Sunday, July 14, 2013

Solving Digits and Place Value Problems: GRE Math Fact

2:28 AM
First let’s get some of the basic concepts out of the way:
Digits and Place Values
All numbers are composed of digits and each of these digits contributes to the value of the number based on their position.
Example: Number 26 has 2 in the tenths place and 6 in the ones place. As the following table indicates, 26 has 2 sets of 10’s and 6 set of 1s.

Therefore you can write 26 as:  26 = 2 × 10 + 6
This concept can also be applied to larger numbers as well as fractions. For example, the number 8345 has number 8 in the thousands place, number 3 in hundreds place, number 4 in tens place and number 5 in one’s place. We can write 8345 as:
8345 = 8 × 1000 + 3 × 100 + 4 × 10 + 5
The following table summarizes the place value and the effective value of each of the digits.

Numbers and Digits in Decimals
The concept is also applicable for decimals. Each of the digits to the right of the decimal point has a value. For example: Number 12.65 has number 1 in the tens place, number 2 in ones place, number 6 in the tenths place and number 5 in the hundredth place.  We can write 12.65 as:
12.65 = 1 × 10 + 2 × 1 + 6 ÷ 10 + 5 ÷ 100

In a decimal system, any number can be written as the sum of  integral powers of 10 multiplied by numbers from 1 to 9.
Fractional numbers can also be written using this convention. For fractions, negative powers will be involved.
• The digits to the left correspond to increasingly positive integral powers of 10.
• The digits to the right of the decimal point correspond to increasingly negative integral powers of 10.

Wrapping Up
We can use the above concepts to represent 4981.623
1. The 4 at the left has a place value $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 4 \times10^3=4000$(also referred as thousands place)
2. The 9 at the left has a place value $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 9 \times10^2=900$(also referred as hundreds place)
3. The 8 at the left has a place value $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 8 \times 10^1 =80$(also referred as tens place)
4. The 1 at the left has a place value $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 1 \times 10^0=1$(also referred as ones place)
5. The decimal point "."
6. The 6 at the right has a place value $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 6 \times 10^{-1}= 6 \times 0.1 = 0.6$(also referred as tenths place)
7. The 2 at the left has a place value $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 2 \times 10^{-2}=0.02$(also referred as hundredths place)
8. The 3 at the right has a place value $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 3 \times 10^{-3} = 3 \times 0.01 = 0.003$(also referred as thousandths place)
The place value table can be summarized as follows:
 Digit 4 9 8 1 . 6 2 3 Place Multiple 1000 100 10 1 NA 0.1 0.01 0.001 Effective Value 4000 900 80 1 . 0.6 0.02 0.003 Place Value Thousands Place Hundreds Place Tens Place One’s Place Decimal Point Tenths Place Hundredths Place Thousandths Place
The place value concept is very useful to solve more complex digits problem in GRE. Using this concept, any 3 digit numbers can be written as $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 100x + 10 y + z$ , where $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} x$$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} y$, and   $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} z$  can be any integer from 0 to 9.
Similarly, any 2 digit decimal less than 1 can be expressed as $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \frac{x}{10} + \frac{y}{100}$, where  $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} x$ and $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} y$ can be any integers between 0 to 9.

Example:
The digit in the hundreds place of a 4 digit number is increased by 4. What is the difference between the new number and the old number?
A) 10
B ) 16
C) 300
D) 400
E) 1600
Solution:

The digit at the hundreds place of any number has a place value of $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} x \times10^2$(where  $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} x$ is the digit at the hundreds place).  If this digit is increased by 4, the new number will increase by $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 4 \times 10^2=400$
Therefore the difference between the new number and the old number is = 400