Multiplication of Decimals
Multiplication of Decimals by Power of 10
Case 1: When a decimal is multiplied by 10. Like\[\text{8}.\text{6542}\times \text{1}0\] The decimal is shifted one digit right. Thus\[\text{8}.\text{6542}\times \text{1}0=\text{86}.\text{542}\]
Case 2: When a decimal is multiplied by 100. Like\[\text{8}.\text{6542}\times \text{1}00\] The decimal is shifted two digit right. Thus\[\text{8}.\text{6542}\times \text{1}00=\text{865}.\text{42}\]
Case 3: When a decimal is multiplied by 1000. Like\[~\text{8}.\text{6542}\times \text{1}000\] The decimal is shifted three digit right. Thus \[\text{8}.\text{6542}\times \text{1}000=\text{8654}.\text{2}\]
Note: As you increase the power of 10 by one, the decimal point shift one digit right.
For example: \[\text{2}.\text{35682}\times \text{1}{{\text{0}}^{2}}=\text{235}.\text{682}\] \[2.35682\times {{10}^{3}}=2356.82\] \[2.35682\times {{10}^{4}}=23568.2\]
\[\mathbf{Multiply:394}\mathbf{.5607\times 1000}\]
Solution:
\[~\text{394}.\text{56}0\text{7}\times \text{1}000=\text{39456}0.\text{7}\]
\[\mathbf{Multiply:100}\mathbf{.15\times 100}\]
Solution:
\[\text{1}00.\text{15}\times \text{1}00=\text{1}00\text{15}\]
\[\mathbf{Multiply:494}\mathbf{.3607\times 1000}\]
(a) 234343.787
(b) 446676.9786
(c) 23424.56757
(d) 494360.7
(e) None of these
Answer (d)
Explanation-\[\text{494}.\text{36}0\text{7}\times \text{1}000=\text{49436}0.\text{7}\]
Multiplication of a Decimal and a Whole Number
Step 1: Remove the point from the decimal and multiply the numbers like multiplication of whole numbers.
Step 2: Insert a point in the product such that the given decimal and the product have same number of decimal places.
Find the product of 42 and 75.62.
Solution:
Multiply 42 and 7562 Thus \[\text{42}\times \text{7562}=\text{3176}0\text{4}\] Now place a point in 317604 such that 317604 and 75.62 have same decimal places. Thus the product of\[~\text{42}\times \text{7562}=\text{3176}.0\text{4}\] 
Subtraction of Decimals
Step 1: Arrange the decimals one below other such that decimal points come in same column.
Step 2: Now subtract the digits.
Step 3: Place the decimal in the decimal column in the difference.
Subtract 96.042 from 231.289
Solution:
Subtract\[\text{231}.\text{289}-\text{96}.0\text{42}=\text{135}.\text{247}\]
A = 10.468, B = 9.234. Find\[\mathbf{A}-\mathbf{B}\]
Solution:
\[\text{A}-\text{B }=\text{ 1}0.\text{468}-\text{9}.\text{234 }=\text{1}.\text{234}\] 
Addition of Decimals
Step 1: Arrange the decimals one below other such that decimal points come in same column.
Step 2: Now add the digits.
Step 3: Place a point in the sum in point column.
Add: \[\mathbf{231}.\mathbf{289}+\mathbf{96}.\mathbf{042}+\mathbf{1}.\mathbf{468}+\mathbf{9}.\mathbf{234}\]
Solution:
\[\begin{align} & \,\text{231}.\text{289} \\ & \,\,\,\text{96}.0\text{42} \\ & \,\,\,\,\,\,\,\text{1}.\text{468} \\ & +\text{ 9}.\text{234} \\ & \underline{\overline{\text{338}.0\text{33}}} \\ \end{align}\]
\[\mathbf{2312}\mathbf{.899 + 604}\mathbf{.2146 + 89}\mathbf{.2342}\]
Solution:
\[\begin{align} & \,\text{2312}.\text{899}0 \\ & \,\,\,\,\text{6}0\text{4}.\text{2146} \\ & +\text{ }\,\text{89}.\text{2342} \\ & \underline{\overline{\text{3}00\text{6}.\text{3478}}} \\ \end{align}\] 
Comparison of Decimals
Step 1: Compare the whole parts. The decimal having greater whole part is greater.
Note : In case of equal whole parts, follow the step 2.
Step 2: Compare the decimal part, the decimal having greater decimal part is greater.
Compare between 9.2 and 7.9 and find which is greater?
Solution:
Whole part of 9.2 = 9 Whole part of 7.9 =7 \[\text{9}>\text{7}.\text{ So 9}.\text{2}>\text{7}.\text{9}\]
Compare 14.7 and 14.3, which is greater?
Solution:
Whole part of 14.7 =14
Whole part of 14.3 = 14
\[\because \]Both the decimals have same whole part, therefore, compare decimal parts.
