This one is driving me crazy...The explanation in the back of the book doesnt help either...
OG11th ed (DS #131)
Is 5^k less than 1000?
(1) 5^k+1 > 3000
(2) 5^k-1 = 5^k-500
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My steps
_________________________________________________________
Step 1: Rephrase the question Is 5^k less than 1000?
possible rephrases
a) 5^k < 1000
b) 5^k < (2x5)^3
c) 5^k < 2^3 5^3
Step 2: What underlying concepts can I use to rephrase/understand the question/statements?
USE THE LAW OF INDICES
" a^m x a^n = a^m+n "
" a^-m = 1/a^m "
Step 3: This is where I get lost...can anyone finish my steps and provide an explanation?
_______________________________________________________
Here's how the OG broke the statements in the back of the book...
_______________________________________________________
OG explanation of Statement (1) 5^k+1 > 3000
step 1 - divide both sides of the given inequality by 5 (*see my note*)
5^k+1 ÷ 5 > 3000 ÷ 5 = 5^k > 600
step 2: Statement 1 - INSUFFICIENT
Although 5^k > 600, it is unknown if 5^k < 1000
** The book confuses me here I thought when you divide you're suppose to change the direction of inequality sign **
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OG explanation of Statement (2) 5^k-1 = 5^k-500
step 1- subtract 5^k from both sides
5^k-1 - 5^k = 5^k - 5^k - 500
step 2 - divide all terms by -1
(5^k ÷ -1) - (5^k-1 ÷ -1) = (-500 ÷ -1)
(5^k) - (5^k-1) = (500) **
** The book confuses me here - am I missing something? I thought when you divide all terms by -1, that 5^k will become - 5^k, that -5^k-1 ÷ -1 will be come +5^k-1
step 3 - property of exponents
5^k-5^k (5^-1) = 500
step 4 - substitute for 5^-1
5^k-5^k(1/5) = 500
step 5 - factor out 5^k
5^k(1-1/5) = 500
step 6 - simplify
5^k(4/5) = 500
step 7 - multiply both sides by 5/4
5^k = 500 (5/4)
5^k = 625 which is less than 1000 - SUFFICIENT
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This sure seems like a lot of calculation for a data sufficiency problem. I know my other alternative was to pick a number for k, but that seems like a cop out plus its not getting at the root of the concept behind the problem....
Is there an alternative approach to this problem that still applies the concept that is tested ie, the laws of indices?
OG11th ed (DS #131)
Is 5^k less than 1000?
(1) 5^k+1 > 3000
(2) 5^k-1 = 5^k-500
_________________________________________________________
My steps
_________________________________________________________
Step 1: Rephrase the question Is 5^k less than 1000?
possible rephrases
a) 5^k < 1000
b) 5^k < (2x5)^3
c) 5^k < 2^3 5^3
Step 2: What underlying concepts can I use to rephrase/understand the question/statements?
USE THE LAW OF INDICES
" a^m x a^n = a^m+n "
" a^-m = 1/a^m "
Step 3: This is where I get lost...can anyone finish my steps and provide an explanation?
_______________________________________________________
Here's how the OG broke the statements in the back of the book...
_______________________________________________________
OG explanation of Statement (1) 5^k+1 > 3000
step 1 - divide both sides of the given inequality by 5 (*see my note*)
5^k+1 ÷ 5 > 3000 ÷ 5 = 5^k > 600
step 2: Statement 1 - INSUFFICIENT
Although 5^k > 600, it is unknown if 5^k < 1000
** The book confuses me here I thought when you divide you're suppose to change the direction of inequality sign **
___________________________________________
OG explanation of Statement (2) 5^k-1 = 5^k-500
step 1- subtract 5^k from both sides
5^k-1 - 5^k = 5^k - 5^k - 500
step 2 - divide all terms by -1
(5^k ÷ -1) - (5^k-1 ÷ -1) = (-500 ÷ -1)
(5^k) - (5^k-1) = (500) **
** The book confuses me here - am I missing something? I thought when you divide all terms by -1, that 5^k will become - 5^k, that -5^k-1 ÷ -1 will be come +5^k-1
step 3 - property of exponents
5^k-5^k (5^-1) = 500
step 4 - substitute for 5^-1
5^k-5^k(1/5) = 500
step 5 - factor out 5^k
5^k(1-1/5) = 500
step 6 - simplify
5^k(4/5) = 500
step 7 - multiply both sides by 5/4
5^k = 500 (5/4)
5^k = 625 which is less than 1000 - SUFFICIENT
____________________________________
This sure seems like a lot of calculation for a data sufficiency problem. I know my other alternative was to pick a number for k, but that seems like a cop out plus its not getting at the root of the concept behind the problem....
Is there an alternative approach to this problem that still applies the concept that is tested ie, the laws of indices?
