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4*. Doris invests some money in a savings account.
In the first year the annual percentage rate is 3%.
In the second year the annual percentage rate is 2%.
At the end of the second year, there is £4044.81 in the account.
How much did she invest?
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Sagot :

Darian

Answer:

[tex]\huge\boxed{\£3850.00}[/tex]

Identify variables and constants:

  • y is the initial amount invested
  • x is the amount in the account at the end of the first year

Useful Information:

  • [tex]\text{Final Amount}=\text{Initial Amount} *(1+\frac{\text{Interest Rate}}{100})[/tex]

Step-by-step explanation:

To determine how much Doris initially invested, we need to work backwards from the amount in the account at the end of the second year, taking into account the interest rates for both years. We can first create two equations. As we know the amount in the account is £4044.81 at the end of the two years and the interest rate for the second year is 2%, we can substitute this into the equation [tex]\text{Final Amount}=\text{Initial Amount} *(1+\frac{\text{Interest Rate}}{100})[/tex], with the final amount, in this case, being 4044.81, and the initial amount is represented by x, this gives us the equation [tex]4044.81=1.02x[/tex]. We can then create an equation for the percentage increase in the first year, given that x in this case is the final amount and y is the initial amount invested, this gives us the equation of [tex]x=1.03y[/tex]. We can then substitute x from the equation [tex]x=1.03y[/tex] into the equation [tex]4044.81=1.02x[/tex], this gives us the amount at the end of year 2 in terms of the initial amount invested. The equation for this is, [tex]4044.81=1.02(1.03y)[/tex]. This can be simplified by multiplying 1.02 by 1.03y, giving us [tex]4044.81=1.0506y[/tex]. The final step to work out the value of y is to divide both sides by 1.0506, giving us [tex]y=3850.00[/tex].

1) Create an equation for the 2nd year.

[tex]4044.81=1.02x[/tex]

2) Create an equation for the 1st year.

[tex]x=1.03y[/tex]

3) Substitute x into the 2nd year equation to give it in terms of y.

[tex]4044.81=1.02(1.03y)[/tex]

4) Simplify by multiplying 1.02 by 1.03y

[tex]4044.81=1.0506y[/tex]

5) Divide 4044.81 by 1.0506.

[tex]y=\frac{4044.81}{1.0506}[/tex]

[tex]y=3850.00[/tex]

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