Decimals
A decimal is a number that represents a fraction of something larger. It is another way of representing a fraction. For example the fraction \(\frac{1}{2}\) can be written in decimal form as 0.5.
Each digit in a decimal number has a value depending upon its place or position in the number. This is known as place value. The table below provides the place value of each digit in a decimal number of 6 digits.
Hundreds | Tens | Ones | Decimal Point | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|
1 x 100 = 100 | 1 x 10 = 10 | 1 x 10 = 10 | . | \(\frac{1}{10}\) = 0.1 | \(\frac{1}{100}\) = 0.01 | \(\frac{1}{1000}\) = 0.001 |
When writing decimals in expanded form, each digit is expressed as per its place value.
This is accomplished by multiplying each digit by its place value and then adding the products.
Consider the following example: Using the place value table, express each digit in 236.482 as per their place values.
Hundreds | Tens | Ones | Decimal Point | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|
2 x 100 = 200 | 3 x 10 = 30 | 6 x 1 = 6 | . | \(\frac{4}{10}\) = 0.4 | \(\frac{8}{100}\) = 0.08 | \(\frac{1}{1000}\) = 0.002 |
Therefore, the decimal 236.482 can be written as; 2 hundreds + 3 tens + 6 ones + 4 tenths + 8 hundredths + 2 thousandths.
Using Models to Divide Decimals by Whole Numbers
Division of a decimal by a whole number can be modeled using a grid of 100 squares. 100 squares together equals one whole, one column of squares (10 squares) equals 1 tenths or 0.1 and one square equals 1 hundredths or 0.01. Depending upon the whole number value of the decimal we might have to use more than 1 grid.
A larger value can be regrouped as a smaller value in the grid. For example; 1 whole can be regrouped as 10 tenths ( 0.1) or 100 hundredths (0.01).
We can also use multiplication to check the answers because multiplication is the inverse of division. We can do this by multiplying the divisor by the quotient.
To divide a decimal by a whole number, shade the model as per the value of the dividend and then divide the shaded portion into equal groups of the same size as the whole number or the divisor. Then, the quotient will be the size of each equal group.
For example, \(0.8\div 2=?\)
0.8 is 8 tenths
8 columns can be used to represent 8 tenths. We divide the 8 columns into equal groups. Then determine the size of each group.
Each group has 4 columns, that is, 4 tenths.
Therefore, \(0.8\div 2=0.4\).
Check: 0.4 x 2 = 0.8.
(To check the answer, use the inverse operation. To ensure that the quotient is correct, we can multiply the divisor by the quotient. We’ll know the quotient is correct if the product equals the dividend.
Pay attention to the decimal point when multiplying to ensure that it is placed correctly in the quotient.)
Using Models to Divide Decimals by Decimals
To divide a decimal by a decimal we can use the same model as used above. Observe the value of the dividend and the divisor in the division equation or problem. Shade the model as per the value of the dividend and then divide the shaded portion into equal groups of the same size as the divisor. Then, the quotient will be the size of each equal group.
For example, \(0.8\div 0.2=?\)
Shade 8 columns to represent 8 tenths.
Divide the shaded portion into groups of 0.2.
We have 4 groups containing 2 tenths.
Hence, \(0.8\div 0.2=4\).
Solved Examples
Example 1: Use the model to find \(2.25\div 5=?\)
Solution: 2.25 can be regrouped as 225 hundredths. Shade 225 hundredths to represent 2.25. Divide the model into 5 equal groups.
- 225 hundredths can be divided equally into 5 groups each of size 45 hundredths.
- 45 hundredths is 0.45.
Therefore, \(2.25\div 5=0.45\)
Example 2: To make a beaded wind chime, you cut a 2.75-feet long string into five equal-length pieces. What is the total length of each piece of string?
Solution: To determine the length of each piece of string divide the total length of the string by the total number of pieces, that is, \(2.75\div 5=?\).
2.75 can be regrouped as 275 hundredths. Shade 275 hundredths to represent 2.75.
Divide the model into 5 equal groups.
- 275 hundredths can be divided equally into 5 groups each of size 55 hundredths.
- 55 hundredths is 0.55.
Therefore, \(2.75\div 5=0.55\).
Hence, length of each string piece = 0.55 foot.
Example 3: Use a model to find \(1.72\div 0.86=?\)
Solution: Shade 17.2 columns to represent 1.72. Divide the model into groups of 86 tenths, that is 0.86.
There are 2 groups of 0.86.
Therefore, \(1.72\div 0.86=2\).
Example 4: You buy a bag of peanuts, pay $5 to the cashier and get $2.93 in change. If the price of the peanuts is $0.23 per pound then what quantity of peanuts did you buy?
Solution: Money spent on the peanuts = $5 – $2.93 = 2.07$.
The cost of peanuts per pound is $0.23.
Therefore, the quantity of peanuts brought = \((2.07\div 0.23)\) pounds.
Shad 20.7 columns to represent 2.07. Divide the model into groups of 23 tenths, that is,0.23. There are 9 groups of 0.23.
\(2.07\div 0.23=9\)
Therefore, the quantity of peanuts bought is 9 pounds.
Example 5: A 1.5-feet piece of scrapbook paper has been given to you. You’re going to cut it into 0.5-foot-long sheets . How many scrapbook sheets do you have now?
Solution: We have to find \(1.5\div 0.5\)
Shade 15 columns to represent 1.5 feet. Divide the model to show groups of 0.5 or 5 tenths.
There are 3 groups of 5 tenths.
\(1.5\div 0.5=3\).
Therefore, there are 3 sheets of scrapbook paper.
The place value of the fourth digit after the decimal point in a decimal number is 10 thousandths.
For example, in 2.4531, the place value of the digit 1 is ten thousandths.
The tenths, hundredths and thousandths can be related as:
10 thousandths = 1 hundredths.
10 hundredths = 1 tenth.