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Decimals
Decimals
A decimal is a number that is written using the base-ten place value system. A decimal point separates the ones and tenths digits
Add and subtract decimals
Addition of decimals
You have learnt addition of decimals upto two decimal places in class 4.
Let us review it.
We have seen that in addition of decimals the following points are kept in mind.
- We line up the decimal point.
- The decimals are made equal decimal places by adding zeros.
- We add the decimals as we add the whole numbers.
- The decimal points are in column.
We proceed ahead to learn the addition of decimal up to four places.
Subtraction of Decimals
We have learnt subtraction of decimals upto two decimal places in class 4.
Let us review it
We have seen that in subtraction of decimals the following points are kept in mind..
- We line up the decimal points
- The decimals are made equal decimal places by adding zeros.
- We subtract the decimals as we subtract the whole numbers.
- The decimal points are in column
We proceed ahead to learn the subtraction of decimals upto four decimal places
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Decimals
Recognize like and unlike decimals
Consider the following decimals:
I) 33.2 It has one decimal place
ii) 124.35 It has two decimal places.
iii) 41.237 It has three decimal places.
iv) 29.1345 It has four decimal places.
Like decimals:
Decimals with the same number of decimal places are called like decimals.
For example, 12.345, 2.127 are like decimals, each has three decimal places.
Unlike decimals:
Decimals with different number of decimal places are called unlike decimals.
For example, 9.72, 13.5, 321.578 and 3.1245 are unlike decimals.
Each has a different number of decimal places.
Note:
We can transform unlike decimals to like decimals by adding placeholder zeros on right side of decimal part.
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Decimals
Multiplication of decimals by 10, 100 and 1000
(a)Multiplication of decimals by 10
Multiplying a decimal by 10 is equivalent to forming a new number by moving the decimal point of the given decimal to the right 1 place.
Examples
- 3.57 × 10 = 35.7 iii. 97.23 × 10 = 972.3
- 15.453 × 10 = 154.53 iv. 321.4 × 10 = 3214
(b) Multiplication of decimals by 100
Multiplying a decimal by 100 is equivalent to forming a new number by moving the decimal point of the given decimal to the right 2 places.
Examples
i. 38.241 × 100 = 3824.1 ii. 4.1532 × 100 = 415.32
iii. 65.32 × 100 = 6532 iv. 987.5 × 100 = 98750
(c) Multiplication of decimals by 1000
Multiplying a decimal by 1000 is equivalent to forming a new number by moving the decimal point of the given decimal to the right 3 places
Examples
i. 2.3781 × 1000 =2378.1 ii. 8.23451 × 1000 = 8234.51
ili. 7.32 × 1000 = 7320 iv. 5.7 × 1000 = 5700
More Information
1. Decimal And Whole Numbers Multiplication
2. Multiplication of Decimal Numbers -
Decimals
Division of decimals by 10, 100 and 1000
(a)Division of decimals by 10
Dividing a decimal by 10 is equivalent to forming a new number by moving the decimal point of the given decimal to the left 1 place.
Examples
i. 51.23 ÷ 10 = 5.123 ii. 321.25 ÷ 10 = 32.12
iii. 7.98 ÷ 10 = 0.798 iv. 0.275 ÷ 10 = 0.0275
(b) Division of decimals by 100
Dividing a decimal by 100 is equivalent to forming a new number by moving the decimal point of the given decimal to the left 2 places.
Examples
i. 321.25 ÷ 100 = 3.215 ii. 98.2 ÷ 100 = 0.982
iii. 8.34 ÷ 100 =0.0834 iv. 0.391 ÷ 100 = 0.00391
Rule:
Multiplication of the decimal by the whole number ignoring the decimal point. See the decimal point in the given decimal and mark the decimal point in the product with the same number of places.