The digit at the tenths place in 14.7 = 7
The digit at the tenths place in 14.3 = 3
\[\text{7}>\text{3}.\text{ So 14}.\text{7}>\text{14}.\text{3}\] 
Decimals
A decimal number is broadly divided into two parts.
(i) Whole number part
(ii) Decimal part The two parts are separated by a dot (.) called the decimal point. From the decimal point as you move on the left the place value is multiplied by 10 and as you move on the right it is divided by 10.
| Decimal Fraction | Whole Part | Decimal Part |
| 5.3 | 5 | 3 |
| 147.81 | 147 | 81 |
| 15.679 | 15 | 679 |
| .8 | 0 | 8 |
| 1.004 | 1 | 004 |
It is divided into ten equal parts. Out of which two parts are shaded. We say, two tenth of the whole is shaded. Two tenth is written as\[\frac{2}{10}\]. We also write\[\frac{2}{10}\]as more... 
Operation on the Fractions
Addition of Like Fractions
\[\text{Sum of like fractions}=\frac{\text{Sum of numerators}}{\text{common denominator}}\] In addition of like fractions, sum of the numerators will be the numerator for the resulting fraction and the common denominator will be the denominator. \[\frac{P}{R}+\frac{Q}{R}=\frac{P+Q}{R}\]
Add the following fractions.
Solution: \[\frac{3}{11}+\frac{2}{11}+\frac{5}{11}=\frac{10}{11}\]
Add the following fractions:
\[\frac{3}{10}+\frac{2}{10}\]
Solution:
\[\frac{3}{10}+\frac{2}{10}=\frac{5}{10}\]
Addition of Unlike Fractions
Add\[\frac{5}{7}\] and\[\frac{5}{8}\]
Step 1: Convert the fractions into like fractions. \[\frac{5}{7}=\frac{5\times 8}{7\times 8}=\frac{40}{56}\] And \[\frac{5}{8}=\frac{5\times 7}{8\times 7}=\frac{35}{56}\]
Step 2: Add numerator of the fractions\[\text{4}0+\text{35}=\text{75}\].
Step 3: Write the sum as numerator for the required fraction and common denominator as denominator\[\frac{75}{56}\]
Add\[\frac{12}{19}\]and\[\frac{12}{11}\]
Solution: \[\frac{12}{19}=\frac{12\times 11}{19\times 11}=\frac{132}{209}\]
\[\frac{12}{11}=\frac{12\times 19}{19\times 19}=\frac{228}{209}\]
\[\frac{12}{19}+\frac{12}{11}=\frac{132}{209}+\frac{228}{209},=\frac{132+228}{209}=\frac{360}{209}\]
Subtraction of Like Fractions
\[\text{Difference of like fractions}=\frac{\text{Difference of numerators}}{\text{Common denominator}}\]
In subtraction of like fractions, the difference of the numerators will be the numerator and the common denominator will be the denominator for the required fraction.
\[\frac{p}{q}-\frac{r}{q}=\frac{p-r}{q}\]
Solve the following:
\[\frac{5}{7}-\frac{3}{7}\]
Solution: \[\frac{5}{7}-\frac{3}{7}=\frac{5-3}{7}=\frac{2}{7}\]
Represent the shaded part in the above figures as a fraction and find their difference.
Solution:
\[\frac{3}{5}-\frac{2}{5}=\frac{3-2}{5}=\frac{1}{5}\]
Subtraction of Unlike
Fractions Subtract\[\frac{5}{6}-\frac{4}{5}\]
Step 1: Convert the fractions into like fractions.
\[\frac{5}{6}=\frac{5\times 5}{6\times 5}=\frac{25}{30}\]
\[\frac{4}{5}=\frac{4\times 6}{5\times 6}=\frac{24}{30}\]
Step 2: Find difference of the numerator.
\[\text{25}-\text{24}=\text{1}\].
Step 3: Write the difference as numerator and common denominator as denominator for the required fraction.
\[\frac{1}{30}\]
Solve\[\frac{7}{9}-\frac{5}{8}\]. Solution:
\[\frac{7}{9}=\frac{7\times 8}{9\times 8}=\frac{56}{72}\]
\[\frac{5}{8}=\frac{5\times 9}{8\times 9}=\frac{45}{72}\]
Now \[\frac{56}{72}-\frac{45}{72}=\frac{56-45}{72}=\frac{9}{72}=\frac{1}{8}\] 
Comparison of Fraction
Comparison of Like Fractions
Let\[\frac{p}{q}\]and\[\frac{r}{q}\]are like fractions.
If p is greater than \[q,\frac{p}{q}>\frac{r}{q}\]
Compare between\[\frac{\mathbf{7}}{\mathbf{9}}\]and\[\frac{5}{\mathbf{9}}\].Which is greater?
Solution:
\[7>5\] \[\frac{7}{9}>\frac{5}{9}\]
Comparison of Fractions Having Same Numerator
If the two fractions have same numerator, the fraction which has smaller denominator is greater. Like\[\frac{P}{Q}\]is greater than\[\frac{P}{R}\]if\[\text{Q}<\text{R}\].
