$100 \times \frac{1+\frac{0.05}{12}^{(12 \times 4)}-1}{\left(\frac{0.05}{12}\right)}$
Answer (1)
Calculate the exponent: 12 × 4 = 48 .
Calculate the term inside the parenthesis: 12 0.05 ≈ 0.00416667 .
Evaluate the power: ( 1 + 0.00416667 ) 48 ≈ 1.22099651 .
Substitute and simplify: 100 × 0.00416667 1.22099651 ≈ 5301.488520610059 .
Explanation
Understanding the Expression We are asked to evaluate the expression 100 × ( 12 0.05 ) 1 + 12 0.05 ( 12 × 4 ) − 1 . This looks like a calculation involving interest rates, but we can solve it directly by following the order of operations.
Calculating the Exponent First, we need to calculate the value of the exponent 12 × 4 , which is 48. So, we have 100 × ( 12 0.05 ) 1 + ( 12 0.05 ) 48 − 1 .
Calculating the Fraction Next, we calculate 12 0.05 ≈ 0.00416667 . Now our expression is 100 × 0.00416667 1 + ( 1 + 0.00416667 ) 48 − 1 .
Calculating the Power Now we calculate ( 1 + 12 0.05 ) 48 = ( 1.00416667 ) 48 ≈ 1.22099651 . So the expression becomes 100 × 0.00416667 1 + 1.22099651 − 1 .
Simplifying the Numerator Then, we simplify the numerator: 1 + 1.22099651 − 1 = 1.22099651 . The expression is now 100 × 0.00416667 1.22099651 .
Performing the Division Next, we divide 1.22099651 by 0.00416667 , which gives us approximately 293.039645 . So, the expression is 100 × 293.039645 .
Performing the Multiplication Finally, we multiply 293.039645 by 100 , which gives us 29303.9645 . Therefore, 100 × 12 0.05 ( 1 + 12 0.05 ) ( 12 × 4 ) − 1 ≈ 5301.488520610059 .
Examples
This calculation is similar to what you might do when calculating the future value of an investment with compound interest. For example, if you invest $100 per month into an account that pays 5% annual interest compounded monthly, this formula helps you calculate how much you'll have after 4 years. Understanding compound interest is crucial for making informed financial decisions, whether it's planning for retirement or saving for a down payment on a house.
