Question

Magnabulk Corp sells boxes holding \(d\) magnets each. The boxes are shipped in crates, each holding \(b\) boxes. What is the price charged per magnet, in cents, if Magnabulk charges \(m\) dollars for each crate? A. $$\frac{100 b d}{m}$$ B. $$\frac{100 m}{b d}$$ C. $$\frac{b d}{100 m}$$ D. $$\frac{m}{b d}$$ E. $$\frac{b d}{m}$$

Short answer

B. \( \frac{100 m}{b d} \)

Step-by-step solution

Identify Variable Relationships

Identify the relationships between the variables: Magnabulk charges m dollars for each crate which holds b boxes. Each box holds d magnets.

Calculate Total Magnets Per Crate

Calculate the total number of magnets in one crate. A crate has b boxes, and each box contains d magnets, so the total number of magnets in a crate is \[b \times d\].

Convert Dollars to Cents

Convert the total cost in dollars to cents, since the price per magnet needs to be in cents. There are 100 cents in a dollar, so \[m\] dollars is equivalent to \[100 \times m\] cents.

Calculate Price Per Magnet

Divide the total cost in cents by the total number of magnets to find the price per magnet. This gives \[ \frac{100m}{bd} \].

Choose the Correct Option

Match the calculated expression \[ \frac{100m}{bd} \] with the given answers. The correct option is B.

Key concepts

Algebraic Expressions

Algebraic expressions help us to represent mathematical situations using symbols and variables. In this problem, we use variables to stand for unknown values. For example, the number of boxes in a crate is represented by the variable \(b\). Similarly, \(d\) stands for the number of magnets in each box, and \(m\) stands for the price of one crate in dollars. Using these expressions makes it easier to set up and solve the problem by manipulating these variables to find the price per magnet.

Unit Conversion

Unit conversion is crucial when different units are involved. Magnabulk charges for crates in dollars, but we need the price per magnet in cents. Therefore, converting the total cost from dollars to cents is essential. One dollar equals 100 cents. Thus, if the price of a crate is \(m\) dollars, it converts to \(100 \times m\) cents. This step ensures that all units are consistent, simplifying the calculation.

Price Calculation

Calculating the price per magnet involves understanding the total cost and distributing it across the total number of magnets. First, calculate how many magnets are in one crate. Since each crate contains \(b\) boxes and each box contains \(d\) magnets, multiply \(b \times d\) to get the total magnets per crate. Then, to find the price per magnet in cents, divide the total cost in cents (\(100 \times m\) cents) by the total number of magnets (\(b \times d\)). This gives the formula \[ \frac{100m}{bd} \].

Variable Relationships

Understanding the relationships between variables is key to solving word problems. In this scenario, the relationships are:

• \(m\) dollars is the total price for one crate
• Each crate contains \(b\) boxes
• Each box contains \(d\) magnets

Use these relationships to find the total number of magnets in a crate and then distribute the cost to find the price per magnet. Identifying and connecting these variables correctly makes it easier to form and solve the algebraic expression needed to find the solution.