More Examples
Example 3 Solve 7.324 × 5
Solution : 7.324 × 5
= 7.324 × 5 [3 decimal places]
= 36.620 [3 decimal places]
Example 4 Solve 1.4235 × 67
Solution : 1.4235 × 67
= 1.4235 × 67 [4 decimal places]
=95.3745 [4 decimal places]
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Decimals
Division of a decimal with a whole number
Consider the following examples:
Example 1 Divide 782.25 by 21
Solution 782.25 ÷ 21
Divide as you would with whole numbers.
Example 2 Divide 725.772 by 31
Solution 725.772 ÷ 31
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Decimals
Multiplication of Decimal Numbers
Multiplication of a decimal with tenth, and hundredths onlyConsider the following example
Example 1
Find the product of 7.5 and 0.6
Solution :7.5 × 0.6
\(=\frac{75}{10}\times \frac{6}{10}=\frac{450}{100}=4.50\)
Example 2
Find the product of 12.3 and 0.5
Solution : 12.3 × 0.5
\(=\frac{123}{10}\times \frac{5}{10}=\frac{123}{20}=6.15\)
Example 3
Solve 2.3 × 0.05
Solution 2.3 × 0.05
\(=\frac{23}{10}\times \frac{5}{100}=\frac{115}{1000}=0.115\)
Example 4
Solve 37.3 × 0.05
\(=\frac{373}{10}\times \frac{5}{100}=\frac{1865}{1000}=1.865\)
Multiplication of decimal by a decimal (with three decimal places)
Example 1
Solve 4.2 × 0.004
Solution 4.2 × 0.004
\(=\frac{42}{10}\times \frac{4}{1000}=\frac{168}{10,000}=0.0168\)
Example 2
Find the product of 15.6 and 0.423
Solution 15.6 × 0.423
\(=\frac{156}{10}\times \frac{423}{1000}=\frac{65988}{10,000}=6.5988\)
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Decimals
Division of Decimal by Decimal (by converting decimals to fractions)
Example 1
Divide 0.8 by 0.4
Solution 0.8 ÷ 0.4
\(\frac{8}{10}\div \frac{4}{10}=\frac{8}{10}\times \frac{10}{4}=\frac{8}{4}=2\)
Example 2
Divide 0.05 by 0. 005
Solution 0.05 ÷ 0. 005
\(\frac{5}{100}\div \frac{5}{1000}=\frac{5}{100}\times \frac{1000}{5}=10\)
Example 3
Solve 1.575 ÷ 4.5
Solution
1.575 ÷ 4.5
\(\frac{1575}{1000}\div \frac{45}{10}=\frac{1575}{1000}\times \frac{10}{45}=\frac{35}{100}=0.35\)
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Decimals
Use of division to change fractions into decimals
Example 1
Convert \(1 \over 4\) to decimal.
Thus \(1 \over 4\)=0.25
Remember:
- Dividend 1 is smaller than divisor 4. We cannot divide it.
- Put decimal point on the right of 1 and put decimal point in the quotient.
- Now 4 × 2 =8, we add one more zero on the right of already taken zero.
- Now wecomplete the division process. Leave it when remainder in zero.
- Always line up the decimal points.
Note: we can take 3 as whole number and change \(1 \over 8\) to decimal.
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Decimals
Simplify decimal expressions involving brackets (applying one or more basic operations)
Example 1 Simplify: 2.1 + (1.3 × 2.1 ÷ 0.7)
Solution 2.1 + (1.3 × 2.1 ÷ 0.7)
= 2.1 + (1.3 × 3)
= 2.1 + 3.9
=6.0
Working
2.1 ÷ 0.7
=\({21 \over10}\times{10 \over7}\)
=3
Example 2 Simplify: 8.2 - (2.2 × 1.1 + 3.1)
Solution 8.2 + (2.2 × 1.1 + 3.1)
= 8.2 + (2.42 + 3.1)
= 8.2 + 5.52
=2.68
Working
Example 3 Simplify: 2.2 (6.4 - 2.52 ÷ 2.1)
Solution 2.2 (6.4 - 2.52 ÷ 2.1)
= 2.2 (6.4 - 1.2)
= 2.2 × 5.2
= 11.44
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Decimals
Round off decimals upto specified number of decimal places
Definition:
To round a number means to approximate the number to a given value. When rounding look at the digit to the right of the given place value. If the digit to the right is less than 5, round down. If the digit to the right is 5 or greater than 5, round up.
