| Ingredient | Number of Servings | Amount in 1 Serving | Amount of Sugar 1 Serving |
|---|---|---|---|
| Icing Sugar | \(3\) | \(100 \; [\text{g}]\) | \(98 \; [\text{g}]\) |
| Chocolate Chips | \(6\) | \(15 \; [\text{g}]\) | \(8 \; [\text{g}]\) |
First, we can express the total amount of sugar in the mixture from three servings of icing sugar and six servings of chocolate chip cookies as:
where \(x\) represents the number of servings of icing sugar and \(y\) represents the number of servings of chocolate chips.
Next, we can substitute each serving amount into the equation:
\(A = 98(3) + 8(6)\)
\(A = 294 + 48\)
\(A = 342 \; [\text{g}]\)
Since the recipe makes a dozen (\(12\)) cookies, we divide the total amount of sugar by this value to determine how much sugar is in a single cookie:
\(A_{cookie} = \cfrac{342}{12} = 28.5\)
\(A_{cookie} = 28.5 \; [\text{g}]\)
Therefore, we can determine that there are \(\textbf{28.5 [g]}\) of sugar in each cookie. This is more than your target of \(20\) grams!
The question wants us to use less icing sugar and chocolate chips to make smaller cookies with less sugar. For example, if we made a half recipe, but made \(12\) smaller cookies, each cookie would have \(14.25 \;[\text{g}]\) of sugar. Since each cookie in the dozen contains \(20\;[\text{g}]\) of sugar, we can determine there will be \(240\;[\text{g}]\) of sugar in total.
First, we can write an equation to represent the total amount of sugar in the mixure:
We want the total amount of sugar to be \(20\;[\text{g}]\) times \(12\) cookies. Since this equation has two unknowns variable, we need another equation to solve.
In the originial recipe, the number of servings of chocolate chips is twice as much as the number of servings of icing sugar. We should keep this ratio so the cookies aren't gross (just a little healither!). This can be expressed as:
Now we have two equations each with two unknowns. We can first use substitution to solve for \(x\):
\(240 = 98x + 8y\)
\(240 = 98x + 8(2x)\)
\(240 = 98x + 16x\)
\(240 = 114x\)
\(x = \cfrac{240}{114}\)
\(x \approx 2.1\)
Next, we can plug \(x\) into the second equation to solve for \(y\):
\(y = 2x\)
\(y = 2(2.1)\)
\(y \approx 4.2\)
Therefore, we should use \(\textbf{2}\) servings of icing sugar and \(\textbf{4}\) servings of chocolate chips. We should make about a \(\mathbf{2/3}\) recipe and make smaller cookies.