Question

A manufacturer of bicycles has 4815 wheels, 2305 frames, and 2255 handlebars. (a) How many bicycles can be manufactured using these parts? (b) How many parts of each kind are left over? (c) Which part limits the production of bicycles?

Short answer

(a) 2255 bicycles can be manufactured using the available parts. (b) There will be 305 wheels, 50 frames, and 0 handlebars left over. (c) The handlebars limit the production of bicycles.

Step-by-step solution

Identify the parts needed to manufacture a bicycle

To make a single bicycle, we need 2 wheels, 1 frame, and 1 handlebar.

Calculate the number of bicycles each part can produce separately

We have the following parts available: - 4815 wheels - 2305 frames - 2255 handlebars Divide the number of each part by the requirement for a single bicycle: - Wheels: 4815 wheels / 2 wheels per bicycle = 2407.5 bicycles - Frames: 2305 frames / 1 frame per bicycle = 2305 bicycles - Handlebars: 2255 handlebars / 1 handlebar per bicycle = 2255 bicycles

Determine the maximum number of bicycles that can be produced

To find the maximum number of bicycles that can be manufactured, we need to find the minimum value from our results in Step 2 because that will be the limiting factor. The minimum value is 2255 bicycles. Hence, the manufacturer can produce 2255 bicycles using the available parts.

Calculate the number of leftover parts for each component

To find the leftover parts, we use the number of bicycles that could be produced by each component and subtract the number of bicycles that were actually produced (2255): - Wheels: 4815 wheels - (2255 bicycles * 2 wheels per bicycle) = 305 wheels left - Frames: 2305 frames - 2255 bicycles = 50 frames left - Handlebars: 2255 handlebars - 2255 bicycles = 0 handlebars left

Determine which part limits the production of bicycles

As calculated in Step 2, the limiting factor that resulted in the minimum number of bicycles produced is the handlebars. Therefore, the handlebars limit the production of bicycles. To summarize: (a) The manufacturer can produce 2255 bicycles with the available parts. (b) There are 305 wheels, 50 frames, and 0 handlebars left over. (c) The handlebars limit the production of bicycles.

Key concepts

Limiting Reagent

In the context of manufacturing, the term 'limiting reagent' is analogous to the 'limiting component' or 'limiting part' — an essential concept that determines the maximum output of a production process. When producing an item, there are often multiple components required in specific ratios. If one or more components are not available in the proper ratios, the one in shortest supply (relative to the production requirements) becomes the limiting reagent.

For example, in the case of bicycle production, a manufacturer needs 2 wheels, 1 frame, and 1 handlebar to build a single bicycle. Given a certain number of each component, the limiting reagent is the one that will run out first, halting production. Here, it is the handlebars that limit the production capacity to 2255 bicycles, because they are available in the smallest quantity in comparison to how many are needed per bicycle. This concept ensures that the process of parts-to-product calculation is precise, ultimately affecting the efficiency and profitability of manufacturing operations.

Parts-to-Product Calculation

The 'parts-to-product calculation' is a fundamental stoichiometric process in manufacturing which involves determining how many complete products can be made from a set of given components. It involves a basic mathematical operation where the number of available components is divided by the number of each required for one complete product.

In the case of our bicycle manufacturer, the calculation was performed as follows:

Excess Components

After manufacturing products, companies often deal with 'excess components', which are the leftover parts after the limiting reagent has been fully utilized. Knowledge of excess components is critical for inventory management, cost control, and planning for future production cycles.

In the bicycle manufacturing example, once the maximum number of bicycles is produced (which is limited by the quantity of handlebars), the leftover wheels and frames become excess components - 305 wheels and 50 frames, to be specific. These excesses may be used in subsequent production cycles, but it is paramount for the manufacturer to manage them wisely to prevent overstocking and associated costs. Proper tracking and accounting for excess components helps in making informed decisions about ordering supplies and in reducing waste within the production process.