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At the end of each year, the value of a certain antique watch is "c" percent more than its value one year earlier, where "c" has the same value each year. If the value of the watch was "k" dollars on January 1, 1992, and "m" dollars on January 1, 1994, then in terms of "m" and "k", what was the value of the watch, in dollars, on January 1, 1995?
A. \(m+\frac{1}{2}(m-k)\)
B. \(m+\frac{1}{2}*\frac{m-k}{k}*m\)
C. \(\frac{m√m}{√k}\)
D. \(\frac{m^2}{2k}\)
E. \(km^2\)
A. \(m+\frac{1}{2}(m-k)\)
B. \(m+\frac{1}{2}*\frac{m-k}{k}*m\)
C. \(\frac{m√m}{√k}\)
D. \(\frac{m^2}{2k}\)
E. \(km^2\)
Thiagaraj
Yes, it should.
At the end of each year, the value of a certain antique watch is "c" percent more than its value one year earlier, where "c" has the same value each year. If the value of the watch was "k" dollars on January 1, 1992, and "m" dollars on January 1, 1994, then in terms of "m" and "k", what was the value of the watch, in dollars, on January 1, 1995?
A. \(m+\frac{1}{2}(m-k)\)
B. \(m+\frac{1}{2}*\frac{m-k}{k}*m\)
C. \(\frac{m√m}{√k}\)
D. \(\frac{m^2}{2k}\)
E. \(km^2\)
Price in 1992 - \(k\);
Price in 1993 - \(k*(1+\frac{c}{100})\);
Price in 1994 - \(k*(1+\frac{c}{100})^2=m\). From this we can get that \((1+\frac{c}{100})=\sqrt{\frac{m}{k}}\);
Price in 1995 - \(m*(1+\frac{c}{100})=m*\sqrt{\frac{m}{k}}\).
Answer: C.
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Joined: 04 Oct 2011
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buckkitty
I went with smart numbers.
c= 10 %
1992 => k = 100
1994 => m = 121
obviously,
1995 => 133.1
Only choice c gives desired result.
\((m*sqrt(m))/sqrt(k)\) = \(121 \sqrt{121} / \sqrt{100}\) = 133.1
General Discussion
buckkitty
m = k*(1+c)^2
m/k = (1+c)^2
(m/k)^(1/2) = 1+c
c= (m/k)^(1/2) - 1
the value of the watch in jan 1 1995 is:
m(1+c)
so m(1+(m/k)^(1/2) - 1)
= m(m/k)^(1/2))
that is C
@Conocieur: Shouldn't the equation read
m = k*(1+c/100)^2 ? Since it says in the question, 'c percent more'..
m = k*(1+c/100)^2 ? Since it says in the question, 'c percent more'..
Bunuel
Shouldn't this year be raised by the third power? since its the third year.
fozzzy
It is actually.
Price in 1994 is \(k*(1+\frac{c}{100})^2\) which is \(m\), so the price in 1995 is \(k*(1+\frac{c}{100})^2*(1+\frac{c}{100})\) or \(m*(1+\frac{c}{100})\).
Hope it's clear.
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4112019
Value on Jan 1, 1992 = k
Value on Jan 1, 1993 = k(1+c/100)
Value on Jan 1, 1994 = \(k(1 + c/100)^2 = m\)
So, \((1 + c/100) = \sqrt{\frac{m}{k}}\)
Value on Jan 1 1995 = \(k(1+c/100)^3 = k(1 + c/100)^2 * (1 + c/100)\)
= \(m*\sqrt{\frac{m}{k}}\)
Yes, I generally prefer plugging in numbers but the calculations here are a little painful (with squares and roots) so using algebra is not a bad idea.
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ScottTargetTestPrep
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buckkitty
We can make the following expressions:
Value of the watch in 1992 = k
Value of the watch in 1993 = k*(1 + c/100)
Value of the watch in 1994 = k*(1 + c/100)^2 = m
Value of the watch in 1995 = k*(1 + c/100)^3 = m(1 + c/100)
Since k*(1 + c/100)^2 = m (the value of the value in 1994), we have:
(1 + c/100)^2 = m/k
1 + c/100 = √(m/k)
Thus, the value of the watch in 1995 is:
m(1 + c/100) = m*√(m/k) = m*√m/√k
Answer: C
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buckkitty
For an algebraic approach, we need to recognize that the value of the watch increases by the same factor each year. So, for the ease of calculations, let's say that the value increases by a factor of F.
Aside: Notice that the answer choices do not include the variable c. This tells me that I don't need to keep that variable in my solution.
