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Chapter 5: Problem 67

Leo is planning his spring flower garden. He wants to plant tulip and daffodil bulbs. He will plant 6 times as many daffodil bulbs as tulip bulbs. If he wants to plant 350 bulbs, how many tulip bulbs and how many daffodil bulbs should he plant?

Short Answer

Expert verified
Leo should plant 50 tulip bulbs and 300 daffodil bulbs.

Step by step solution

01

Define Variables

Let the number of tulip bulbs be represented by the variable \( t \). The number of daffodil bulbs is 6 times the number of tulip bulbs, so it can be represented as \( 6t \).
02

Set Up the Equation

According to the problem, the total number of bulbs (tulips and daffodils) is 350. Therefore, the equation is: \[ t + 6t = 350 \]
03

Combine Like Terms

Combine the terms involving \( t \) on the left side of the equation: \[ 7t = 350 \]
04

Solve for \( t \)

Divide both sides of the equation by 7 to solve for \( t \): \[ t = \frac{350}{7} = 50 \]
05

Calculate Number of Daffodil Bulbs

Since the number of daffodil bulbs is 6 times the number of tulip bulbs, we calculate: \[ 6t = 6 \times 50 = 300 \]
06

Verify the Solution

Add the number of tulip bulbs and daffodil bulbs to ensure it matches the total: \[ 50 + 300 = 350 \] This confirms the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equations
Solving problems often involves setting up and solving equations. An equation is a mathematical statement that asserts the equality of two expressions. In this problem, we set up an equation to represent the relationship between the tulip bulbs and daffodil bulbs. The equation used is: \[ t + 6t = 350 \] where \( t \) is the number of tulip bulbs. This equality shows that the sum of the number of tulip bulbs and six times the number of tulip bulbs equals the total number of bulbs, which is 350. By solving this equation, we can find the value of \( t \). Breaking down and writing these equations properly is crucial for finding the correct solution to the problem.
Variables
Variables are symbols used to represent unknown values. In algebra, we use variables to make complex problems more manageable. In this problem, the variable \( t \) is used to represent the number of tulip bulbs. We also represented the number of daffodil bulbs with the expression \( 6t \) because there are six times more daffodil bulbs than tulip bulbs. By defining these variables clearly:
  • \( t \): Number of tulip bulbs
  • \( 6t \): Number of daffodil bulbs
it becomes easier to set up and solve the equations we need to find the solution.
Linear Equations
A linear equation is an equation between two variables that gives a straight line when plotted on a graph. In our problem, the equation \( t + 6t = 350 \) is a linear equation because it involves a single power of the variable \( t \). Linear equations are of the form \( ax + by = c \). Our equation simplifies to \( 7t = 350 \), which is straightforward to solve.
  • Combine like terms to simplify: \( t + 6t = 7t \).
  • Solve for \( t \) by dividing both sides by 7: \( 7t = 350 \) thus \( t = 50 \).
Knowing how to handle linear equations is a fundamental skill in algebra.
Problem-Solving
Effective problem-solving in algebra involves several steps. Here's how we tackled the flower bulb problem:
  • Define the Variables: Clearly state what each variable represents.
  • Set Up the Equation: Create an equation based on the problem's conditions.
  • Simplify and Solve: Combine like terms and solve for the variable.
  • Verify the Solution: Check that your solution makes sense and satisfies the original conditions.
In this case, we checked that the final numbers of tulip and daffodil bulbs add up to the total specified. Being systematic and thorough will lead to accurate solutions.

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Most popular questions from this chapter

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