select all the correct answers.\ntammy deposits $1,850 in an individual retirement account earning 2.6%…

select all the correct answers.\ntammy deposits $1,850 in an individual retirement account earning 2.6% interest, compounded annually. she also deposits $2,015 in business interest - bearing account earning 1.5% interest, compounded annually.\nselect the equation and the number of years, x, it will take for the amount of money in both accounts to be equal. round to the nearest whole year.\n□ 9 years\n□ 1,850(1.026)^x = 2,015(1.015)^x\n□ 6 years\n□ 1,850(1.126)^x = 2,015(1.115)^x\n□ 1,850(1.26)^x = 2,015(1.15)^x\n□ 8 years

select all the correct answers.\ntammy deposits $1,850 in an individual retirement account earning 2.6% interest, compounded annually. she also deposits $2,015 in business interest - bearing account earning 1.5% interest, compounded annually.\nselect the equation and the number of years, x, it will take for the amount of money in both accounts to be equal. round to the nearest whole year.\n□ 9 years\n□ 1,850(1.026)^x = 2,015(1.015)^x\n□ 6 years\n□ 1,850(1.126)^x = 2,015(1.115)^x\n□ 1,850(1.26)^x = 2,015(1.15)^x\n□ 8 years

Answer

Explanation:

Step1: Recall compound - interest formula

The compound - interest formula is $A = P(1 + r)^x$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (as a decimal), and $x$ is the number of years. For the individual retirement account, $P_1=1850$ and $r_1 = 0.026$, so $A_1=1850(1 + 0.026)^x=1850(1.026)^x$. For the business interest - bearing account, $P_2 = 2015$ and $r_2=0.015$, so $A_2=2015(1 + 0.015)^x=2015(1.015)^x$.

Step2: Set the two amounts equal

We want to find $x$ when $A_1 = A_2$, so the equation is $1850(1.026)^x=2015(1.015)^x$.

Step3: Solve for $x$

$\frac{(1.026)^x}{(1.015)^x}=\frac{2015}{1850}$ Using the property $\frac{a^x}{b^x}=(\frac{a}{b})^x$, we have $( \frac{1.026}{1.015})^x=\frac{2015}{1850}$ $1.010837^x = 1.089189$ Take the natural logarithm of both sides: $x\ln(1.010837)=\ln(1.089189)$ $x=\frac{\ln(1.089189)}{\ln(1.010837)}\approx 8$

Answer:

  1. $1,850(1.026)^x = 2,015(1.015)^x$
  2. 8 years