If a machine depreciates each year at the rate of 50% of its previou

Question

If a machine depreciates each year at the rate of 50% of its previous value, was the machine worth less than $12,000 at the end of three years?

(1) The machine’s starting value was more than $95,000

(2) The machine depreciated more than $24,000 during its second year.

IanStewart · Answer

The stem doesn't make sense in English, but I think it's trying to say that the machine loses 50% of its value each year. So after one year, it will be worth 1/2 its starting value, after two years 1/4 of its starting value, and after three years 1/8 of its starting value.

For it to be worth less than $12,000 after three years, initially it would need to be worth less than (8)(12,000) = $96,000. So that's the question we're trying to answer: was the original value of the machine less than $96,000?

Statement 1 is now clearly insufficient.

There's a conceptual way to look at Statement 2. If you lose half your money, the amount you lose equals the amount you'll have left over. For example, if you have $12, and lose half of it, you lose $6 and have $6 left. So if in the second year, the machine loses more than $24,000 in value, it must be worth more than $24,000 after that, and thus more than $12,000 when it loses another half of its value, and Statement 2 is sufficient to give a "no" answer to the question.

Or you could look at it more algebraically. At the start of the second year, the machine is worth half of its initial value. It then loses half of its value again, so it loses (1/2)(1/2) = 1/4 of its initial value during the second year. Statement 2 thus tells us that 1/4 of the machine's original value is greater than $24,000, so its original value exceeds (4)(24,000) = $96,000, and Statement 2 is sufficient (the answer to the question must be "no"). So the answer is B.

newyork2012 · Answer

Let’s say the original value is a
End of 1st: 1/2 a
End of 2nd:1/4 a
End of 3rd:1/8 a

Is 1/8 a < 12000? ....
Well then we need a< 96000, or 1/2a<48000, or 1/4a< 24000

A..a>95000
Not sufficient ...a could be 95500 or 96500

B..1/4 a > 24000 .. sufficient

Posted from my mobile device

Kritisood · Answer

This is a successive change problem
let x be the value of the machine before depreciating.
depreciating for three years basically means successively losing 50% of its value for three years:

Ques asks $x*\frac{1}{2}*\frac{1}{2}*\frac{1}{2}<12,000?$
=> x<96,000 (simplified)

st 1: x is more than 95,000 NS
st 2: x*1/2*1/2>24000
x>96000
sufficient

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If a machine depreciates each year at the rate of 50% of its previou

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