In 1992, the watch is valued at k dollars.
In 1993, the watch is valued at kF dollars (applying our constant increase of F)
In 1994, the watch is valued at kF² dollars
In 1995, the watch is valued at kF³ dollars
GREAT, we now know the value in 1995. However, when we check the answer choices, none match the expression kF³. So, we have some more work to do.
The question tells us that, in 1994, the watch is valued at m dollars.
So, we now know that kF² = m.
Let's solve this equation for F (you'll see why in a moment)
We get: F² = m/k
F = √(m/k)
We can rewrite this as: F = (√m)/(√k)
We know that the 1995 value = kF³ dollars.
Rewrite, to get the 1995 value = (kF²)(F)
If we replace kF² with m and replace F with (√m)/(√k), we get:
1995 value = (m)((√m)/(√k))
= (m√m)/(√k)
= C
Cheers,
Brent
Bunuel
I followed you up until half of the work for 1994.
Why are we isolating for the percentage (100 + c / 100) on on side?
And for 1995, why are we multiplying by m?
CEdward
We are told that the value of the watch was "m" dollars on January 1, 1994 and since the value of the watch is "c" percent more than its value one year earlier, then in 1995 the value becomes \(m*(1+\frac{c}{100})\).
The question asks to express the value of the watch in 1995 (\(m*(1+\frac{c}{100})\)), in terms of "m" and "k". So, we need to express \(1+\frac{c}{100}\) in terms of "m" and "k". That's why we are expressing it as \((1+\frac{c}{100})=\sqrt{\frac{m}{k}}\) for 1994.
Hope it's clear.
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Given: At the end of each year, the value of a certain antique watch is "c" percent more than its value one year earlier, where "c" has the same value each year.
Asked: If the value of the watch was "k" dollars on January 1, 1992, and "m" dollars on January 1, 1994, then in terms of "m" and "k", what was the value of the watch, in dollars, on January 1, 1995?
Let us assume that value of the watch becomes x times in 1 year; where x = (1+c/100)
January 1, 1992
Value of the watch = k $
January 1, 1994
2 year later
Value of the watch = m $
m = kx^2
\(x = \sqrt{\frac{m}{k}}\)
January 1, 1995
1 year later
Value of the watch = mx = m\sqrt{\frac{m}{k}}
IMO C
Asked: If the value of the watch was "k" dollars on January 1, 1992, and "m" dollars on January 1, 1994, then in terms of "m" and "k", what was the value of the watch, in dollars, on January 1, 1995?
Let us assume that value of the watch becomes x times in 1 year; where x = (1+c/100)
January 1, 1992
Value of the watch = k $
January 1, 1994
2 year later
Value of the watch = m $
m = kx^2
\(x = \sqrt{\frac{m}{k}}\)
January 1, 1995
1 year later
Value of the watch = mx = m\sqrt{\frac{m}{k}}
IMO C
BrentGMATPrepNow
BrentGMATPrepNow hi Brent, i used number plugging
K = 1
c=100%
so 1992 = 1
93 = 2
94= 4
95 = 8
so m= 4
but both C and D are correct answers in this case, arent they ?
Posts: 6,727
dave13
At this point, you can conclude that the correct answer is either C or D.
From here, you'll have to test another set of values to eliminate one of the two remaining out two choices.
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each year value of watch increased by C%
1992 = K $
1993= K (1+c/100)
1994 = K (1+ c/100)^2
this value of 1994 is m $
m = K (1+ c/100)^2
1995= K (1+ c/100)^3
this can be written as,
K (1+c/100)^2 (1+c/100)
we know first part of it is equal to m. so put m in place of it.
m(1+c/100)
now we need k in terms of C.
we will use value of m to find c.
m= k(1+c/100)^2
m/k = (1+c/100)^2
1 +c/100 = [square_root m ][/square_root k]
now put this value in 1995 price,
m * [square_root m ][/square_root k]
option C
1992 = K $
1993= K (1+c/100)
1994 = K (1+ c/100)^2
this value of 1994 is m $
m = K (1+ c/100)^2
1995= K (1+ c/100)^3
this can be written as,
K (1+c/100)^2 (1+c/100)
we know first part of it is equal to m. so put m in place of it.
m(1+c/100)
now we need k in terms of C.
we will use value of m to find c.
m= k(1+c/100)^2
m/k = (1+c/100)^2
1 +c/100 = [square_root m ][/square_root k]
now put this value in 1995 price,
m * [square_root m ][/square_root k]
option C
buckkitty
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