Problem Solving

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 1st, 1995?
a. M +1/2(m-k)
b. M+1/2(m-k/k)m
c. [M*root(m)]/k
d. M^2/2k
e. Km^2

I think the options may be incorrect because the answer is coming out to be

m * (Sq.rt. of (m/k)

Explanation:

Price on 1st Jan 1992 = k
Price on 1st Jan 1993 = k(1+ c/100) {as per the question}
Price on 1st Jan 1994 = k(1+c/100)* (1+ c/100) = k (1+c/100) ^2

As per the question, this is equal to m => m = k (1+c/100)^2 => (1+c/100) = Sq.rt. of (m/k)

Price on 1st Jan 1995 = k (1+c/100)^3 = k (sq.rt of m/k)^3 {as 1+c/100 = Sq.rt. of (m/k)}
= k* (m/k)^1/2*3 = k* (m/k)3/2 = m^(3/2) / k (1/2) = m * (m/k)^1/2.

This is the answer.

fangtray wrote: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 1st, 1995?

m + (1/2)(m-k)

m + (1/2)((m-k)/k)m

(m√m)/√k

m²/2k

km²

I've corrected the answer choices, which had been transcribed incorrectly.

Let k=1 and c=200%.
Value in 1993 = 1 + 2(1) = 3.
m = value in 1994 = 3 + 2(3) = 9.
Value in 1995 = 9 + 2(9) = 27. This is our target.

Now we plug k=1 and m=9 into the answers to see which yields our target of 27.

Only answer choice C works:
(m√m)/√k = (9√9)/√1 = 27.

The correct answer is C.

fangtray wrote: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 1st, 1995?
a. M +1/2(m-k)
b. M+1/2(m-k/k)m
c. [M*root(m)]/k
d. M^2/2k
e. Km^2

Hi! First, let's make sure we understand the essence of the question - since we're increasing by a set percent each year, what we really have here is a compound interest problem. The problem basically boils down to:

Principle investment on Jan 1, 1992: $k
interest rate per year: c%
total value of investment after 2 years: $m

Q: What's the value of the investment, in terms of m and k, after 3 years?

As you suggest, picking numbers is a great way to handle this kind of complicated algebra. Let's start by picking numbers for k and c; once we have those, we can find the value of m.

Since it's a percent question, let's let k=100 and c=10%

After 1 year, our investment is worth 1.1*$100=$110.
After 2 years, our investment is worth 1.1*$110=$121

So, we now know that m=121.

After 3 years, our investment is worth 1.1*$121=$133.10

Now we plug in k=100 and m=121 to the choices, seeing which one gives us a result of 133.10.

A) 121 + .5(12.10)... nope!
B) 121 + .5(12.10/100)*121 = 121 + (6.05/100)*121 = 121 + approx 6*120/100 = 121 + approx 7... nope!
C) (121)(11)/100 = 121/100 * 11 = 1.21 * 11= 13.31 (off by 1 decimal place - are you sure you reproduced this answer correctly?)
D) 121^2/200 = 121/100 * 121/2 = 1.21 * 60ish = too small
E) 100(121^2) = way too big.

No matches, so there's an error in the choices - I'm guessing that (C) was typed out incorrectly (if it were m*root(m/k) then it would be correct).

GMATGuruNY wrote:

fangtray wrote: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 1st, 1995?

m + (1/2)(m-k)

m + (1/2)((m-k)/k)m

(m√m)/√k

m²/2k

km²

I've corrected the answer choices, which had been transcribed incorrectly.

Let k=1 and c=200%.
Value in 1993 = 1 + 2(1) = 3.
m = value in 1994 = 3 + 2(3) = 9.
Value in 1995 = 9 + 2(9) = 27. This is our target.

Now we plug k=1 and m=9 into the answers to see which yields our target of 27.

Only answer choice C works:
(m√m)/√k = (9√9)/√1 = 27.

The correct answer is C.

thx so much, but how did you know to pick those particular numbers that would give you something that evenly rooted? like 1 and 9 and something strange like 200%? My problem with these types of questions if finding out the easy numbers to plug in.. sometimes finding the right numbers to use is not so apparent and i end up with all sorts of decimals and long calculations that will end up giving me an error when i'm racing to finish the test on time.

In general, as the problems get harder picking numbers also gets harder, and the time it takes to solve the problem using numbers may exceed the direct algebraic approach. I am willing to bet that the majority of the test scorers who score high on the GMAT would solve this problem directly. The test is written in a way that rewards people who can quickly deal with the underlying algebra. If most students could do this problem using numbers, then statistically it becomes an easy problem and is of no value in differentiating between the high scorers.

Here is how I would do it with numbers, see the attached in age.

Dabral