Problem 330 - Prealgebra

Question

In the following exercises, solve. Gardening Hiroko is putting in a vegetable garden. He wants to have twice as many lettuce plants as tomato plants. If he buys 12 tomato plants, how many lettuce plants should he get?

Step-by-Step Solution

Answer

Hiroko should buy 24 lettuce plants.

Step 1: Understand the relationship

Hiroko wants to have twice as many lettuce plants as tomato plants.

Step 2: Define the quantities

Let the number of tomato plants be represented by \( T \) and the number of lettuce plants by \( L \). According to the problem, \( T = 12 \).

Step 3: Set up the equation

Since Hiroko wants twice as many lettuce plants as tomato plants, we can write the equation \( L = 2T \).

Step 4: Substitute the known value

Substitute \( T = 12 \) into the equation: \( L = 2 \times 12 \).

Step 5: Solve the equation

Calculate \( L = 24 \).

Step 6: Interpret the result

Hiroko should buy 24 lettuce plants.

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental concept in solving word problems. These expressions involve variables, numbers, and operation symbols like plus, minus, multiply, and divide. For instance, in the given problem, we are dealing with variables that represent the number of tomato and lettuce plants. We used the variable \( T \) for tomato plants and \( L \) for lettuce plants. The expression \( L = 2T \) is an example of an algebraic expression. It tells us that the number of lettuce plants is twice the number of tomato plants.

Understanding algebraic expressions enables you to translate real-world situations into mathematical statements. This translation is crucial for setting up equations to solve word problems effectively. Practice forming and interpreting expressions to enhance this skill.

Equations

Equations are mathematical statements that show the equality between two expressions. In our gardening problem, the equation helps us find out how many lettuce plants Hiroko needs. The main equation here is \( L = 2T \).

To solve an equation, you often need to substitute known values. For instance, we know the number of tomato plants: \( T = 12 \). Substituting this value into the equation gives us \( L = 2 \times 12 \). Solving it, we get \( L = 24 \). Thus, Hiroko needs to buy 24 lettuce plants.

Equations help you find unknown quantities based on known relationships. Always write down what you know and translate the word problem into an equation. Then, solve step by step while keeping track of your substitutions.

Proportions

Proportions deal with the relative sizes of quantities and how they change in relation to each other. In the gardening problem, the proportion is between the tomato and lettuce plants. Hiroko wants twice as many lettuce plants as tomato plants, giving us a simple proportion.

The equation \( L = 2T \) represents a proportional relationship. This means that however many tomato plants Hiroko buys, he should buy twice that number of lettuce plants to maintain the desired proportion.

Recognizing and working with proportions is essential in solving many real-world problems. Whether you're cooking, planning a garden, or working on any other task, understanding how quantities relate to one another simplifies the process and ensures accuracy.