A laboratory processes two types of chemical samples, Type A and Type B. Processing each T… — GRE General Test

A laboratory processes two types of chemical samples, Type A and Type B. Processing each Type A sample requires $40$ minutes and costs $\$30$ , while processing each Type B sample requires $30$ minutes and costs $\$50$ . On a given day, the laboratory spent a total of $20$ hours processing these two types of samples at a total cost of $\$1,450$ . How many more Type B samples were processed than Type A samples?

5Cevap
B
10
C
15
D
20
E
35

Cevap

5 more Type B samples were processed than Type A samples.

The correct answer is 5. By defining

x

as the number of Type A samples and

y

as the number of Type B samples, we construct the time equation

40x + 30y = 1,200

and cost equation

30x + 50y = 1,450

. Solving this system yields

x = 15

and

y = 20

. The difference

y - x = 20 - 15 = 5

Adım Adım Çözüm

Define variables and convert units to maintain consistency.

Let

x

be the number of Type A samples and

y

be the number of Type B samples. Total time available is

20 \text{ hours} \times 60 \text{ minutes/hour} = 1,200 \text{ minutes}

Time specifications for individual samples are given in minutes, so total time must also be in minutes.

Set up a system of two linear equations representing total time and total cost.

Time equation:

40x + 30y = 1,200 \implies 4x + 3y = 120

. Cost equation:

30x + 50y = 1,450 \implies 3x + 5y = 145

The total processing time is the sum of time spent on each sample type, and total cost is the sum of costs for each sample type.

Solve the system of equations using elimination.

Multiply the time equation by 3:

12x + 9y = 360

. Multiply the cost equation by 4:

12x + 20y = 580

. Subtracting the first from the second gives

11y = 220 \implies y = 20

. Substitute

y = 20

into

4x + 3(20) = 120 \implies 4x = 60 \implies x = 15

Eliminating

x

allows direct calculation of

y

, which then yields

x

Calculate the difference requested by the problem.

Difference =

y - x = 20 - 15 = 5

The question specifically asks for how many more Type B samples were processed than Type A samples.

Anahtar Kavram

Formulating and solving systems of linear equations from real-world rate and budget constraints.

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