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Gold depreciated at a rate of \(X\%\) per year between 2000 and 2005. If 1 kg of gold cost \(S\) dollars in 2001 and \(T\) dollars in 2003, how much did it cost in 2002 in terms of \(S\) and \(T\)?

A. \(T\frac{S}{2}\) B. \(T\sqrt{\frac{T}{S}}\) C. \(T\sqrt{S}\) D. \(T\frac{S}{\sqrt{T}}\) E. \(\sqrt{ST}\)

Gold depreciated at a rate of \(X\%\) per year between 2000 and 2005. If 1 kg of gold cost \(S\) dollars in 2001 and \(T\) dollars in 2003, how much did it cost in 2002 in terms of \(S\) and \(T\)?

A. \(T\frac{S}{2}\) B. \(T\sqrt{\frac{T}{S}}\) C. \(T\sqrt{S}\) D. \(T\frac{S}{\sqrt{T}}\) E. \(\sqrt{ST}\)

Price of 1kg gold in 2001 - \(S\)

Price of 1kg gold in 2002 - \(S(1-\frac{x}{100})\)

Price of 1kg gold in 2003 - \(S(1-\frac{x}{100})^2=T\). So, \((1-\frac{x}{100})=\sqrt{\frac{T}{S}}\)

Price of 1kg gold in 2002 - \(S(1-\frac{x}{100})=S*\sqrt{\frac{T}{S}}=\sqrt{ST}\).

Gold depreciated at a rate of \(X\%\) per year between 2000 and 2005. If 1 kg of gold cost \(S\) dollars in 2001 and \(T\) dollars in 2003, how much did it cost in 2002 in terms of \(S\) and \(T\)?

A. \(T\frac{S}{2}\) B. \(T\sqrt{\frac{T}{S}}\) C. \(T\sqrt{S}\) D. \(T\frac{S}{\sqrt{T}}\) E. \(\sqrt{ST}\)

Price of 1kg gold in 2001 - \(S\)

Price of 1kg gold in 2002 - \(S(1-\frac{x}{100})\)

Price of 1kg gold in 2003 - \(S(1-\frac{x}{100})^2=T\). So, \((1-\frac{x}{100})=\sqrt{\frac{T}{S}}\)

Price of 1kg gold in 2002 - \(S(1-\frac{x}{100})=S*\sqrt{\frac{T}{S}}=\sqrt{ST}\).

Answer: E

I'm not following this last step...\(S(1-\frac{x}{100})=S*\sqrt{\frac{T}{S}}=\sqrt{ST}\).

Gold depreciated at a rate of \(X\%\) per year between 2000 and 2005. If 1 kg of gold cost \(S\) dollars in 2001 and \(T\) dollars in 2003, how much did it cost in 2002 in terms of \(S\) and \(T\)?

A. \(T\frac{S}{2}\) B. \(T\sqrt{\frac{T}{S}}\) C. \(T\sqrt{S}\) D. \(T\frac{S}{\sqrt{T}}\) E. \(\sqrt{ST}\)

Step 1: Price of 1kg gold in 2001 - \(S\)

Step 2: Price of 1kg gold in 2002 - \(S(1-\frac{x}{100})\)

Step 3: Price of 1kg gold in 2003 - \(S(1-\frac{x}{100})^2=T\). So, \((1-\frac{x}{100})=\sqrt{\frac{T}{S}}\)

Step 4: Price of 1kg gold in 2002 - \(S(1-\frac{x}{100})=S*\sqrt{\frac{T}{S}}=\sqrt{ST}\).

Answer: E

I'm not following this last step...S*\sqrt{\frac{T}{S}}=\sqrt{ST}[/m].

Any help?

From step 2: Price of 1kg gold in 2002 - \(S(1-\frac{x}{100})\).

From step 3: \((1-\frac{x}{100})=\sqrt{\frac{T}{S}}\). Substitute this into the above equation: \(S(1-\frac{x}{100})=S*\sqrt{\frac{T}{S}}=(\sqrt{S})^2*\sqrt{\frac{T}{S}}=\sqrt{ST}\).

Gold depreciated at a rate of \(X\%\) per year between 2000 and 2005. If 1 kg of gold cost \(S\) dollars in 2001 and \(T\) dollars in 2003, how much did it cost in 2002 in terms of \(S\) and \(T\)?

