Decimals - GRE Quantitative Reasoning

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Question

Find the square root of the following decimal:

$\sqrt{.00081}=$

Answer

The easiest way to find the square root of a fraction is to convert it into scientific notation.

$\dpi{100} \small .00081 = 8.1 \times 10^{-4}$

The key is that the exponent in scientific notation has to be even for a square root because the square root of an exponent is diving it by two. The square root of 9 is 3, so the square root of 8.1 is a little bit less than 3, around 2.8

$\dpi{100} \small \sqrt{8.1 \times 10^{-4}} \approx 2.8 \times 10^{-2} \approx 0.028$

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Question

Find the square root of the following decimal:

$\small \sqrt{0.0049}$

Answer

To find the square root of this decimal we convert it into scientific notation.

$\small 0.0049 = 49 \cdot10^{-4}$

Because $\small 10^{-4}$ has an even exponent, we can divide the exponenet by 2 to get its square root.

$\small \sqrt{0.0049} = \sqrt{49}\cdot\sqrt{10^{-4}} = 7\cdot10^{-2} = 0.07$

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Question

Find the square root of the following decimal:

$\small \sqrt{0.025}$

Answer

This problem can be solve more easily by rewriting the decimal into scientific notation.

$\small 0.025 = 2.5 \times 10^{-2}$

Because $\small 10^{-2}$ has an even exponent, we can take the square root of it by dividing it by 2. The square root of 4 is 2, and the square root of 1 is 1, so the square root of 2.5 is less than 2 and greater than 1.

$\small \sqrt{0.025} = \sqrt{2.5}\times \sqrt{10^{-2}} = 1.58\times 10^{-1} = 0.158$

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Question

Find the square root of the following decimal:

$\small \sqrt{0.00036}$

Answer

This problem becomes much simpler if we rewrite the decimal in scientific notation

$\small 0.00036 = 3.6\times 10^{-4}$

Because $\small 10^{-4}$ has an even exponent, we can take its square root by dividing it by two. The square root of 4 is 2, and because 3.6 is a little smaller than 4, its square root is a little smaller than 2, around 1.9

$\small \sqrt{0.025} = \sqrt{3.6}\times \sqrt{10^{-4}} \approx 1.9\times 10^{-2} = 0.019$

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Question

Find the square root of the following decimal:

$\small \sqrt{0.00000025}$

Answer

To find the square root of this decimal we convert it into scientific notation.

$\small \small 0.00000025 = 25 \times 10^{-8}$

Because $\small \small 10^{-8}$ has an even exponent, we can divide the exponenet by 2 to get its square root. $\small 25$ is a perfect square, whose square root is $\small 5$ .

$\small \small \sqrt{0.00000025} = \sqrt{25}\times \sqrt{10^{-8}} = 5\times10^{-4} = 0.0005$

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Question

Find the square root of the following decimal:

$\small \sqrt{0.0169}$

Answer

To find the square root of this decimal we convert it into scientific notation.

$\small \small 0.0169 = 169\times10^{-4}$

Because $\small 10^{-4}$ has an even exponent, we can divide the exponenet by 2 to get its square root. $\small 169$ is a perfect square, whose square root is $\small 13$ .

$\small \small \sqrt{0.0169} = \sqrt{169}\times \sqrt{10^{-4}} =13\times10^{-2} = 0.13$

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Question

Find the square root of the following decimal:

$\small \sqrt{0.00064}$

Answer

To find the square root of this decimal we convert it into scientific notation.

$\small \small 0.00064 = 6.4 \times10^{-4}$

Because $\small 10^{-4}$ has an even exponent, we can divide the exponenet by 2 to get its square root. The square root of 9 is 3, and the square root of 4 is two, so the square root of 6.4 is between 3 and 2, around 2.53

$\small \small \sqrt{0.00064} = \sqrt{6.4}\times\sqrt{10^{-4}} \approx 2.53 \times10^{-2} = 0.0253$

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Question

Find the square root of the following decimal:

$\small \sqrt{0.00001}$

Answer

To find the square root of this decimal we convert it into scientific notation.

$\small \small \small 0.00001 = 10 \times10^{-6}$

Because $\small \small 10^{-6}$ has an even exponent, we can divide the exponenet by 2 to get its square root. The square root of 9 is 3, so the square root of 10 should be a little larger than 3, around 3.16

$\small \small \sqrt{0.00001} = \sqrt{10}\times \sqrt{10^{-6}} = 3.16\times10^{-3} = 0.00316$

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Question

Find the square root of the following decimal:

$\small \sqrt{0.004}$

Answer

To find the square root of this decimal we convert it into scientific notation.

$\small \small 0.004 = 40 \times10^{-4}$

Because $\small 10^{-4}$ has an even exponent, we can divide the exponenet by 2 to get its square root. The square root of 36 is 6, so the square root of 40 should be a little more than 6, around 6.32.

$\small \small \small \sqrt{0.004} = \sqrt{40}\times\sqrt{10^{-4}} \approx 6.32 \times10^{-2} = 0.0632$

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Question

Solve for :

$\frac{x}{\sqrt{0.04}}=\sqrt{0.16}$

Answer

Just like any other equation, isolate your variable. Start by multiplying both sides by $\sqrt{0.04}$ :

$x=\sqrt{0.16}\sqrt{0.04}$

Now, this is the same as:

$x=\sqrt{0.0064}$

You know that $\sqrt{64}$ is . You can intelligently rewrite this problem as:

$x=\sqrt{0.0064}=\sqrt{\frac{1}{10\textup{,}000}64}$ , which is the same as:

$\sqrt{\frac{1}{10\textup{,}000}64}=\sqrt{\frac{1}{10\textup{,}000}}\sqrt{64}=\frac{1}{100}*8=0.08$

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Question

Simplify.

