# 6-5 Decimal Ops - Concepts and Explanations

## Choosing an Operation and Estimating

Students encounter various situations involving decimals. They decide which operations to use to find a solution. Students estimate to choose operations and check their work.

### Examples

Chakara makes a rectangular tablecloth that is 3.5 meters long and 1.5 meters wide. What is the area of the tablecloth?

The dimensions are about 4 meters by about 2 meters. To find the approximate area, multiply 4 x 2. To find the exact area, multiply 3.5 x 1.5.

## Addition and Subtraction of Decimals

**Decimals As Fractions** Write the decimals as fractions. Find common denominators and add or subtract the
fractions. Then express the answer as a decimal.

### Example

Zeke buys cider for $1.97 and pretzels for $.89. What is the total cost?

Written as fractions with denominator 100, the cost is ^{197}/_{100} + ^{89}/_{100} or ^{286}/_{100}, or 2.86. This is comparable to thinking of the cost in pennies and writing the sum
in dollars.

**Place Value Interpretation** Students analyze the meaning of each digit of a number. They see that they must compute
with digits that occupy like places when adding or subtracting numbers.

### Example

To find the difference 3.725 - 0.41, subtract thousandths from thousandths (0.005 - 0.000), hundredths from hundredths (0.02 - 0.01), tenths from tenths (0.7 - 0.4), and ones from ones (3 - 0).## Multiplication of Decimals

**Decimals As Fractions** Write the decimals as fractions. Multiply the fractions. Then write the answer as
a decimal. The number of decimal places in the factors relates to the number of decimal
places in the answer.

### Example

Find the product 0.3 x 2.3. As fractions, this is ^{3}/_{10} x 2 ^{3}/_{10} = ^{3}/_{10} x ^{23}/_{10}; the product is ^{69}/_{100}, or 0.69, The denominator of the fraction tells the place value of the decimal.

**Place Value Interpretation** Students find patterns in sets of problems to see why counting decimal places makes
sense.

### Example

Find the product 0.25 x 0.31. Use the fact that 25 x 31 = 775. Tenths x tenths results in hundredths in the product, so 2.5 x 3.1 = 7.75. Tenths * hundredths results in thousandths, so 2.5 x 0.31 = 0.775. Hundredths x hundredths results in ten-thousandths, so 0.25 x 0.31 = 0.0775.

## Division of Decimals

**Decimals As Fractions** Express decimals as fractions. Find common denominators. Then divide the numerators.

### Example

Find the quotient 3.25 , 0.5.

Rewrite the expression as ^{325}/_{100} ÷ ^{5}/_{10} = ^{325}/_{100} ÷ ^{50}/_{100}. This is the same as 325 ÷ 50, which is 6 ^{1}/_{2} or 6.5.

**Place Value Interpretation** Write an equivalent problem: multiply the dividend and the divisor by the same power
of ten until both are whole numbers.

### Example

This approach explains why moving decimal points works.

## Decimal Forms of Rational Numbers

**Finite (Terminating) Decimals** Rational numbers with decimal forms that “end” are finite decimals. The simplified
fraction form has only 2s or 5s in the prime factorization of the denominator.

### Example

^{1/}_{2} = 0.5; ^{3}/_{4} = 0.75; ^{1}/_{8} = 0.125; ^{12}/_{75} = 0.16

**Infinite (Repeating) Decimals** Rational numbers with decimal forms that “continue forever” but repeat are infinite
decimals. The simplified fraction form has numbers other than 2 or 5 in the prime
factorization of the denominator.

### Example

^{1}/_{3} = 0.3333...; ^{2}/_{3} = 0.6666...; ^{8}/_{15} = 0.5333...; ^{3}/_{7} = 0.4285714285714...

## Finding Percents

This Unit includes primarily three types of percent problems:

Find a percent of a number, based on the total and the percent rate

### Example

Jill buys a $7.50 CD. Sales tax is 6%. How much is the tax?

1% of $7.50 = ^{1}/_{100} of $7.50, or 0.075. So 6% of $7.50 is 0.075 x 6, or $.45

Find the total amount, based on the percent of the amount and the percent rate

### Example

20% of some number is $2.50. It takes five 20,s to make 100,. 5 x $2.50 = $12.50, so the total bill was $12.50.

Sam got a $12 discount on a $48 shirt. What percent was the discount?

Find the percent rate, based on the percent of the amount and the total.

## Example

Sam got a $12 discount on a $48 shirt. What percent was the discount?

There are four 12s in 48, or the percent is ^{1}/_{4} of 100%, or 25%.