10.1: Recipes

Lesson

Let's explore how ratios affect the way a recipe tastes.

Exercise \(\PageIndex{1}\): Flower Pattern

This flower is made up of yellow hexagons, red trapezoids, and green triangles.

Figure \(\PageIndex{1}\)

1. Write sentences to describe the ratios of the shapes that make up this pattern.
2. How many of each shape would be in two copies of this flower pattern?

Exercise \(\PageIndex{2}\): Powdered Drink Mix

Here are diagrams representing three mixtures of powdered drink mix and water:

Figure \(\PageIndex{2}\): A discrete diagram for three quantities, labeled A, B, and C. Diagram A has 4 white squares and 1 blue square. Diagram B has 4 white squares and 1 blue square. Diagram C has 8 white squares and 2 blue squares. Below diagram Key shows 1 white square equals 1 teaspoon drink mix and 1 blue square equals 1 cup water.

1. How would the taste of Mixture A compare to the taste of Mixture B?
2. Use the diagrams to complete each statement:
   1. Mixture B uses ______ cups of water and ______ teaspoons of drink mix. The ratio of cups of water to teaspoons of drink mix in Mixture B is ________.
   2. Mixture C uses ______ cups of water and ______ teaspoons of drink mix. The ratio of cups of water to teaspoons of drink mix in Mixture C is ________.
3. How would the taste of Mixture B compare to the taste of Mixture C?

Are you ready for more?

Sports drinks use sodium (better known as salt) to help people replenish electrolytes. Here are the nutrition labels of two sports drinks.

Figure \(\PageIndex{3}\): Two nutrition fact labels, A and B. Drink A: serving size 8 fluid ounces. Servings per container, 4. Amount per serving: calories, 50. Total Fat, 0 grams; Percent Daily Value, 0. Sodium, 110 milligrams; Percent Daily Value, 5. Potassium, 130 milligrams; Percent Daily Value, 1. Total Carbohydrate, 14 grams; Percent Daily Value, 5. Sugars, 14 grams. Protein, 0 grams. Percent Daily Values are based on a 2,000 calorie diet. Drink B: serving size 12 fluid ounces. Servings per container, 2.5. Amount per serving: calories, 80. Total Fat, 0 grams; Percent Daily Value, 0. Sodium, 150 milligrams; Percent Daily Value, 6. Potassium, 35 milligrams; Percent Daily Value, 1. Total Carbohydrate, 21 grams; Percent Daily Value, 7. Sugars, 20 grams. Protein, 0 grams. Percent Daily Values are based on a 2,000 calorie diet.

1. Which of these drinks is saltier? Explain how you know.
2. If you wanted to make sure a sports drink was less salty than both of the ones given, what ratio of sodium to water would you use?

Exercise \(\PageIndex{3}\): Batches of Cookies

A recipe for one batch of cookies calls for 5 cups of flour and 2 teaspoons of vanilla.

1. Draw a diagram that shows the amount of flour and vanilla needed for two batches of cookies.
2. How many batches can you make with 15 cups of flour and 6 teaspoons of vanilla? Show the additional batches by adding more ingredients to your diagram.
3. How much flour and vanilla would you need for 5 batches of cookies?
4. Whether the ratio of cups of flour to teaspoons of vanilla is \(5:2\), \(10:4\), or \(15:6\), the recipes would make cookies that taste the same. We call these equivalent ratios.
   1. Find another ratio of cups of flour to teaspoons of vanilla that is equivalent to these ratios.
   2. How many batches can you make using this new ratio of ingredients?

Summary

A recipe for fizzy juice says, “Mix 5 cups of cranberry juice with 2 cups of soda water.”

To double this recipe, we would use 10 cups of cranberry juice with 4 cups of soda water. To triple this recipe, we would use 15 cups of cranberry juice with 6 cups of soda water.

This diagram shows a single batch of the recipe, a double batch, and a triple batch:

Figure \(\PageIndex{4}\): Equivalent ratios of a recipe for fizzy juice using cranberry juice and soda water. One batch equals 5 cups cranberry juice and 2 cups soda water. Two batches equal 10 cups cranberry juice and 4 cups soda water. Three batches equal 15 cups cranberry juice and 6 cups soda water.

We say that the ratios \(5:2\), \(10:4\), and \(15:6\) are equivalent. Even though the amounts of each ingredient within a single, double, or triple batch are not the same, they would make fizzy juice that tastes the same.

Practice

Exercise \(\PageIndex{4}\)

A recipe for 1 batch of spice mix says, “Combine 3 teaspoons of mustard seeds, 5 teaspoons of chili powder, and 1 teaspoon of salt.” How many batches are represented by the diagram? Explain or show your reasoning.

Figure \(\PageIndex{5}\): A discrete diagram for three quantities labeled mustard seeds, in teaspoons, chili powder, in teaspoons, and salt, in teaspoons. The data are as follows: mustard seeds, 9 yellow squares. chili powder, 15 red squares. salt, 3 blue squares.

Exercise \(\PageIndex{5}\)

Priya makes chocolate milk by mixing 2 cups of milk and 5 tablespoons of cocoa powder. Draw a diagram that clearly represents two batches of her chocolate milk.

Exercise \(\PageIndex{6}\)

In a recipe for fizzy grape juice, the ratio of cups of sparkling water to cups of grape juice concentrate is 3 to 1.

1. Find two more ratios of cups of sparkling water to cups of juice concentrate that would make a mixture that tastes the same as this recipe.
2. Describe another mixture of sparkling water and grape juice that would taste different than this recipe.

Exercise \(\PageIndex{7}\)

Write the missing number under each tick mark on the number line.

Figure \(\PageIndex{6}\)

(From Unit 2.1.1)

Exercise \(\PageIndex{8}\)

At the kennel, there are 6 dogs for every 5 cats.

1. The ratio of dogs to cats is ______ to ______.
2. The ratio of cats to dogs is ______ to ______.
3. For every ______ dogs there are ______ cats.
4. The ratio of cats to dogs is ______ : ______.

(From Unit 2.1.1)

Exercise \(\PageIndex{9}\)

Elena has 80 unit cubes. What is the volume of the largest cube she can build with them?

(From Unit 1.6.1)

Exercise \(\PageIndex{10}\)

Fill in the blanks to make each equation true.

1. \(3\cdot\frac{1}{3}=\underline{\qquad}\)
2. \(10\cdot\frac{1}{10}=\underline{\qquad}\)
3. \(19\cdot\frac{1}{19}=\underline{\qquad}\)
4. \(a\cdot\frac{1}{a}=\underline{\qquad}\)

(As long as \(a\) does not equal \(0\).)

1. \(5\cdot\underline{\qquad}=1\)
2. \(17\cdot\underline{\qquad}=1\)
3. \(b\cdot\underline{\qquad}=1\)

(From Unit 2.1.1)