Question

A bus company always keeps 3 tires in stock for every bus it owns, plus an additional 30 tires in stock for emergencies. According to this policy, the bus company needs to have a total of 120 tires in stock. How many buses does the company own? F. 30 G. 35 H. 40 I. 45 K. 50

Short answer

Answer: The company owns 30 buses.

Step-by-step solution

Define variables and equation

Let 'b' represent the number of buses the company owns. The bus company keeps 3 tires in stock for every bus, so it keeps 3 * b tires. It also has an additional 30 tires for emergencies. In total, the company has 120 tires in stock. We can form the equation: 3 * b + 30 = 120.

Solve the equation

Now, we need to solve the equation from step 1 to find the number of buses the company owns: 3 * b + 30 = 120 Subtract 30 from both sides: 3 * b = 90 Divide by 3: b = 30

Choose the correct answer

The company owns 30 buses. The correct answer is (F).

Key concepts

Algebraic Word Problems

Algebraic word problems require translating a real-world scenario into a mathematical equation using variables to represent unknown quantities. Let's take our bus company problem as an example. We're given a practical situation - the company's tire stock policy - and asked a question about it: how many buses does the company own based on their tire stock?

First, identify the key information: the ratio of tires to buses (3:1), the emergency stock (30 tires), and the total stock (120 tires). These pieces become parts of our equation. To improve understanding, always break down the problem and identify what each number in the question represents. This makes it easier to form an equation that models the problem.

Identifying Variables

In our problem, the number of buses is unknown, so we define a variable, typically 'b' or 'x', to represent this unknown. By translating the words into an equation, we set up a framework to find the answer mathematically.

Part of the strategy is to look for keywords that indicate mathematical operations. In this case, 'for every' suggests multiplication, and 'additional' refers to an extra amount we need to add to the total. Remember that understanding the language of word problems is crucial in setting up the correct equation.

Equation Solving

Once an algebraic word problem is translated into an equation, the next step is to solve for the unknown variable. Solving equations is a fundamental skill in math, involving manipulation of the equation to isolate the variable.

In the bus company problem, we have the equation:
\[3b + 30 = 120\].
The goal is to solve for 'b', which represents the number of buses. The process typically involves simplifying and performing inverse operations to get 'b' by itself on one side of the equation.

Simplification Steps

The simplification process may include several steps:

• Combining like terms
• Using inverse operations (addition/subtraction, multiplication/division)

For example, we subtract 30 from both sides to get rid of the extra tires not associated with the buses directly, and then divide by 3 to find the number of buses. These steps are fundamental and are repeated across many types of equation problems.

Practicing different types of equations (linear, quadratic, exponential, etc.) will build a student's confidence and skill in solving various algebraic problems.

Math Problem Solving Strategies

Successful math problem-solving requires a set of strategies that help break down and navigate through complex problems. These strategies can be applied to algebraic word problems and beyond, enhancing overall understanding and efficiency.

Key Strategies Include:

• Understanding the problem: Read it carefully and determine what is being asked.
• Devising a plan: Form a strategy on how to tackle the problem based on what information is given and what is being sought.
• Carrying out the plan: Execute the chosen strategy, which might involve equation solving, drawing diagrams, or logical reasoning.
• Reviewing/reflecting: Once a solution is reached, look back to ensure it makes sense within the context of the problem.

For our bus company problem, the plan involves setting up an equation based on the policy, executing algebraic steps to solve that equation, and then reviewing to make sure the number of buses makes sense with the given tire policy.

Applying these strategies not only leads to the correct answer but also builds a deeper understanding of the mathematical concepts involved. Encourage students to write out their thought process - this reflection helps to reinforce learning and provides a reference for similar problems in the future.