GRE Quantitative Reasoning - How to find decimal fractions

A clothing store can only purchase socks in crates. Each crate has 200 socks and costs $2091.

Quantity A: The amount of socks that can be bought with $12651.

Quantity B: The amount of socks that can be bought with $14574.

The two quantities are equal.

Quantity A is greater.

Quantity B is greater.

The relationship cannot be determined.

Explanation

For this problem, realize that the store cannot buy part of a crate of socks. If they only have enough to pay for part of a crate, they might as well not have any money at all.

For the amount of money listed, figure out how many crates can be purchased:

Quantity A

$\frac{12651}{2091}=6.05$

So six crates can be purchased.

Quantity B:

$\frac{14574}{2091}=6.97$

Not quite enough for seven; only six crates can be purchased.

The two quantities are equal.

A retailer can only order wristbands in bulk cases. Each case has 300 wristbands and costs $1172.

Quantity A: The number of wristbands that can be bought for $10547.

Quantity B: The number of wristbands that can be bought for $10560.

Quantity B is greater.

Quantity A is greater.

The two quantities are equal.

The relationship cannot be determined.

Explanation

The key to this problem is to realize that the store cannot buy partial crates of wristbands--it's all or nothing. Calculate how many crates can be bought with each sum of money by dividing the sum by the price of a crate.

Quantity A:

$\frac{10547}{1172}=8.999$

Although oh so close, only eight crates can be bought with this sum of money. Don't round up!

Quantity B:

$\frac{10560}{1172}=9.010$

There's just enough to buy nine crates.

Quantity B is greater.

Find the fractional equivalent to .

$\frac{1,609}{200}$

$8\frac{45}{100}$

$8\frac{45}{10}$

$\frac{845}{100}$

$\frac{1,600}{200}$

Explanation

This problem requires several conversions. First, convert into a mixed number fraction. This means, the new fraction will have a whole number of and a fraction that represents thousandths. Then, convert the mixed number to an improper fraction by multiplying the denominator by the whole number, and then add that product to the numerator.

Thus, the solution is:

$8.045=8\frac{45}{1,000}=8\frac{45\div 5}{1,000\div 5}=8\frac{9}{200}=\frac{(8\times 200)+9}{200}=\frac{1,609}{200}$

Find the fractional equivalent to . Then simplify.

$3\frac{63}{125}$

$3\frac{120}{250}$

$3\frac{54}{100}$

$3\frac{54}{1,000}$

$3\frac{4}{5}$

Explanation

This problem requires you to first convert the decimal number to a mixed number fraction that has a whole number of and a fraction that represents thousandths. Then reduce the numerator and denominator by common divisors until you can no longer simplify the fraction.

$3.504=3\frac{504}{1,000}=3\frac{504\div 4}{1,000\div 4}=3\frac{126}{250}=3\frac{126\div 2}{250\div 2}=3\frac{63}{125}$

Convert to a fraction. Then simplify.

$\frac{323}{5,000}$

$\frac{323}{500}$

$\frac{646}{1,000}$

$\frac{46}{100}$

$\frac{323}{10,000}$

Explanation

Both fractions and decimals represent part of a whole. The value of this decimal number reaches the ten-thousandths place value. Thus, the fraction must represent ten-thousandths. Then reduce the numerator and denominator by the common divisor of .

$0.0646=\frac{646}{10,000}=\frac{646\div 2}{10,000\div 2}=\frac{323}{5,000}$

Convert ... to a fraction.

$\frac{132}{37}$

$\frac{1189}{333}$

$\frac{3567}{1000}$

$\frac{891}{250}$

$\frac{133}{101}$

Explanation

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 .

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

Quantity A: $\frac{0.0002(0.09)(0.4)}{0.000006(0.2)}$

Quantity B:

The two quantities are equal.

Quantity A is greater.

Quantity B is greater.

The relationship cannot be determined.

Explanation

To compare these two quantities, we'll want to simplify Quantity A.

The fraction

$\frac{0.0002(0.09)(0.4)}{0.000006(0.2)}$

may be a bit daunting; let's convert it to scientific notation:

$\frac{2\cdot10^{-4}(9\cdot10^{-2})(4\cdot 10^{-1})}{6\cdot10^{-6}(2\cdot10^{-1})}$

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

$\frac{72\cdot(10^{-4-2-1})}{12\cdot10^{-6-1}}$

$\frac{72\cdot10^{-7}}{12\cdot10^{-7}}$

Now cancel like factors in the numerator and denominator:

$\frac{6(12)\cdot10^{-7}}{12\cdot10^{-7}}$

The two quantities are equal.

Simplify.

$\frac{0.6025}{8}$

$\frac{241}{3200}$

$\frac{6025}{8}$

$\frac{8}{6025}$

$\frac{241}{32000}$

$\frac{241}{320}$

Explanation

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 .

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

Simplify.

$\frac{0.78}{0.8}$

$\frac{39}{40}$

$\frac{39}{4}$

$\frac{3.9}{40}$

$\frac{39}{400}$

$\frac{40}{39}$

Explanation

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 .

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

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

$\frac{2}{7}$

$\frac{7}{200}$

$\frac{7}{2}$

$\frac{2}{70}$

$\frac{20}{7}$

Explanation

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 .