Trying to figure out % of a number I don't know

Question

This should be a simple question, but maybe it's not?

These folks have a total of 176,524 fruits. Apples make up 38,195, at 21.64%. BUT they want apples to make up 40%.

Instead of just saying they need 70,609 apples to get to 40%, I'm trying to figure out the incremental number of total fruits needed, so 40% are apples, and the number of apples based on the new total fruits.

I can't seem to put together a formula to determine this.

Answer 1

New number of apples = $0.40\times$ New number of fruit.

New numbers of apples is $38,195 + X$ and New number of fruit is $176,524 + X$.

So Solve:

$38,195 + X = 0.40\times (176,524 + X)$

$X - 0.40 X = 0.40\times(176524)-38195\implies 0.60 X= 32414.6\implies X = \frac{32414.6}{0.6}=54024\frac 13$.

Answer 2

Let's say we have $A$ apples (in your case 38,195) and a total of $F$ fruits (in your case 176,524). We want to add $\delta A$ apples so that we have 40% apples, i.e. $$ \frac{A+\delta A}{F+\delta A} = 0.4. $$ Solving this for $\delta A$, we get $$ A+\delta A = 0.4(F+\delta A), \\ 0.6\delta A = 0.4F+A, \\ \delta A = (0.4F-A)/0.6. $$ Plugging in your numbers, we get $\delta A = (0.4F-A)/0.6=54024.33333333334\approx54024$ apples.

Answer 3

A more concrete solution: if apples are to be 40%, then the non-apples must be 60% of the total. Rather than call them non-apples all the time let’s agree that the other fruits happen to be oranges. There are currently $176524-38195 = 138329$ oranges in the mix, so in order for this to be 60% of the total (without adding any more oranges), the number of fruits should be $138329 /0.6 \approx 230548$. We currently have $176524$ fruits so we need to add $230548-176524 = 54024$ more apples.