Miguel is baking cookies for the bake sale using a recipe that requires flour, sugar, and butter in a ratio of 5 to 2 to 1 by volume in cups. suppose miguel uses c cups of butter. his mother tells him to use one cup less of sugar than the recipe requires. then the expression 5c + (2c - 1) + c represents the total number of cups of ingredients that miguel uses. how many cups of butter would he use to have a total of 27 cups of ingredients?

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To determine the number of butter that is needed in the problem, we need an equation that would relate the three variables since these affects the number of cups of ingredients. We are given the ratio of the three ingredients which is 5 cups of flour per 2 cups of sugar per 1 cup of butter. From this we can set up the equation as:

total number of cups = 5c + 2c +1c

However, it was said that the number of cups of sugar has to be lessened by 1 cup so the equation would change accordingly as

 total number of cups = 5c + (2c -1) +c

We calculate as follows:

27 = 5c + (2c -1) +c
27 = 5c + 2c - 1 + c
28 = 8c
c = 7/2

Therefore, Miguel needs 7/2 or 3 and 1/2 cups of butter.