In Travis County, only full-time and part-time teachers are

Question

In Travis County, only full-time and part-time teachers are employed, and every full-time teacher has the same salary, as does every part-time teacher. The first proposal is to increase the salary of every full-time teacher by 10% and reduce that of every part-time teacher by 2%. The second proposal is to reduce the salary of every full-time teacher by 10% and increase that of every part-time teacher by 2%. If the total payroll under the first proposal would be 8 1⁄3% higher than it would be under the second, what fraction of the current payroll is paid to full-time teachers?

(A) 1/4
(B) 1/3
(C) 2/5
(D) 1/2
(E) 3/5

(A) 1/4
(B) 1/3
(C) 2/5
(D) 1/2
(E) 3/5

QuizBall · Answer

Kevin I'm not even close to knowing how to solve this one haha

So when you increase full time by 10%, and lower part time by 2% - it is 8 1/3% higher than: Reducing full time by 2%, and increasing part time by 10%.....

1.1F + 0.98 P = 1.083 (0.98F + 1.1P) ?
F = 3P

I don't think I'm approaching this correctly

I eventually just said 1/3 because of 8 1/3% lol

kevincan · Answer

Here’s a good start :

Let x be the fraction of the current payroll paid to full-time teachers.

Then the current payroll can be normalized to 1:

Full-time payroll: x
Part-time payroll: 1 − x

Under the first proposal:

P1 = 1.10x + 0.98(1 − x)
= 0.98 + 0.12x

Under the second proposal:

P2 = 0.90x + 1.02(1 − x)
= 1.02 − 0.12x

kevincan · Answer

What does F stand for ?

QuizBall · Answer

I was going for F = a full time teacher’s salary, and P = a part time teacher’s salary. But now I see that it’s easier to think of X being the total payroll

Regor60 · Answer

The question is ambiguously written as it relates to the language "exactly the opposite", which can be reasonably interpreted as decrease the salary of the full-time teacher by 2% and increase the salary of part-time teachers by 10%, which was already demonstrated by one of the responses.

The question should be rewritten to clarify its intent.

The question should be rewritten to clarify its intent.

kevincan · Answer

That was indeed ambiguous : I ought to spell it out. Thanks

kevincan · Answer

Let’s call the current full time teacher payroll f and the current part time teacher payroll p

We should know that 83 1/3% is 5/6, and thus 8 1/3 % is 5/60= 1/12

1.1f + 0.98p = (1 +1/12)(0.9f + 1.02p)

12(1.1f + 0.98p)= 13(0.9f + 1.02p)

1.5f =13(p +p/50) - 12(p - p/50)= 1.5p

f=p

Answer 1/2

In Travis County, only full-time and part-time teachers are

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In Travis County, only full-time and part-time teachers are

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