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# DECIMALS, PERCENTAGES, AND SQUARE ROOTS

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### Decimals, Percentage, and Square Roots

 4.1 Percentage

### Decimal Fractions

It was in the study of whole numbers that the concept of place value was introduced. This allowed a large number of objects to be represented in a compact way by giving a different meaning to a digit depending on its location relative to the unit's place. Thus, the number 30 means thirty objects, whereas the number 300 means three-hundred objects. This concept can be extended to indicate fractional numbers of objects and requires the introduction of the decimal point. The decimal point separates the units place from the fractional part of a number. The place immediately to the right of the decimal point is called the tenths place, and the digit located there indicates the number of tenths of an object. The place to the right of the tenths is the hundredths place, and so on (see below).

7 3 6 1 . 2 9 8

Thousandths

1/1000

Hundredths

1/100

Tenths

1/10

Units

1

Tens

10

Hundreds

100

Thousands

1000

Numbers with digits to the right of the decimal place are decimal fractions. Numbers consisting of a whole number and a decimal fraction are sometimes called mixed decimals. The magnitude of a decimal fraction or a mixed decimal is the sum of the digits each multiplied by its place value. The procedure is exactly the same as computing the magnitude of a whole number (see below).

Digit

Place Value

In the example above, the LCD of the fractional parts were found so that they could be added directly.

A common fraction can be changed to a decimal by performing the division, which is indicated by the common fractions.

For example, the decimal equivalent of 3/4 can be found as follows:

The number 0.75 is the same as 75/100. Reducing 75/100 to its lowest terms, dividing the numerator and denominator by 25, results in the fraction 3/4.

There are some common fractions that do not have exactly equivalent decimal fractions. The decimal equivalent of 2/3 is:

where the . . . indicates successive 6s.

This type of decimal fraction is normally approximated by rounding.

In a previous example, we found the decimal equivalent of a common fraction by dividing. The reverse process can be done by changing a decimal fraction to a common fraction, by writing the decimal as a common fraction, and then reducing it to lowest terms.

For example, the common fraction equivalent of the decimal fraction 0.375 can be seen below:

Remember that the decimal fraction 0.375 is just a compact way of writing:

 7 x 1000 = 7000 3 x 100 = 300 6 x 10 = 60 1 x 1 = 1 2 x 1/10 = 2/10 = 200/1000 9 x 1/100 = 9/100 = 90/1000 8 x 1/1000 = 8/1000 = 8/1000

The addition and subtraction of decimals is accomplished in exactly the same manner as the addition and subtraction of whole numbers. The numbers are placed in a vertical column, taking care that the place values are aligned. In the case of decimals, this can be ensured by aligning the decimal points.

For example, here is how to find the sum of 39.62, 41.093, and 0.0327:

 39.62 41.093 0.0327 80.7457

Notice that the principle of carrying from one place to the next is also used in adding decimals.

Subtraction is done in the same manner as with whole numbers.

Below, you can see how to find the difference between 32.1 and 16.379:

 32.1 -16.379 15.721

There are two points to notice in this example. First, it is sometimes necessary to add zeros to the right of a decimal fraction as an aid in the subtraction process. This does not change the value of the decimal; that is, 32.1 is exactly equal to 32.100. Second, the principle of borrowing from one place and adding to another was used.

### Multiplying and Dividing Decimals

The multiplication of decimals can be accomplished in two ways. In the first method, the numbers are placed one above the other and the multiplication carried out without regard to the decimal points, just as with whole numbers. The decimal point in the product is located by adding the number of digits to the right of the decimal points in the multiplicand and the multiplier and placing the decimal point in the product with this number of digits to the right of the decimal point.

This is how to multiply 16.2 and 1.15:

There is one digit to the right of the decimal point in the multiplicand and two digits to the right of the decimal point in the multiplier. Thus, the decimal point in the product is placed with three digits (2+1) to its right.

The multiplication could also be carried out by converting the decimals to improper fractions and multiplying directly.

Here is how you multiply 16.2 and 1.15:

In dividing decimals, the decimal point in the divisor is moved all the way to the right, and the decimal point in the dividend is moved to thesamenumber of places to the right. Division is then carried out, and the decimal point in the quotient is located directly above the decimal point in the dividend.

For example, you can see here how to divide 41.05 by 2.5:

Note the movement of the original decimal point.

Notice that the process of moving the decimal points is equivalent to multiplying the dividend and divisor by factors of 10 so that the divisor becomes a whole number. The division can also be carried out by converting the decimals to improper fractions and then dividing.

Here is how you would divide 41.05 by 2.5:

### Percentage

In dealing with everyday problems, statements frequently are made such as a pay raise of 10 percent, or the price increased by 25 percent, and so on. The statement really is a pay raise of ten-one-hundredths of the former salary. The word percent means "hundredths" and is given the symbol %.

25 percent means 25 hundredths:

A percent is changed to a common fraction by omitting the percent sign, placing the number over 100, and reducing the resulting fraction if possible.

Here is how you would change 32% to a common fraction:

Thus, if a quantity changes by 32%, this is equivalent to changing the quantity by an amount equal to 8/25 of its value.

A percent can be changed to a decimal fraction by omitting the percent sign and moving the decimal point two places to the left.

1. To change 92% to a decimal fraction:

92% = 0.92

2. To change 8% to a decimal fraction:

8% = 0.08

### Squares and Square Roots

There is a compact notation that is utilized to indicate when a number is multiplied by itself. This operation is indicated by placing a small number called an exponent above and to the right of another number, which is called the base. The exponent shows how many times the base is multiplied by itself.

1. 62means that 6 is multiplied by itself twice (6 x 6).
2. 53means that 5 is multiplied by itself 3 times (5 x 5 x 5).
3. 24means that 2 is multiplied by itself 4 times (2 x 2 x 2 x 2).

A special name is given for multiplying a number by itself two times. This is called squaring.

62is called the square of 6 or 6 squared

62= 6 x 6 = 36

The square of 6 is 36.

The square root of a number is the quantity which, when multiplied by itself (squared), gives the original number as a product.

The square root of 36 is 6 because 6 x 6 = 36.

A special symbol is used to indicate the operation of taking the square root of a number. This operation is shown by the symbol √, which means that you are to take the square root of whatever is placed under the symbol.

√36 = 6

The following tables list common squares and square roots:

 Number 1 2 3 4 5 6 7 8 9 10 Square 1 4 9 16 25 36 49 64 81 100

 Number 1 4 9 16 25 36 49 64 81 100 Square Root 1 2 3 4 5 6 7 8 9 10

### Calculator Exercises

The calculator is used very often when working problems involving decimals, percentage, and square roots. This is normally much easier than working problems with fractions using the calculator. The following examples show the versatility of using the calculator for these problems.

### Percentage

 1. Changing 25% into a decimal form: 25% = 0.25 2. Changing 8% into a decimal form: 8% = 0.08 3. Finding 25% of 40: 25% x 40 =?25% x 40 = 10

### Square and Square Roots

 1. Finding the square of 6: 62=?6 x 6 = 36 2. Finding the square of 2.5: 2.52=?2.5 x 2.5 = 6.25 3. Finding the square root of 36: √36 = 6 4. Finding the square root of 6.25: √6.25 = 2.5