Answer:
18
Step-by-step explanation:
First I'm making all mixed numbers improper fractions.
Base recipe: 12 muffins = 2/3 milk + 7/4 flour + 1/4 coconut + 3/4 apple
Change: 9/8 apple
On second look, I ddn't need to do that except for the last one, the change in apple. When increasing things proportionally you basically multiply everything by the same number. A simple example.
Say a recipe made 10 of something with 1 of ingredient A and 2 of ingredient B. Well, what if you made the recipe twice? that would be 20 things from 2 of A and 4 of B. This is the same as changing the recipe to make double. It works if you halve things as well, or multiply by any number or fraction.
So the point is, we changed the amount of apples. We want to know how much we changed it by, then that much is how much the total number of muffins changes. So if we doubled apples the muffins would double, but I don't think it's that simple.
Started at 3/4 cup of chopped apple and increased to 9/8 cups. We want to know what to multiiply by to do this. So we set up a simple algebraic equation.
(3/4)x = 9/8
where x is what we multiply by. Now, to do this you could do one step, divide by 3/4 or you could do it in two steps, multiply both sides by 4 then divide by 3. Or the other way around. Either way you will get the same number.
(3/4)x = 9/8
3x = 9/2
x = 3/2
And you can (and should) double check
(3/4)x = 9/8
(3/4)(3/2) = 9/8
9/8 = 9/8
Yay! If you don't gt fraction multiplication let me know. But the recipie is an increase of 3/2 or 1.5, so there will be 1.5 as many muffins.
12*1.5 = 12(3/2) = 18
![54, a_2, a_3, 128 \\ \\ a_1=54 \\ a_4=128 \\ \\ a_n=a_1 \times r^{n-1} \\ a_4=a_1 \times r^3 \\ 128=54 \times r^3 \\ \frac{128}{54}=r^3 \\ \frac{128 \div 2}{54 \div 2}=r^3 \\ \frac{64}{27}=r^3 \\ \sqrt[3]{\frac{64}{27}}=\sqrt[3]{r^3} \\ \frac{\sqrt[3]{64}}{\sqrt[3]{27}}=r \\ r=\frac{4}{3}](https://tex.z-dn.net/?f=54%2C%20a_2%2C%20a_3%2C%20128%20%5C%5C%20%5C%5C%0Aa_1%3D54%20%5C%5C%0Aa_4%3D128%20%5C%5C%20%5C%5C%0Aa_n%3Da_1%20%5Ctimes%20r%5E%7Bn-1%7D%20%5C%5C%0Aa_4%3Da_1%20%5Ctimes%20r%5E3%20%5C%5C%0A128%3D54%20%5Ctimes%20r%5E3%20%5C%5C%0A%5Cfrac%7B128%7D%7B54%7D%3Dr%5E3%20%5C%5C%20%5Cfrac%7B128%20%5Cdiv%202%7D%7B54%20%5Cdiv%202%7D%3Dr%5E3%20%5C%5C%0A%5Cfrac%7B64%7D%7B27%7D%3Dr%5E3%20%5C%5C%0A%5Csqrt%5B3%5D%7B%5Cfrac%7B64%7D%7B27%7D%7D%3D%5Csqrt%5B3%5D%7Br%5E3%7D%20%5C%5C%0A%5Cfrac%7B%5Csqrt%5B3%5D%7B64%7D%7D%7B%5Csqrt%5B3%5D%7B27%7D%7D%3Dr%20%5C%5C%0Ar%3D%5Cfrac%7B4%7D%7B3%7D)
