3 is ten-thousands digit 4 is thousands digit 5 is the hundreds digit 6 is tens digit 7 is units digits
The GMAT commonly asks questions where it makes specifications of the units digit.
If x is the units digit and y is the tens digit and z is in the hundreds digit, then show the three digit number zyx.
To solve, since z is in the hundreds digit, y is the tens digit, use 100z + 10y + x = zyx
How to Round Decimals
To round decimals: eliminate the digit on the right, but if it is 5 or greater, increase the right digit by one.
What is 6.785 rounded to the nearest
tenths? = 6.8
hundredths? = 6.79
Example
Write this expression as a decimal:
31.2
100,000
Solution
When we divide by 100,000, we move the decimal point
5 places to the left. Then 31.2 = 0.000312.
A mixed decimal is the sum of an integer and a decimal fraction, much like the mixed fraction. The integer 5 added to the decimal fraction 35/100 is 5 + 35/100 = 5 + 0.35 = 5.35. The mixed decimal, or a decimal fraction, is usually simply called a decimal. (Note: the leading 0 in 0.35 is simply convention; it does not have mathematical significance: 0.35 = .35).
Example
Write the fraction 649/100 as a decimal.
Solution
The numerator 649 can be written as the decimal "649.0". When we divide by 100, we move the decimal point 2 places to the left so that
649/100 = 6.49
Note: It is not necessary to write any zeros after the 9; that is, 6.490 is equivalent to 6.49.
Example
Express 0.075 as a fraction in lowest terms.
Solution
The decimal is expressed as a fraction .075 = 75/1000 The denominator and numerator are factored resulting in:
75
1000
=
(25 × 3)
(25 × 40)
=
3
40
Adding and Subtracting Decimals
To add or subtract decimals, we write the decimals in a column with the decimal points aligned vertically.
Combine the decimals with a plus sign.
Combine the decimals with a minus sign.
Subtract the sum of the negative decimals from the sum of the positive decimals.
In performing these tasks, we add zeros to the right of the decimal point so that each number has an entry in each column. For example, if we subtract 3.021 from 5, we write 5 as 5.000 so that there is an entry in each of the 3 places to the right of the decimal in both numbers.
Example
Add 5 + 2.783 + 3.04.
Solution
Write the decimals in a column, with zeros added if none exist:
..5.000 ...2.783 + 3.040
10.823
Example
Compute 6.98 + 3.217 + 3 - 3.637
Solution
There are two possible ways to
solve this problem:
(1) Add the positive decimals and the negative
decimals:
+ 6.980
+ 3.217
+ 3.000 - 3.637
9.56
(2) Subtract the sum of the negative decimals
from the sum of the positive decimals:
13.197 - 3.637
9.560
Multiplying and Dividing Decimals
Multiply two decimals just like you would multiply two integers. The number of decimal places in the product is then equal to the total of the decimal places in the two decimals.
Example
Compute 0.05 × 12.
Solution
Set up the multiplication as though the decimals were integers:
12 × .05 = .60
The answer must have a total of 2 decimal places: .60
To divide two decimals, move the decimal point in the divisor (the number doing the dividing) to the right so that the divisor is an integer. Move the decimal point in the dividend to the right the same number of places. Now perform the division, placing the decimal point in the answer directly above the decimal point in the dividend.
Example
Compute .6/12
Solution
Change .6 to .60 and divide .60 by 12.
.60/12 =.05
Shortcut: Multiplying/Dividing by 10
Rule #1: If you multiply a decimal by 10, you would move the decimal 1 place to the right; 100, move 2 to the right; 1000, move 3 to the right, and so on.
For instance, 100,000 × 0.0054 = 540.
You see that there are 5 zeros in 100,000; therefore, we moved the decimal 5 places to the right.
Rule # 2: When you divide by 10, you move the decimal one place to the left; 100, move 2 to the left; 1000, move 3 to the left.