Example 1 Round 7.12 to the nearest tenth
Solution
We want to round to the nearest tenth.
7.12 Because the hundredths’ digit 2 is less than 5, round down and drop the remaining digits
The decimal 7.12 rounded to the nearest tenth is 7.1
Example 2 Round 6.237 to the nearest hundredth.
Solution
6.237 Because the thousandths’ digit is greater than 5, round up.
The decimal 6.237 rounded to the nearest hundredths is 6.24
Example 3
Round 17.5678 to the nearest thousandths
Solution
17.5678 Because 8 > 5, we round up
The decimal 17.5678 rounded to the nearest thousandths is 17.568
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Decimals
Convert fractions to decimals and vice versa
We have learnt how to convert fraction to decimals in article 4.1.1.
We know that:
\({1 \over 4}=0.25\) ii)\({ {4} \over 5}=0.8\) iii)\(1 {3 \over 4}=1.75\) iv)\(3 {1 \over 8}=3.125\) and \(2 {1 \over 80}=2.0125\)
Let us learn how to convert decimals to fractions
Example 1
Convert 0.25 to fraction
Solution
0.25 consist of 2 tenths and 5 hundredths
\(\begin{align} & \text{Thus},\text{ }0.\text{25 }=\frac{2}{10}+\frac{5}{100} \\ & =\frac{2\times 10+5\times 1}{100} \\ & =\frac{20+5}{100}=\frac{25}{100} \\ & =\frac{1}{4} \\ \end{align} \)
(simplest form)
Alternatively
0.25(it has 25 hundredths)
\(=\frac{25}{100}=\frac{1}{4}\)
(simplest form)
Example 2
Convert 3.125 to fraction.
Solution
3.125 to fraction
\(3.125=\frac{3125}{1000}=\frac{25}{8}=3\frac{1}{8}\)
Rule to convert decimals to fractions
i.Remove the decimal point.
ii. In the denominator put 1 under decimal point.
iii. Add as many zeros on the right side of 1 as decimal places in the given decimal.
iv. Simplify the fraction to its simplest form.
More examples:
i. Convert 2.73 to fraction
Solution
=2.73
\(=\frac{273}{100}=2\frac{73}{100}\)
ii. Convert 1.65 to fraction
Solution
1.65
\(=\frac{165}{100}=\frac{33}{20}=1\frac{13}{20}\)
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Decimals
Solve real life problems involving decimal
Example 1
Noureen bought 6 note books at the rate of Rs. 22.75 per notebook. How much did she pay?
Solution
Cost of one notebook = Rs. 22.75
Number of notebooks = 6 Cost of 6 notebooks = 22.75 × 6 = Rs. 136.50
Example 2
Javeria bought 13.1 meters of cloth and paid Rs. 238.42 to the shopkeeper. Find the cost per metre of the cloth?
Solution
Number of metres the cloth was bought = 13.1
Money paid to the shopkeeper = Rs. 238.42
Rate of the cloth per metre = 238.42 ÷ 13.1
\(\begin{align} & 238.42\times \frac{1}{13.1}=\frac{238.42}{13.1} \\ & =Rs.18.20 \\ \end{align} \)
Example 3
Mehwish is 1.91m tall and Nazli is 0.03m smaller than Mehwish. Find Nazli’s height
Solution
Mehwish’s height = 1.91m
Nazli’s height = 1.91 - 0.03 = 1.88m