A parent raises a child's allowance by 12% each year. If the allowance is $3.50 now, when will it reach $18?
18 = 3.5(1.12)^n
5.142857 = 1.12^n
take log of both sides and use log rules....
n = log 5.142857/log 1.12 = 14.45
so after 14 years the allowance will be below $18
but after 15 years it will be over $18
To find out when the child's allowance will reach $18, we can set up an equation using the given information.
Let's denote the number of years as 'x'.
The parent raises the allowance by 12% each year, which means the child's allowance will increase by 12% of its previous value each year. So, the child's allowance after x years can be calculated using the formula:
Allowance after x years = Initial allowance + (Percentage increase × Initial allowance)
In this case, the initial allowance is $3.50, and the percentage increase is 12%. Thus, the equation becomes:
Allowance after x years = $3.50 + (0.12 × $3.50)
We can simplify this equation as:
Allowance after x years = $3.50(1 + 0.12)
Now, we need to solve for 'x' when the allowance reaches $18. So, we can set up the equation:
$3.50(1 + 0.12)^x = $18
To solve for 'x', we can rearrange the equation as:
(1 + 0.12)^x = $18 / $3.50
Now, we can solve this equation using logarithms, specifically the natural logarithm (ln). Taking the natural logarithm of both sides, we have:
ln[(1 + 0.12)^x] = ln($18 / $3.50)
Now, we can simplify further:
x ln(1 + 0.12) = ln($18 / $3.50)
Finally, to isolate 'x', we divide both sides of the equation by ln(1 + 0.12):
x = ln($18 / $3.50) / ln(1 + 0.12)
Using a calculator, we can evaluate the right side of the equation to find 'x'.