A cupcake requires 35 grams of sugar and 50 grams of flour, and a muffin requires 30 grams of sugar and 65 grams of flour. Shawna needs to use up at least 460 grams of sugar to make cupcakes and muffins, and she wants to use at most 970 grams of flour. Let C denote the number of cupcakes she makes and M the number of muffins she makes. Write an inequality that represents the condition based on the number of grams of sugar. Write an inequality that represents the condition based on the number of grams of flour.
The correct answer and explanation is:
Let’s define the variables:
- Let CC be the number of cupcakes.
- Let MM be the number of muffins.
Sugar Constraint:
Each cupcake requires 35 grams of sugar, and each muffin requires 30 grams of sugar. Shawna needs to use at least 460 grams of sugar.
Inequality: 35C+30M≥46035C + 30M \geq 460
Flour Constraint:
Each cupcake requires 50 grams of flour, and each muffin requires 65 grams of flour. Shawna wants to use at most 970 grams of flour.
Inequality: 50C+65M≤97050C + 65M \leq 970
Explanation (300 words):
To represent real-life conditions with math, we use inequalities to model constraints. In this problem, Shawna is baking two types of items—cupcakes and muffins—and must manage how much sugar and flour she uses.
We begin by identifying what each item consumes:
- One cupcake uses 35g of sugar and 50g of flour.
- One muffin uses 30g of sugar and 65g of flour.
Let CC and MM represent the number of cupcakes and muffins, respectively. The total amount of sugar used is the sum of sugar in cupcakes and sugar in muffins: 35C+30M35C + 30M. Shawna needs to use at least 460 grams, so this expression must be greater than or equal to 460. This gives the inequality: 35C+30M≥46035C + 30M \geq 460
For flour, the total used is 50C+65M50C + 65M, and Shawna wants to stay within a maximum of 970 grams. So, the total must be less than or equal to 970, resulting in: 50C+65M≤97050C + 65M \leq 970
These inequalities allow Shawna to figure out how many cupcakes and muffins she can bake without exceeding ingredient limits. They are especially useful in optimization problems, where someone may want to maximize or minimize a certain quantity (e.g., profit or cost) subject to such constraints.