An artisan workshop produces custom wooden chairs and tables. Each chair requires 3 hours … — GMAT

An artisan workshop produces custom wooden chairs and tables. Each chair requires $3$ hours of carving and $2$ hours of finishing, while each table requires $5$ hours of carving and $4$ hours of finishing. If the workshop logged a total of $110$ hours of carving and $84$ hours of finishing last week, how many tables were produced?

A
10
B
14
16Cevap
D
18
E
22

Cevap

The workshop produced 16 tables.

By defining

c

as the number of chairs and

t

as the number of tables, we can set up two linear equations representing total hours:

3c + 5t = 110

for carving and

2c + 4t = 84

for finishing. Multiplying the second equation by

1.5

yields

3c + 6t = 126

. Subtracting

3c + 5t = 110

from

3c + 6t = 126

leaves

t = 16

. Thus, the workshop produced 16 tables.

Adım Adım Çözüm

Define variables and translate the word problem into a system of linear equations.

Let

c

be the number of chairs and

t

be the number of tables. Carving equation:

3c + 5t = 110

. Finishing equation:

2c + 4t = 84

The total hours for each activity equal the sum of hours spent on chairs and tables.

Eliminate variable

c

to solve for

t

Multiply the finishing equation by

1.5

to get

3c + 6t = 126

. Subtract the carving equation (

3c + 5t = 110

) from this equation:

(3c + 6t) - (3c + 5t) = 126 - 110 \implies t = 16

Aligning the coefficient of

c

in both equations allows direct elimination of

c

to isolate

t

Verify the solution by calculating

c

and checking both original equations.

Substitute

t = 16

into

2c + 4(16) = 84 \implies 2c + 64 = 84 \implies 2c = 20 \implies c = 10

. Check carving:

3(10) + 5(16) = 30 + 80 = 110

Ensures that

t = 16

and

c = 10

satisfy both resource constraints without calculation errors.

Anahtar Kavram

Setting up and solving a system of two linear equations in two variables

İpuçları

Set up two separate linear equations: one for total carving hours and one for total finishing hours.

Let

c

be the number of chairs and

t

be the number of tables. Your system is

3c + 5t = 110

and

2c + 4t = 84

Multiply

2c + 4t = 84

1.5

to get

3c + 6t = 126

, then subtract

3c + 5t = 110

to find

t

directly.

Daha Fazla Pratik

Try solving a similar problem where the total revenue and total unit count are given to practice standard linear system modeling.

Alternatif Yöntem

Divide the finishing equation

2c + 4t = 84

2

to get

c + 2t = 42 \implies c = 42 - 2t

. Substitute this into the carving equation:

3(42 - 2t) + 5t = 110 \implies 126 - 6t + 5t = 110 \implies -t = -16 \implies t = 16

Tahmini Süre:1m 30s

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Cevap

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Anahtar Kavram

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İpuçları

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