A manufacturing shop produces two models of custom bicycle frames: Standard and Deluxe. Pr… — GMAT

A manufacturing shop produces two models of custom bicycle frames: Standard and Deluxe. Producing each Standard frame requires $2$ hours of welding and $1$ hour of painting. Producing each Deluxe frame requires $3$ hours of welding and $2$ hours of painting. During a single week, the shop logged a total of $130$ hours of welding and $75$ hours of painting for these two models. How many Deluxe bicycle frames were produced during that week?

Answer: 20 frames

Answer

The number of Deluxe bicycle frames produced during that week is 20.

Translating the resource limitations into a system of two linear equations yields

2x + 3y = 130

for welding hours and

x + 2y = 75

for painting hours, where

x

and

y

represent the number of Standard and Deluxe frames respectively. Expressing

x

in terms of

y

from the painting equation gives

x = 75 - 2y

. Substituting this expression into the welding equation gives

2(75 - 2y) + 3y = 130

, which simplifies to

150 - y = 130

, giving

y = 20

Deluxe frames.

Step-by-Step Solution

Define variables for the unknowns.

Let

x

be the number of Standard bicycle frames produced and

y

be the number of Deluxe bicycle frames produced.

Assigning variables allows us to translate the problem into algebraic expressions.

Set up a system of linear equations.

Welding constraint:

2x + 3y = 130

Painting constraint:

x + 2y = 75

Each constraint represents the sum of hours spent on Standard and Deluxe frames for that process.

Solve the system using elimination or substitution.

Multiply the painting equation by

2

2x + 4y = 150

.
Subtract the welding equation (

2x + 3y = 130

) from this result:

(2x + 4y) - (2x + 3y) = 150 - 130 \Rightarrow y = 20

Eliminating

x

directly solves for

y

, which is the requested quantity (Deluxe frames).

Key Concept

Solving systems of two linear equations with two variables by substitution or elimination.

Estimated Time:1m 30s

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