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Math  /  Algebra

Question12. Write the equation of the following and solve algebraically. Round to the nearest tenth. (1) Philip invested $1000\$ 1000 in an account at 12%12 \% per annum, compounded annually. How many years will a take the investment to reach $2000\$ 2000 ?

Studdy Solution

STEP 1

1. Philip invested \$1000 initially.
2. The interest rate is 12% per annum, compounded annually.
3. We want to find the number of years it takes for the investment to grow to \$2000.
4. The formula for compound interest is used: \( A = P(1 + r)^t \), where \( A \) is the amount of money accumulated after \( t \) years, including interest, \( P \) is the principal amount (the initial amount), \( r \) is the annual interest rate (decimal), and \( t \) is the time in years.

STEP 2

1. Identify the known values and the unknown variable.
2. Write the compound interest equation using the known values.
3. Solve the equation algebraically for the unknown variable.
4. Round the solution to the nearest tenth.

STEP 3

Identify the known values and the unknown variable.
- Principal amount, P=1000 P = 1000 - Final amount, A=2000 A = 2000 - Annual interest rate, r=0.12 r = 0.12 (since 12% = 0.12) - Unknown variable: t t (the number of years)

STEP 4

Write the compound interest equation using the known values.
The formula for compound interest is:
A=P(1+r)t A = P(1 + r)^t
Substitute the known values into the equation:
2000=1000(1+0.12)t 2000 = 1000(1 + 0.12)^t

STEP 5

Solve the equation algebraically for the unknown variable t t .
First, divide both sides by 1000 to isolate the exponential expression:
20001000=(1.12)t \frac{2000}{1000} = (1.12)^t
Simplify the left side:
2=(1.12)t 2 = (1.12)^t
Take the natural logarithm of both sides to solve for t t :
ln⁡(2)=ln⁡((1.12)t) \ln(2) = \ln((1.12)^t)
Using the property of logarithms, bring down the exponent:
ln⁡(2)=t⋅ln⁡(1.12) \ln(2) = t \cdot \ln(1.12)
Solve for t t by dividing both sides by ln⁡(1.12) \ln(1.12) :
t=ln⁡(2)ln⁡(1.12) t = \frac{\ln(2)}{\ln(1.12)}

STEP 6

Calculate the value of t t and round to the nearest tenth.
Using a calculator:
t≈0.69310.1133 t \approx \frac{0.6931}{0.1133} t≈6.116 t \approx 6.116
Round to the nearest tenth:
t≈6.1 t \approx 6.1
The number of years it will take for the investment to reach \$2000 is approximately \( \boxed{6.1} \) years.

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