Question

Liz has a recipe that uses 1 1/3 cups of sugar and 2 1/4 cups of flour to make 18 muffins. She has a total of 9 cups of flour. Liz wants to use all of her flour to make as many muffins as possible using this recipe.

Exactly how many cups of sugar will Liz use if she use all 9 cups of flour? & how many muffins will Liz make if she uses all 9 cups of flour?

Answer 1

well, first off let's convert the mixed fractions to improper fractions.

`\stackrel{mixed}{1(1)/(3)}\implies \cfrac{1\cdot 3+1}{3}\implies \stackrel{improper}{\cfrac{4}{3}} ~\hfill \stackrel{mixed}{2(1)/(4)}\implies \cfrac{2\cdot 4+1}{4}\implies \stackrel{improper}{\cfrac{9}{4}}`

now, let's use proportions to bind the sugar and flour that she'll be using, now if she were to use the 9 cups of flour, let's see how much will that be in sugar for this recipe

`\begin{array}{ccll} \stackrel{cups}{sugar}&\stackrel{cups}{flour}\\ \cline{1-2} (4)/(3)&(9)/(4)\\[1em] S&9 \end{array}\implies \cfrac{~~ ( 4)/(3 ) ~~}{S}~~ = ~~\cfrac{~~ ( 9)/( 4) ~~}{9}\implies \cfrac{~~ ( 4)/(3 ) ~~}{(S)/(1)}~~ = ~~\cfrac{~~ ( 9)/( 4) ~~}{(9)/(1)}\implies \cfrac{4}{3S}=\cfrac{1}{4} \\\\\\ 16=3S\implies \cfrac{16}{3}=S\implies \boxed{ {\Large \begin{array}{llll} 5(1)/(3)=S \end{array}}}`

now, we know that 9/4 cups of flour gives Liz 18 muffins, well hell how many for 9 cups of flour then?

`\begin{array}{ccll} \stackrel{cups}{flour}&muffins\\ \cline{1-2} (9)/(4) & 18\\[1em] 9& M \end{array} \implies \cfrac{~~ ( 9)/(4 ) ~~}{9}~~=~~\cfrac{18}{M}\implies \cfrac{~~ ( 9)/(4 ) ~~}{(9)/(1)}~~=~~\cfrac{18}{M} \\\\\\ \cfrac{1}{4}=\cfrac{18}{M}\implies \boxed{M=72}`