A. \(T\frac{S}{2}\) B. \(T\sqrt{\frac{T}{S}}\) C. \(T\sqrt{S}\) D. \(T\frac{S}{\sqrt{T}}\) E. \(\sqrt{ST}\)

Step 2: Price of 1kg gold in 2002 - \(S(1-\frac{x}{100})\)

Step 3: Price of 1kg gold in 2003 - \(S(1-\frac{x}{100})^2=T\). So, \((1-\frac{x}{100})=\sqrt{\frac{T}{S}}\)

Step 4: Price of 1kg gold in 2002 - \(S(1-\frac{x}{100})=S*\sqrt{\frac{T}{S}}=\sqrt{ST}\).

Answer: E

I'm not following this last step...S*\sqrt{\frac{T}{S}}=\sqrt{ST}[/m].

Any help?

From step 2: Price of 1kg gold in 2002 - \(S(1-\frac{x}{100})\).

From step 3: \((1-\frac{x}{100})=\sqrt{\frac{T}{S}}\). Substitute this into the above equation: \(S(1-\frac{x}{100})=S*\sqrt{\frac{T}{S}}=(\sqrt{S})^2*\sqrt{\frac{T}{S}}=\sqrt{ST}\).

Does this make sense?

Step 1: Price of 1kg gold in 2001 - \(S\)

I take the substitution approach

In 2001 , 1 Kg cost $100 . In 2002 cost is $ 90 . In 2003 cost is $81 S = 100, T = 81

Now i have subsituted the value in the options. I checked option a,b,c . All were not tallying. Option D : T * S/root T = 81 * 100/9 = 900 Option E : root of ST = root of 81*100 =90.

For this problem it would be easier to express percent change as simply "x":

Step 1 2001=S 2002=S*x 2003=S*x*x=T

Step 2 We take 2003 and do some algebra (actually to get to this is the harderst thing about the problem) and express x in terms of "S" and "T": T=S*x^2 x^2=T/S x=root (T/S)

Step 3 We substitute the value of x into 2002: 2002=S*root (T/S) 2002=root S*root S*(root T/root S) 2002=root (ST)

2001 - S 2002 - S(1-x/100) 2003 - here the amount from 2002 is reduced by another x percent . so we multiply the amount from 2002 ie. s( 1-x/100) by another (1-x/100) Eg . if each year the reduction was by 40 percent. The steps would be 2001 - S 2002 - S(1-40/100) 2003 - S(1-40/100)*(1-40/100) = S(1-40/100)^2

Similarly since here the percent reduction is x

2001 - S 2002 - S(1-x/100) ---> This is the year we need but without the x ..we need it just in S and T. So we need to find a value for (1-x/100) part in terms of S and T 2003 = S(1 - X/100)^2

Now given 2003 is T .So S(1 - X/100)^2 = T, (1 - X/100)^2 = T/S (1 - X/100) = (T/S)^1/2 ----> We found a value for (1-x/100) in terms of S and T.

Substitute this in the 2002 eqn - S(1-x/100) = S((T/S)^1/2)) = (ST)^1/2

Bunuel im trying to refresh my math memory about rules again, how come the answer is root(ST) and not simply ST . Remember the rule was something about needing to place the answer back to its original form? please help thank you

Bunuel im trying to refresh my math memory about rules again, how come the answer is root(ST) and not simply ST . Remember the rule was something about needing to place the answer back to its original form? please help thank you

Don't know which rule are you referring to... Can you please tell me which step from the solution is unclear?
_________________

thanks for such a quick reply ! im just wondering why the answer in the end is the root(ST), I follow you up until S x Root(T/S) , but then you square both terms which becomes S^2 x T/S which then equals ST, if i follow you right? but why is the final answer root(ST)?

thanks for such a quick reply ! im just wondering why the answer in the end is the root(ST), I follow you up until S x Root(T/S) , but then you square both terms which becomes S^2 x T/S which then equals ST, if i follow you right? but why is the final answer root(ST)?

No, we don't square. We just write S as \((\sqrt{S})^2\):

1) Let X=10, S=100 (cost of gold in 2001), T=81 (cost of gold in 2003). Then gold cost in 2002 will be 90. 2) Plug numbers in each answer option. Only E matches: \sqrt{100*90}=90