$\frac{0.3}{5}$

Answer

Whenever there are decimals in fractions, we remove them by shifting the decimal place over however many it takes to make number an integer.

In this case we have to move the decimal in the numerator to the right one place.

Then, we add just one zero to the denominator.

Final answer becomes:

$\frac{3}{50}$ .

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Question

Simplify.

$\frac{0.78}{0.8}$

Answer

With the numerator having more decimal spots than the denominator, we need to move the decimal point in the numerator two places to the right.

Then in the denominator, we move the decimal point also two to the right. Since there's only one decimal place we just add one more zero.

Then we can reduce by dividing top and bottom by .

$\frac{0.78}{0.8}=\frac{7.8}{8}=\frac{78}{80}=\frac{39}{40}$

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Question

Simplify.

$\frac{0.6025}{8}$

Answer

Since there are four decimal places, we shift the decimal point in the numerator four places to the right.

For the denominator, since there is no decimal point, we just add four more zeroes.

Then reduce by dividing top and bottom by .

$\frac{0.6025}{8}=\frac{6025}{80000}=\frac{241}{3200}$

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Question

What is of $\frac{1}{13}$ ?

Answer

We need to convert the sentence into a math expression. Anytime there is "of" means we need to multiply. Let's first convert the decimal to a fraction. We need to move the decimal point two places to the right.

Since is the same as $\frac{0.26}{1}$ we can add two more zeroes to the denominator.

$\frac{26}{100}\cdot\frac{1}{13}$

We can reduce the to a and the to a .

$\frac{26}{100}\cdot \frac{1}{13}=\frac{2}{100}\cdot \frac{1}{1}$

Then reduce the to and the to .

$\frac{2}{100}=\frac{2 \cdot 1}{2 \cdot 50}=\frac{1}{50}$ .

Then dividing into and we get .

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Question

of is $\frac{1}{10}$ . What is ?

Answer

We need to convert this sentence into a math expression. Anytime there is "of" in a sentence it means we need to multiply. Let's convert $\frac{1}{10}$ into a decimal which is .

Thus our mathematical expression becomes:

.

Divide both sides by .

$x=\frac{0.1}{0.35}$

Move decimal point two places to the right. The numerator will become . Then simplify by dividing top and bottom by .

$x=\frac{10}{35}=\frac{2}{7}$

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Question

Solve for .

Answer

Let's convert the decimal into a fraction.

$\frac{1}{2}x^2+x+\frac{1}{2}=0$

If we multiply everything by , we should have an easier quadratic.

Remember, we need to find two terms that are factors of the c term that add up to the b term.

This is the only value.

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Question

Evaluate.

$\frac{\frac{0.48}{3}}{5}$

Answer

Let's actually simplify the top of the fraction. divides into .

We should have:

$\frac{\frac{0.48}{3}}{5}=\frac{0.16}{5}$ .

Then move the decimal two spots to the right and add two zeroes to the denominator.

$\frac{16}{500}$

Let's actually multiply top and bottom by to get: $\frac{32}{1000}$ .

Now we want to eliminate those zeroes. By dividing, the decimal point in the numerator moves to the left three places to get an answer of $\frac{0.032}{1}$ or .

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Question

Evaluate and express in a fraction.

${\frac{0.39}{0.81}}\div{\frac{0.52}{0.27}}$

Answer

Since each decimal has two digits, we can convert easily to integers.

$\frac{\frac{0.39}{0.81}}{\frac{0.52}{0.27}}=\frac{\frac{39}{81}}{\frac{52}{27}}$

Then multiply top and bottom by $\frac{27}{52}$ to get: $\frac{39}{81}\cdot \frac{27}{52}$

is reduced to and is reduced to

Then and can be divided by to get and respectively.

$\frac{39}{81}\cdot\frac{27}{52}=\frac{3}{3}\cdot\frac{1}{4}=\frac{1}{4}$

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Question

Convert ... to a fraction.

Answer

Let be . Let's multiply that value by . The reason is when we subtract it, we will get us an integer instead and the repeating decimals will disappear.

If we subtract, we get .

Divide both sides by and we get $\frac{3564}{999}$ .

If you divide by on top and bottom, you should get the answer. Otherwise, just divide top and bottom by three times based on the divisibility rules for . If the sum is divisible by , then the number is divisible by .

$\frac{3564}{999}=\frac{27 \cdot 132}{27\cdot 37}=\frac{132}{37}$

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Question

Simplify the fraction:

$\frac{300(0.00002)(70)}{0.0021(40)}$

Answer

To begin, it can be useful to convert the values in the fraction

$\frac{300(0.00002)(70)}{0.0021(40)}$

into a modified scientific notationnotation:

$\frac{3\cdot10^2(2\cdot10^{-5})(7\cdot10^1)}{21\cdot10^{-4}(4\cdot10^1)}$

Now multiply the ten terms (adding exponents together) and the non-ten terms:

$\frac{42\cdot 10^{-2}}{84\cdot10^{-3}}$

From here, reduce the terms, subtracting the bottom tens exponent from the top tens exponent:

$\frac{1\cdot 10^{-2-(-3)}}{2}$

$\frac{1\cdot 10^{1}}{2}$

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