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Problem 13

# 9 dozen cookies can be made with $$\frac{27}{2}$$ cups of sugar. Write and solve an equation to see how many cups each dozen cookies needs.

Expert verified
It takes $$\frac{3}{2}$$ cups of sugar to make a dozen cookies.
See the step by step solution

## Step 1: Understanding the problem

We are given that 9 dozen cookies can be made with $$\frac{27}{2}$$ cups of sugar. We need to find out how many cups of sugar are needed for 1 dozen cookies. Let x be the number of cups of sugar needed for 1 dozen cookies.

## Step 2: Set up a proportion

We will use a proportion to compare the number of cups of sugar used for 9 dozen cookies and 1 dozen cookies: $\frac{9 \text{ dozen cookies}}{\frac{27}{2} \text{ cups of sugar}} = \frac{1 \text{ dozen cookies}}{x \text{ cups of sugar}}$

## Step 3: Solve the proportion

To solve the proportion for x, we first cross-multiply: $9 \text{ dozen cookies} * x \text{ cups of sugar} = 1 \text{ dozen cookies} * \frac{27}{2} \text{ cups of sugar}$ Now, we have: $9x = \frac{27}{2}$

## Step 4: Isolate x

To isolate x, we need to divide both sides of the equation by 9: $x = \frac{\frac{27}{2}}{9}$

## Step 5: Simplify the expression

To further simplify the expression, we can rewrite the division as a multiplication and invert the divisor: $x = \frac{27}{2} * \frac{1}{9}$ Now, multiply the numerators together and the denominators together: $x = \frac{27 * 1}{2 * 9}$ $x = \frac{27}{18}$ Finally, simplify the fraction by dividing both the numerator and denominator by 9: $x = \frac{27 \div 9}{18 \div 9}$ $x = \frac{3}{2}$

## Step 6: Conclusion

Therefore, it takes $$\frac{3}{2}$$ cups of sugar to make a dozen cookies.

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