**Decimals**

If a fraction’s denominator is a power of 10, it can be written in a special form called a decimal fraction.

Some common decimals are $\frac{1}{10}$ = .1, $\frac{2}{100}$ = .02, $\frac{3}{1000}$ = .003. Notice that the number of decimal places corresponds to the number of zeros in the denominator of the fraction. Also note that the value of the decimal place decreases to the right of the decimal point:

This decimal can be written in expanded form as follows:

.1234 = $\frac{1}{10}$ + $\frac{2}{100}$ + $\frac{3}{1000}$ + $\frac{4}{10000}$

Sometimes a zero is placed before the decimal point to prevent misreading the decimal as a whole number. The zero has no affect on the value of the decimal.

For example: .2 = 0.2.

Fractions can be converted to decimals by dividing the denominator into the numerator.

For example, to convert $\frac{5}{8}$ to a decimal, divide 8 into 5 (note, a decimal point and as many zeros as necessary are added after the 5):

The procedures for adding, subtracting, multiplying, and dividing decimals are the same as for whole numbers, except for a few small adjustments.

• **Adding and Subtracting Decimals:**To add or subtract decimals, merely align the decimal points and then add or subtract as you would with whole numbers.

$\stackrel{1.369}{\frac{+9.7}{11.069}}\stackrel{12.45}{\frac{-6.367}{6.083}}$

• **Multiplying Decimals:**Multiply decimals as you would with whole numbers. The answer will have as many decimal places as the sum of the number of decimal places in the numbers being multiplied.

• **Dividing Decimals:**Before dividing decimals, move the decimal point of the divisor all the way to the right and move the decimal point of the dividend the same number of spaces to the right (adding zeros if necessary). Then divide as you would with whole numbers.

**Example:**

$\frac{1}{5}$ of .1 percent equals:

(A) 2

(B) .2

(C) .02

(D) .002

(E) .0002

Recall that percent means to divide by 100. So .01 percent equals $\frac{1}{100}$ = .01. To convert $\frac{1}{5}$to a decimal, divide 5 into 1:

In percent problems, “of” means multiplication. So multiplying .2 and .01 yields

$\stackrel{.01}{\frac{\times .2}{.002}}$

Hence, the answer is (D). Note, you may be surprised to learn that the GRE would consider this to be a hard problem.

**Example:**

The decimal .1 is how many times greater than the decimal (. 001) ^{3}?

(A) 10 (B) 10^{2 }(C) 10^{5 }(D) 10^{8 }(E) 10^{10}

Converting .001 to a fraction gives $\frac{1}{1000}$. This fraction, in turn, can be written as $\frac{1}{{10}^{3}}$ or10^{-3}. Cubing this expression yields (. 001)^{3} = (10^{-3})^{3} = 10^{-9}. Now, dividing the larger number, .1, by the smaller number, (. 001)^{3}, yields

$\frac{.1}{{\left(.001\right)}^{3}}$ = $\frac{{10}^{-1}}{{10}^{-9}}$= 10^{-1- (-9)}= 10^{-1+9}= 10^{8}

Hence, .1 is 10 ^{8}times as large as (. 001) ^{3}. The answer is (D).

**Example:**

Let *x*= .99, *y*=√ .99, and *z*= (. 99)^{2}. Then which of the following is true?

(A) *x < z < y*

(B) *z < y < x*

(C) *z < x < y*

(D) *y < x < z*

(E) *y < z < x*

Converting .99 into a fraction gives $\frac{99}{100}$. Since $\frac{99}{100}$ is between 0 and 1, squaring it will make it smaller and taking its square root will make it larger. Hence, (.99)^{2} <.99 √ .99. The answer is (C). Note, this property holds for all proper decimals (decimals between 0 and 1) just as it does for all proper fractions.