Question

determine the present value p that must be invested to have the future value a at simple interest rate r after time t. a = $6000, r = 2.5%, t = 2 years the present value p that must be invested is $\square (round up to the nearest cent.)

Explanation:

Step1: Recall simple interest formula

The formula for future value \( A \) with simple interest is \( A = P(1 + rt) \), where \( P \) is present value, \( r \) is annual interest rate (in decimal), and \( t \) is time in years. We need to solve for \( P \), so rearrange the formula: \( P=\frac{A}{1 + rt} \).

Step2: Convert rate to decimal

Given \( r = 2.5\%=0.025 \), \( A = 6000 \), \( t = 2 \). Substitute values into the formula: \( P=\frac{6000}{1+(0.025\times2)} \).

Step3: Calculate denominator

First, calculate \( rt = 0.025\times2 = 0.05 \). Then \( 1 + rt=1 + 0.05 = 1.05 \).

Step4: Solve for P

Now, \( P=\frac{6000}{1.05}\approx5714.29 \) (rounded to nearest cent). Wait, wait, no: wait, \( 6000\div1.05 = 5714.2857\cdots \), but wait, the problem says "round up to the nearest cent". Wait, 5714.2857... rounded up to nearest cent: the hundredth place is 8, thousandth is 5, so when rounding up, 5714.28 + 0.01 = 5714.29? Wait, no, 5714.2857...: the third decimal is 5, so when rounding to nearest cent (two decimals), we look at third decimal. 5 is equal to or more than 5, so we round up the second decimal: 8 becomes 9? Wait, no: 5714.2857... is 5714.28 when truncated to two decimals, but the third decimal is 5, so we round up the second decimal: 8 + 1 = 9? Wait, no, 5714.2857: the first decimal is 2, second is 8, third is 5. So rounding to two decimals: 5714.29? Wait, but let's recalculate \( \frac{6000}{1.05} \). 1.05 times 5714.28 is 1.055714.28 = 5714.28 + 5714.280.05 = 5714.28 + 285.714 = 6000? Wait, 5714.281.05 = 5714.28 + 285.714 = 6000.0 (approx)? Wait, 5714.281.05 = 5714.281 + 5714.280.05 = 5714.28 + 285.714 = 6000.0 (exactly? Wait, 5714.280.05 = 285.714, and 5714.28 + 285.714 = 5999.994, which is close to 6000. Wait, maybe my initial formula is wrong? Wait, no, the simple interest formula: \( A = P + Prt = P(1 + rt) \), so \( P = A/(1 + rt) \). Let's compute \( 1 + rt = 1 + 0.025*2 = 1.05 \). Then \( 6000 / 1.05 = 5714.285714... \). So when rounding up to the nearest cent, we take the ceiling of the value to two decimal places. The ceiling of 5714.285714... to two decimal places is 5714.29? Wait, no, ceiling function: ceiling(5714.2857, 2 decimals) is 5714.29? Wait, 5714.2857 is between 5714.28 and 5714.29. The ceiling (round up) would be 5714.29. Wait, but let's check: if we invest 5714.28, then the interest is 5714.280.0252 = 5714.280.05 = 285.714, so total is 5714.28 + 285.714 = 5999.994, which is less than 6000. If we invest 5714.29, then interest is 5714.290.05 = 285.7145, total is 5714.29 + 285.7145 = 6000.0045, which is more than 6000. So to have at least 6000, we need to round up to 5714.29? Wait, but the problem says "round up to the nearest cent", so we take the smallest amount that when rounded up (i.e., the ceiling) to nearest cent gives enough. Wait, maybe my initial calculation was wrong. Wait, \( P = A/(1 + rt) = 6000/(1 + 0.0252) = 6000/1.05 = 5714.285714... \). So when rounding up to the nearest cent, we look at the third decimal, which is 5, so we round up the second decimal: 8 becomes 9, so 5714.29. Wait, but 5714.2857 is 5714.28 when truncated, but the third decimal is 5, so we round up the second decimal: 8 + 1 = 9, so 5714.29.

Answer:

\( \$5714.29 \)