贷款余额公式

贷款是从朋友、银行或其他金融机构借来的钱，以换取未来的资本和利息。借入的金额称为本金，而利息是您为获得贷款而支付的费用。由于债权人冒着您可能拖欠贷款的风险，他们必须通过收取称为利息的费用来补偿这种风险。担保和无担保贷款是最常见的贷款类型。担保贷款是借款人以物品（如汽车、船或房屋）作为担保的贷款。

贷款余额

银行和债权人通常会提供方便的还款方式，例如每月分期付款，债务人将每月向贷方偿还等额的款项，直到债务完成。仍需偿还的债务数量称为贷款余额。它的计算方法是从贷款总额中减去所有先前支付的本金总额。

贷款余额计算公式如下：

$B=A(1+r)^n-\frac{p}{r}[(1+r)^n-1]$

where, 
B is the balance to be paid, 
A refers to the principal, 
P is the payment made, 
r is the compounded rate of interest and 
n is the number of time periods.

示例问题

问题 1. 如果本金为 10000 美元，每月还款为 200 美元，年利率为 5%，一年后的贷款余额是多少。

解决方案：

Given: A = $10000, P = $200, r = 2% or 5/1200 = 0.0041, n = 1 year = 12 months

Let the balance loan amount after one year be B.

As per the loan balance formula, $B=A(1+r)^n-\frac{p}{r}[(1+r)^n-1]$.

Substituting the given values in the formula, we have:

B = $10000(1+0.0041)^{12}-\frac{200}{0.0041}[(1+0.0041)^{12}-1]$

= 10000(1.0503) − 48780.48(1.0503 − 1)

= 10503 − 2453.658

= $ 8049.34

问题 2. 如果本金为 20000 美元，每月还款为 200 美元，年利率为 5%，1 年后的余额是多少。

解决方案：

Given: A = $20000, P = $200, r = 2% or 5/1200 = 0.0041, n = 1 year = 12 months

Let the balance loan amount after one year be B.

As per the loan balance formula, B = $A(1+r)^n-\frac{p}{r}[(1+r)^n-1]$.

Substituting the given values in the formula, we have:

B = $20000(1+0.0041)^{12}-\frac{200}{0.0041}[(1+0.0041)^{12}-1]$

= 20000(1.0503) − 48780.48(1.0503 − 1)

= 21006 − 2453.658

= $ 18552.342

问题 3. 如果本金为 $30000，每月还款为 $200，年利率为 5%，一年后的贷款余额是多少。

解决方案：

Given: A = $30000, P = $200, r = 2% or 5/1200 = 0.0041, n = 1 year = 12 months

Let the balance loan amount after one year be B.

As per the loan balance formula,  $B = A(1+r)^n-\frac{p}{r}[(1+r)^n-1]$.

Substituting the given values in the formula, we have:

$B = 30000(1+0.0041)^{12}-\frac{200}{0.0041}[(1+0.0041)^{12}-1]$

= 30000(1.0503) − 48780.48(1.0503 − 1)

= 31509 − 2453.658

= $ 29055.342

问题 4. 如果本金为 40000 美元，每月还款为 200 美元，年利率为 5%，则 1 年后的贷款余额是多少。

解决方案：

Given: A = $40000, P = $200, r = 2% or 5/1200 = 0.0041, n = 1 year = 12 months

Let the balance loan amount after one year be B.

As per the loan balance formula, $B = A(1+r)^n-\frac{p}{r}[(1+r)^n-1]$.

Substituting the given values in the formula, we have:

B = $40000(1+0.0041)^{12}-\frac{200}{0.0041}[(1+0.0041)^{12}-1]$

= 40000(1.0503) − 48780.48(1.0503 − 1)

= 42012 − 2453.658

= $ 39558.34

问题 5. 如果本金为 50000 美元，每月还款为 200 美元，年利率为 5%，一年后的贷款余额是多少。

解决方案：

Given: A = $50000, P = $200, r = 2% or 5/1200 = 0.0041, n = 1 year = 12 months

Let the balance loan amount after one year be B.

As per the loan balance formula, $B = A(1+r)^n-\frac{p}{r}[(1+r)^n-1]$

Substituting the given values in the formula, we have:

B = $50000(1+0.0041)^{12}-\frac{200}{0.0041}[(1+0.0041)^{12}-1]$

= 50000(1.0503) − 48780.48(1.0503 − 1)

= 52515 − 2453.658

= $ 50061.342

问题 6. 如果本金为 60000 美元，每月还款为 200 美元，年利率为 5%，则 1 年后的贷款余额是多少。

解决方案：

Given: A = $60000, P = $200, r = 2% or 5/1200 = 0.0041, n = 1 year = 12 months

Let the balance loan amount after one year be B.

As per the loan balance formula, $B = A(1+r)^n-\frac{p}{r}[(1+r)^n-1]$

Substituting the given values in the formula, we have:

B = $60000(1+0.0041)^{12}-\frac{200}{0.0041}[(1+0.0041)^{12}-1]$

= 60000(1.0503) − 48780.48(1.0503 − 1)

= 63018 − 2453.658

= $ 60564.342

问题 7. 如果本金为 70000 美元，每月还款额为 200 美元，年利率为 5%，一年后的贷款余额是多少。

解决方案：

Given: A = $70000, P = $200, r = 2% or 5/1200 = 0.0041, n = 1 year = 12 months

Let the balance loan amount after one year be B.

As per the loan balance formula, $B = A(1+r)^n-\frac{p}{r}[(1+r)^n-1]$.

Substituting the given values in the formula, we have:

B = $70000(1+0.0041)^{12}-\frac{200}{0.0041}[(1+0.0041)^{12}-1]$

= 70000(1.0503) − 48780.48(1.0503 − 1)

= 73521 − 2453.658

= $ 71067.342