Problem 25 - Algebra 2 and Trigonometry

Question

A cookie recipe uses flour and sugar in the ratio of 9 : 4. If Nicholas uses 1 cup of sugar, how much flour should he use?

Step-by-Step Solution

Answer

Nicholas should use 2.25 cups of flour.

Step 1: Understanding the Ratio

Nicholas' cookie recipe has a ratio of flour to sugar as 9:4. This means for every 9 parts of flour, there are 4 parts of sugar used in the cookie recipe.

Step 2: Setting Up the Proportion

Let the amount of flour be represented as \( x \) cups. We can set up the ratio proportionally: \( \frac{x}{1} = \frac{9}{4} \). This equation states that \( x \) cups of flour is to 1 cup of sugar, as 9 cups of flour is to 4 cups of sugar.

Step 3: Solving the Equation

To find \( x \), the amount of flour, we solve the equation \( \frac{x}{1} = \frac{9}{4} \). Cross-multiplying gives \( 4x = 9 \).

Step 4: Calculating the Amount of Flour

Solve for \( x \) by dividing both sides of the equation by 4: \( x = \frac{9}{4} \). Simplify \( \frac{9}{4} \) to 2.25.

Step 5: Conclusion

Nicholas should use 2.25 cups of flour when using 1 cup of sugar, following the original ratio of 9:4.

Key Concepts

Understanding Ratios

Ratios are a way to compare two or more quantities. They tell us how much of one thing there is compared to another. In our cookie recipe example, the ratio of flour to sugar is 9:4. This means that for every 9 parts of flour, there are 4 parts of sugar.

Understanding ratios can be really useful in recipes, as it helps keep the consistency and flavor when scaling ingredients up or down. When you see a ratio like 9:4, it can be useful to think of it in terms of parts. If refining the taste or size of the recipe, you can adjust the number of cups accordingly while maintaining the ratio.

• Consider the ratio as a guide to scale ingredients.
• Each number in the ratio can be thought of as 'parts' of the ingredient it represents.
• This helps in maintaining the balance of a recipe, no matter how much of it you make.

Cross-multiplication Basics

Cross-multiplication is a method used to solve equations that involve fractions. It's extremely useful when working with proportions or when equating two ratios. Here’s how it works:

Suppose you have a proportion like \( \frac{a}{b} = \frac{c}{d} \). To solve for one of the unknowns, you can "cross-multiply." Multiply the numerator of one fraction by the denominator of the other and do the same for the other pair, so you get \( a \times d = b \times c \).

In our cookie recipe problem, we had \( \frac{x}{1} = \frac{9}{4} \). By cross-multiplying, we end up with \( 4x = 9 \). This gives us a simple equation to solve.

• Cross-multiplication helps in simplifying the solving process.
• It's especially effective in problems involving proportions.
• Always remember to equalize the products of the cross-multiplied terms.

Fraction Simplification

Simplifying fractions is a mathematical method of reducing the fraction to its simplest form, where numerator and denominator have no common divisors other than 1. Doing so helps in making numbers more manageable and interpretations clearer.

In our example, after using cross-multiplication, Nicholas solved \( x = \frac{9}{4} \). Here, there's no common divisor between 9 and 4, so the fraction is already simplified. However, it's essential to convert this to a more understandable number when necessary, such as turning \( \frac{9}{4} \) into 2.25, which tells us Nicholas needs 2.25 cups of flour.

• Start by finding the greatest common divisor for numerator and denominator.
• Divide both by this number to simplify.
• Convert improper fractions to mixed numbers if the context requires clarity.