What is the percent of increase from $147 annually to $414?

Question

Cameron Arnold · Accepted Answer

#color(red)("The time span not defined as 1 year")#
For 1 year only #color(blue)(-> "percentage is " 187.5%)#
Or simple interest over whole period
'.....................................................
For more than 1 year where #n# is number of years
and annually compounded interest.

#" "color(blue)(x%=log^(-1)((log(414)-log(144))/n + 2) - 100)# #color(red)("You have not stated the number of years.")# #color(red)("The use of the word 'annually' implies more than 1 year")# Let the number of years be #n# #" "$144(1+x/100)^n=$414# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Suppose the number of years is 1 then")# #" "144+(144xx ?/100)=414# #color(brown)("Subtract 144 from both sides")# #" "144xx ?/100=270# #color(brown)("Multiply both sides by "100/144)# #color(blue)(" "?%=270xx100/144 = 187.5%)# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Suppose "n>1)color(magenta)(" "->" Needs algebra")# #color(brown)("Notice that " (1+x/100) ->((100+x)/100))# #color(brown)("Taking logs of both sides")# #" "log($144)+nlog((100+x)/100)=log($414)# #" "nlog((100+x)/100)=log(414)-log(144)# #" "log((100+x)/100)=(log(414)-log(144))/n# #" "log(100+x)-log(100)=(log(414)-log(144))/n# #" "log(100+x)=(log(414)-log(144))/n + log(100)# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(brown)("But "log(100)=2# #" "log(100+x)=(log(414)-log(144))/n + 2# #" "100+x=log^(-1)((log(414)-log(144))/n + 2)# #" "color(blue)(x%=log^(-1)((log(414)-log(144))/n + 2) - 100)#

Genesis Amaya · Answer

undefined To find the percent increase from $147 annually to $414, you can use the following formula:

$$ \text{Percent Increase} = \left( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \right) \times 100\% $$

Substituting the given values:

$$ \text{Percent Increase} = \left( \frac{414 - 147}{147} \right) \times 100\% $$

$$ \text{Percent Increase} = \left( \frac{267}{147} \right) \times 100\% $$

$$ \text{Percent Increase} = 1.816 \times 100\% $$

$$ \text{Percent Increase} = 181.6\% $$

So, the percent increase from $147 annually to $414 is 181.6%.