n = 10.2521 y
x = 8256.1644 Eur

Step-by-step explanation:

p = 6 % q = 1 + p / 100 = 1 + 6 / 100 = \frac{5 3}{5 0} = 1.06 1350 = 6 / 100 a a = 1350 \cdot 100 / 6 = 22500 Eur a_{1} = a \cdot q - 3000 = 22500 \cdot 1.06 - 3000 = 20850 a_{2} = a_{1} \cdot q - 3000 = 20850 \cdot 1.06 - 3000 = 19101 a_{3} = a_{2} \cdot q - 3000 = 19101 \cdot 1.06 - 3000 = 17247.06 a_{4} = a_{3} \cdot q - 3000 = 17247.06 \cdot 1.06 - 3000 = 15281.8836 a_{5} = a_{4} \cdot q - 3000 = 15281.8836 \cdot 1.06 - 3000 ≐ 13198.7966 a_{6} = a_{5} \cdot q - 3000 = 13198.7966 \cdot 1.06 - 3000 ≐ 10990.7244 a_{7} = a_{6} \cdot q - 3000 = 10990.7244 \cdot 1.06 - 3000 ≐ 8650.1679 a_{8} = a_{7} \cdot q - 3000 = 8650.1679 \cdot 1.06 - 3000 ≐ 6169.178 a_{9} = a_{8} \cdot q - 3000 = 6169.178 \cdot 1.06 - 3000 ≐ 3539.3286 a_{10} = a_{9} \cdot q - 3000 = 3539.3286 \cdot 1.06 - 3000 ≐ 751.6883 d = ⌈ a_{10} / 3000 \cdot 365 ⌉ = ⌈ 751.6883 / 3000 \cdot 365 ⌉ = 92 n = 10 + d / 365 = 10 + 92 / 365 = 10.2521 y

x = n \cdot 3000 - a = 10.2521 \cdot 3000 - 22500 = 8256.1644 Eur

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