A sum of money is invested in a business. In each year, this investment earns 11 over 2 times as much as in the preceding year. If the investment earned a total of $29,250 in four years, how much did it earn in the fourth year?
earning in first year --- x
earining in 2nd year = 5.5x
earning in 3rd year = 5.5^2 x
earning in 4th year = 5.5^3 x
x(1 + 5.5 + 5.5^2 + 5.5^3) = 29250
inside the bracket is a GP, but since there are only 4 times, we might as well just add them up
x(203.125) = 29250
x = 144
so in the 4th year they earned 5.5^3(144) or 23,958
check:
in first year --- 144
in 2nd year --- 792
in 3rd year --- 4356
in 4th year --- 23958
add them up to get 29,250
1. A sum of money is invested in a business. In each year this investment earns 1 times as much as in the preceding year. If the investment earned a total of $29,250 in four years, how much did it earn in the fourth year?
• CORRECT: $12,150
How to get it is to find out the original investment and do that by the following steps
amount= origamount*1.5^timeInYears
so in four years, amount=orig*1.5^4
earned=amount-orig=orig^1.5^4-orig
29250=orig (1.5^4-1)
29250=x(1.5^4-1)
Let's solve your equation step-by-step.
29250=x(1.54−1)
Step 1: Simplify both sides of the equation.
29250=4.0625x
Step 2: Flip the equation.
4.0625x=29250
Step 3: Divide both sides by 4.0625.
4.0625x
4.0625
=
29250
4.0625
x=7200
Witch is 7200 then times it by 1.5^4 then
7200(1.54−1.53)
=(7200)(1.54+−3.375)
=(7200)(1.54)+(7200)(−3.375)
=36450−24300
=12150
To solve this problem, let's start by assigning variables to the different unknowns. Let's call the amount of money invested in the business "x", and let's call the amount earned in the first year "y".
According to the problem, the investment earns 11/2 times as much as in the preceding year. So, in the second year, the amount earned is (11/2) * y, in the third year, it is (11/2) * (11/2) * y, and in the fourth year, it is (11/2) * (11/2) * (11/2) * y.
Now, we can set up an equation using the information given in the problem. The total amount earned in four years is $29,250. So, the equation is:
y + (11/2) * y + (11/2) * (11/2) * y + (11/2) * (11/2) * (11/2) * y = 29,250
To simplify this, we can write it as:
y + (11/2)y + (121/4)y + (1331/8)y = 29,250
To add the fractions on the left side, we need a common denominator. The common denominator is 8. So, we rewrite the equation as:
(8/8)y + (44/8)y + (121/4)y + (1331/8)y = 29,250
Now, we can combine the terms:
(1504/8)y = 29,250
To isolate "y", we can multiply both sides of the equation by 8/1504:
y = 29,250 * (8/1504)
Calculating this, we find that y ≈ $155.23 (rounded to the nearest cent).
Therefore, the amount earned in the fourth year is (11/2) * (11/2) * (11/2) * y:
(11/2) * (11/2) * (11/2) * y ≈ (11/2) * (11/2) * (11/2) * 155.23 ≈ $2,211.24 (rounded to the nearest cent).
So, the investment earned approximately $2,211.24 in the fourth year.