Question

Luis is making a dessert that requires three eggs for each cup of flour. a. How many eggs does he need for three cups of flour? b. For a party, Luis made a large batch of his dessert using a dozen eggs. How much flour did he use?

Short answer

a. 9 eggs, b. 4 cups of flour

Step-by-step solution

Identify the ratio of eggs to flour

The problem states that Luis needs three eggs for each cup of flour. Therefore, the ratio of eggs to flour is 3:1.

Calculate the number of eggs for three cups of flour (part a)

Since Luis needs three eggs for each cup of flour, for three cups of flour, he will need: \(3 \text{ cups of flour} \times 3 \text{ eggs per cup} = 9 \text{ eggs}\).

Determine the amount of flour used when a dozen eggs are used (part b)

A dozen eggs equates to 12 eggs. Using the ratio of 3 eggs per cup of flour, set up the equation: \(3x = 12\). Solve for \(x\): \( x = \frac{12}{3} = 4\). This means Luis used 4 cups of flour.

Key concepts

ratios and proportions

Imagine you have a recipe that says you need three eggs for each cup of flour. This relationship between eggs and flour can be described using ratios. A **ratio** shows how much of one thing there is compared to another. For example, the ratio of eggs to flour in Luis's recipe is 3:1. This means for every three eggs, there's one cup of flour.
Ratios are useful because they help us understand and maintain balance in quantities. If you increase one part of the ratio, you must proportionately increase the other to keep the same relationship. Understanding ratios is crucial for solving problems like how many ingredients are needed to scale a recipe.
In Luis's problem, the steps show how you can use the given ratio (3 eggs per cup of flour) to find out both: how many eggs are needed for more flour and how much flour is needed for more eggs.
Remember: To maintain the ratio, always multiply or divide both parts of the ratio by the same number.

algebraic equations

An algebraic equation is a fancy way of saying a math statement with an equals sign that shows the relationship between numbers and variables. In the exercise, the ratio of eggs to flour (3:1) can be represented as an algebraic equation when working out how much flour Luis used with a dozen (12) eggs.
Here's how:
When Luis uses a dozen eggs, you set up the equation based on the given ratio.
You know 3 eggs need 1 cup of flour. So for 12 eggs, you write the equation as:
\(3x = 12\)
Here, \(x\) represents the cups of flour needed. To solve for \(x\), you divide both sides of the equation by 3:
\( x = \frac{12}{3} = 4\)
This tells Luis he needs 4 cups of flour for a dozen eggs. Algebraic equations make solving ratio problems easier by turning them into simple steps.

basic arithmetic

Basic arithmetic is the foundation of all math. It includes four main operations: addition, subtraction, multiplication, and division. In Luis's problem, basic arithmetic helps us break down the steps and solve for the unknowns.
Let’s look at part (a):
If Luis needs 3 eggs per cup of flour, and he’s using 3 cups of flour, you use multiplication:
\(3 \text{ cups of flour} \times 3 \text{ eggs per cup} = 9 \text{ eggs}\)
Similarly, in part (b) when given a dozen eggs (12), you use division to find out cups of flour:
\(12 \text{ eggs} \text{ ÷ 3 eggs per cup} = 4 \text{ cups of flour}\)
Using multiplication and division makes it easier to manage and solve problems efficiently and accurately. So, keep practicing these basic operations, and you'll find math problems much easier to handle!