How to square a decimal - GRE Quantitative Reasoning

Card 0 of 5

Question

Which quantity is greater? When

Quantity A

Quantity B

1

Answer

X must equal a number between 0 and 1.

If you square any decimal number in this range you will get an answer less than 1. Try = 0.998.

Therefore Quantity B is always larger.

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Question

Which is greater, when $\small \small \small 0<x<1?$

Quantity A

$\small x^2$

Quantity B

$\small \frac{x}{2}$

Answer

When $\small 0<x<1$ , $\small x$ must be a decimal. Therefore it will decrease when it is squared. To find this answer we can substitute two values for $\small x$ .

$\small x = 0.9$ and $\small x = 0.1$

When $\small x = 0.9$

$\small \small 0.9^2 = 0.81 > \frac{0.9}{2} = 0.45$

But when $\small x = 0.1$

$\small 0.1^2 = 0.01 < \frac{0.1}{2} = 0.05$

Therefore we cannot tell which is greater based on the information given.

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Question

Which is greater, when $\small \small 0<x<1?$

Quantity A

$\small \small x$

Quantity B

$\small 2x^2$

Answer

When $\small 0<x<1$ , $\small x$ must be a decimal. Therefore it will decrease when it is squared. To find this answer we can substitute two values for $\small x$ .

$\small x = 0.9$ and $\small x = 0.1$

When $\small x = 0.9$

$\small \small \small 2\cdot0.9^2 = 1.62 > 0.9$

But when $\small x = 0.1$

$\small \small 2\cdot0.1^2 = 0.02 < 0.1$

Therefore we cannot tell which is greater based on the information given.

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Question

Which is greater, when $\small -1<x<0?$

Quantity A

$\small \left | x\right |$

Quantity B

$\small x^2$

Answer

When $\small \small -1<x<0$ , $\small x$ must be a negative decimal. Therefore it will decrease when it is squared and become positive. The first quantity $\small \left | x\right |$ is the distance to 0 from $\small x$ , in other words the positive value of $\small x$ . The second quantity $\small x^2$ will always be positive, but because it is a decimal it will always be less than $\small x$ , if $\small x$ was a positive quantity. Since $\small \left | x\right |\equiv x$ , then $\small \left | x\right |$ wiill always be greater than $\small x^2$ .

Quantity A is greater.

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Question

Which is greater, when $\small \small -1<x<0?$

Quantity A

$\small x^2$

Quantity B

$\small 1$

Answer

When $\small \small -1<x<0$ , $\small x$ must be a negative decimal. Therefore it will decrease when it is squared and become positive. Because $\small x$ must be a decimal, $\small x^2$ will also be a decimal. Therefore $\small x^2$ will never be greater than 1.

Quantity B is greater.

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