If 10^2x/100^3y

Question

If  \frac{10^{2x}}{100^{3y}} < 1 , which of the following must be true?

A x < 3y
B 2x < 3y
C 2x < y
D 2x + 3y < 1
E 2x + 3y < 0

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MaitriD · Answer

(10^2x/ 100^3y) < 1
(10^2x/ 10^2*3y) < 1
(2x/2*3y )<1
x/ 3y < 1
x < 3y

Option A

luisdicampo · Answer

To determine which statement must be true, we need to simplify the given inequality using exponent rules. Given inequality: \frac{10^{2x}}{100^{3y}} < 1 Method 1: Algebraic Approach (Base Conversion) The key is to express both the numerator and the denominator with the same base. We know that 100 = 10^2 . Substitute this into the denominator: \frac{10^{2x}}{(10^2)^{3y}} < 1 Apply the power rule (a^b)^c = a^{b \cdot c} : \frac{10^{2x}}{10^{6y}} < 1 Apply the quotient rule \frac{a^m}{a^n} = a^{m-n} : 10^{2x - 6y} < 1 We know that 1 can be written as 10^0 . 10^{2x - 6y} < 10^0 CRITICAL RULE: Since the base (10) is greater than 1, we can drop the bases and compare the exponents directly without flipping the inequality sign. 2x - 6y < 0 Add 6y to both sides: 2x < 6y Divide by 2: x < 3y This matches Option (A). *** Method 2: Picking Numbers Let's find values for x and y that make the original inequality true, then test the options to eliminate incorrect ones. We need \frac{10^{2x}}{100^{3y}} < 1 . Let's try y = 1 . The denominator becomes 100^3 = (10^2)^3 = 10^6 . To make the fraction less than 1, the numerator 10^{2x} must be smaller than 10^6 . So, let's choose 2x = 4 , which means x = 2 . Check validity: \frac{10^4}{10^6} = 10^{-2} < 1 . (Valid set: x=2, y=1 ). Now test the options: A. x < 3y \implies 2 < 3(1) -> 2 < 3 ( True ) B. 2x < 3y \implies 2(2) < 3(1) -> 4 < 3 ( False ) C. 2x < y \implies 4 < 1 ( False ) D. 2x + 3y < 1 \implies 4 + 3 < 1 -> 7 < 1 ( False ) E. 2x + 3y < 0 \implies 7 < 0 ( False ) Only Option A remains. Answer: A

Dereno · Answer

Given that :  \frac{10^{2x}}{100^{3y}} < 1

This can be re written as :

100 can be written as 10^2. Then, 100 ^3y = 10^ (2*3y) = 10 ^6y

\frac{10^{2x}}{10^{6y}} < 1

10 ^ (2x -6y) < 1

For a number of the form a^n to be less than 1. This can occur, only when n is negative or less than zero.

Thus, (2x - 6y) <0

2x < 6y

x < 3y

Option A

impeditfugiat · Answer

Can you write it this way as well ? : 10^2x < 100^3y --> 100^x <100^3y --> x<3y ?

Bunuel · Answer

________________
Yes, that's correct.

If 10^2x/100^3y < 1, which of the following must be true?

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