Calvin deposits $400 in a savings account that accrues 5% interest compounded monthly. After c years, Calvin has $658.80. Makayla deposits $300 in a different savings account that accrues 6% interest compounded quarterly. After m years, Makayla has $613.04. What is the approximate difference in the number of years that Calvin and Makayla have their money invested?

To find the approximate difference in the number of years that Calvin and Makayla have their money invested, we need to compare the values of c and m.

Let's start by solving for c, the number of years Calvin has his money invested.

We can use the formula for compound interest to find the value of Calvin's investment after t years:

A = P(1 + r/n)^(nt)

Where:
A is the final value of the investment
P is the principal (initial amount)
r is the annual interest rate (expressed as a decimal)
n is the number of times the interest is compounded per year
t is the number of years

In this case, Calvin deposits $400 in an account that accrues 5% interest compounded monthly. So we have:

A = $658.80
P = $400
r = 0.05 (5% expressed as a decimal)
n = 12 (compounded monthly)

Plugging in these values, we get:

$658.80 = $400(1 + 0.05/12)^(12c)

Dividing both sides by $400 and simplifying, we get:

1.647 = (1.00417)^(12c)

Taking the natural logarithm of both sides, we have:

ln(1.647) = ln(1.00417)^(12c)

Using the logarithmic property, we simplify to:

ln(1.647) = 12c ln(1.00417)

Dividing both sides by 12 ln(1.00417), we get:

c ≈ ln(1.647) / (12 ln(1.00417))

Using a calculator, we find that c ≈ 3.20 years.

Now let's solve for m, the number of years Makayla has her money invested.

Using the same formula for compound interest, we can find the value of Makayla's investment after t years:

A = P(1 + r/n)^(nt)

In this case, Makayla deposits $300 in an account that accrues 6% interest compounded quarterly. So we have:

A = $613.04
P = $300
r = 0.06 (6% expressed as a decimal)
n = 4 (compounded quarterly)

Plugging in these values, we get:

$613.04 = $300(1 + 0.06/4)^(4m)

Dividing both sides by $300 and simplifying, we get:

2.043467 = (1.015)^(4m)

Taking the natural logarithm of both sides, we have:

ln(2.043467) = ln(1.015)^(4m)

Using the logarithmic property, we simplify to:

ln(2.043467) = 4m ln(1.015)

Dividing both sides by 4 ln(1.015), we get:

m ≈ ln(2.043467) / (4 ln(1.015))

Using a calculator, we find that m ≈ 2.90 years.

To find the approximate difference in the number of years, we subtract c from m:

Approximate difference = m - c
Approximate difference ≈ 2.90 - 3.20
Approximate difference ≈ -0.30

Therefore, the approximate difference in the number of years that Calvin and Makayla have their money invested is approximately -0.30 years.

To solve this problem, we need to set up two equations based on the given information and then solve for the difference in years.

Let's first consider Calvin's situation:
The formula for compound interest is given by: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

In this case, Calvin deposits $400, the interest rate is 5% (or 0.05), and the interest is compounded monthly (so n = 12).

Using this information, we can set up the equation:
658.80 = 400(1 + 0.05/12)^(12c)

Now let's consider Makayla's situation:
Similarly, the formula for compound interest in her case is: A = P(1 + r/n)^(nt), with P as the principal amount ($300), r as the annual interest rate (6% or 0.06), and n as the number of times the interest is compounded per year (in this case, quarterly, so n = 4).

Using this information, we can set up the equation:
613.04 = 300(1 + 0.06/4)^(4m)

To find the approximate difference in the number of years that Calvin and Makayla have their money invested, we need to solve for c and m and then find the absolute value of their difference.

Now, we can either use trial and error or a numerical method like graphing or using a calculator's solver function to find the values of c and m. Once we find those values, we can take the absolute value of the difference between c and m.

Alternatively, we can use a numerical method like substitution or elimination to find the approximate difference between the number of years c and m:

From the equations:
658.80 = 400(1 + 0.05/12)^(12c)
613.04 = 300(1 + 0.06/4)^(4m)

We can isolate (1 + 0.05/12)^(12c) and (1 + 0.06/4)^(4m) respectively:

(1 + 0.05/12)^(12c) = 658.80/400
(1 + 0.06/4)^(4m) = 613.04/300

Now we can use logarithms to solve for c and m. Taking the logarithm of both sides of the equations, we have:

ln((1 + 0.05/12)^(12c)) = ln(658.80/400)
ln((1 + 0.06/4)^(4m)) = ln(613.04/300)

Simplifying these expressions, we get:

(12c)ln(1 + 0.05/12) = ln(658.80/400)
(4m)ln(1 + 0.06/4) = ln(613.04/300)

Now we can solve for c and m by dividing both sides of the equations by the appropriate factors:

c = ln(658.80/400)/(12 * ln(1 + 0.05/12))
m = ln(613.04/300)/(4 * ln(1 + 0.06/4))

Finally, to find the approximate difference in the number of years, we calculate the absolute value of the difference between c and m:
|c - m|

Note: Since we already have approximate values for c and m, we can plug them into the equation |c - m| and evaluate to find the approximate difference in the number of years.

Future value in year n (FVn)=Present value (PV)(1+(anual %rate))^# of years

Calvin
PV=400
% rate= 5
# of years= 1
FVn= 658.85

Years to get $658.80= 10.227

Makayla
PV=300
% rate= 6
# of years= 4
FVn= 613.04

Years to get $613.04= 12.265

Difference in the number of years that Calvin and Makayla have their money invested: (12.265-10.227)= 2.038