Answer

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**Hint:**

We assume the number of cups of butter to be used as $ 2x $ and the number of cups sugar to be used as $ 3x $. We equate $ 2x $ to a given number of cups of butter and solve for $ x $. We put the value of $ x $ in $ 3x $ to find the number of cups sugar to be used. \[\]

**Complete step by step answer:**

We know that a ratio is a fraction with both numerator and denominator as positive numbers. If $ a $ and $ b $ are two positive numbers then the ratio from $ a $ to $ b $ is given as

\[a:b=\dfrac{a}{b}\]

We can multiply or divide a positive number $ k $ and the value of ratio will not change. It means

\[\begin{align}

& a:b=ka:kb \\

& a:b=\dfrac{k}{a}:\dfrac{k}{b} \\

\end{align}\]

We are given in the question that a recipe for cookies uses butter and sugar in ratio 2:3. Since we can multiply any positive number without changing the value soft the ratio. We multiply positive number $ x $ to have the ratio as

\[2x:3x=2:3\]

Let us assume the number of cups of butter to be used as $ 2x $ and the number of cups sugar to be used as $ 3x $ . We are given that we are using 8 cups of butter. So we have

\[2x=8\]

We divide both sides by 4 to have;

\[\begin{align}

& \Rightarrow \dfrac{2x}{2}=\dfrac{8}{2} \\

& \Rightarrow x=4 \\

\end{align}\]

So the number of cups of sugar to be used is

\[3x=2\times 4=12\]

**Note:**

We note that we can find the total number of cups as $ 3x+2x=5x=4\times 4=20 $. We also note that we use ratios to compare the same type of quantities. Here in this problem, we were comparing a number of cups. We also note that the quantities must have the same unit of measure. Here if we want to compare butter and sugar by weight they must be in the same unit that is either in kilogram or gram but not both. We can alternatively solve using the unitary method.

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