Find the greatest fraction out of the given fractions:
\[\frac{18}{23},\frac{18}{17},\frac{18}{19},\frac{18}{20},\frac{18}{12}\]
Solution:
\[\frac{18}{12}\]is the greatest fraction among the given fractions. As it has smallest denominator.
Comparison of Unlike Fractions
Compare between\[\frac{7}{13}\]and\[\frac{6}{9}\]
Step 1: Convert the fractions into like fractions.
\[\frac{7\times 9}{13\times 9}=\frac{63}{117}\]
And\[\frac{6\times 13}{9\times 13}=\frac{78}{117}\]
Step 2: The fraction having greater numerator is greater.
\[\because \]\[78>63\]
\[\therefore \]\[\frac{78}{117}>\frac{63}{117}\]or\[\frac{6}{9}>\frac{7}{13}\]
Compare between\[\frac{21}{22}\]and\[\frac{22}{23}\], which is greater?
Solution: \[\frac{21}{22}=\frac{21\times 23}{22\times 23}=\frac{483}{506}\]
\[\frac{22}{23}=\frac{22\times 22}{23\times 22}=\frac{484}{506}\]
\[\because \]\[484>483\]
\[\therefore \]\[\frac{484}{506}>\frac{483}{506}\]or\[\frac{22}{23}>\frac{21}{22}\] 
Conversion of mixed Fraction into Improper Fraction
Convert\[11\frac{4}{7}\]into improper fraction
Step 1: Multiply the whole number by the denominator of the fractional part and add the numerator to the resulting number. \[\text{11}\times \text{7}+\text{4}=\text{81}\].
Step 2: Write the resulting number as numerator for the required fraction and denominator is same as the fractional part has\[\frac{81}{7}\].
Convert \[\mathbf{44}\frac{\mathbf{3}}{\mathbf{7}}\] into improper fraction.
Solution: \[\frac{7\times 44+3}{7}=\frac{308+3}{7}=\frac{311}{7}\]
Thus\[44\frac{3}{7}=\frac{311}{7}\] 
Fraction
Fraction is used to indicate a part of a whole. It is represented as \[\frac{a}{b}\] where, a is called numerator and b is called denominator of the fraction. It may be explained as - If a whole is divided into some equal parts, each part is called fraction of the whole and the number which is used to represent the part is called fractional number. Let 4 kg flour is divided into five equal parts. The amount each part will contain is represented as \[\frac{4}{5}\] kg here \[\frac{4}{5}\] is a number which is known as fraction. Thus fraction is a mathematical term which represents part of a whole. Shaded part in the following figures has been represented by fractions.
1 kg corn is divided into five equal parts. Represent the amount contained by each part as a fraction.
Solution:
1 kg corn is divided into five equal parts. Thus each part will contain \[\frac{1}{5}\] kg corn.
Like Fraction
The fractions having same denominator are called like fractions.
\[\frac{1}{4},\frac{6}{4},\frac{7}{4}\] are like fractions.
Choose the like fractions from the following:
\[\frac{2}{5},\frac{7}{8},\frac{4}{15},\frac{3}{8}\]
Solution: \[\frac{7}{8}\]and\[\frac{3}{8}\]are like fractions because they have same denominator.
Unlike Fraction
The fractions having different denominators are called unlike fractions.
\[\frac{4}{5},\frac{5}{9},\frac{6}{3}\]are unlike fractions because they have different denominators.
Are\[\frac{4}{9}\]and\[\frac{5}{7}\]unlike fractions?
Solution: Yes, because\[\frac{4}{9}\]and\[\frac{5}{7}\]have different denominators.
Conversion of Unlike Fraction into Like Fraction
Let\[\frac{p}{q}\]and\[\frac{r}{s}\]are two unlike fractions. To convert them into like fractions first multiply p and q by s then multiply r and s by q.
Thu \[\frac{p\times s}{q\times s}\] and\[\frac{r\times q}{s\times q}\]are like fractions.
Convert\[\frac{7}{11}\]and\[\frac{6}{9}\].into like fractions.
Solution:\[\frac{7}{11}=\frac{7\times 9}{11\times 9}=\frac{63}{99}\]
\[\frac{6}{9}=\frac{6\times 11}{9\times 11}=\frac{66}{99}\]
\[\frac{63}{99}\]and\[\frac{66}{99}\]are like fractions
Equivalent Fraction
The fractions which have same value are called equivalent fractions. Like\[\frac{5}{21}\]and\[\frac{10}{42}\] are equivalent fractions.
Finding an Equivalent Fraction of a given Fraction
To find an equivalent fraction of a given fraction, numerator and denominator of the fraction is multiplied or divided by a same number.
Find an equivalent fraction of\[\frac{\mathbf{16}}{\mathbf{21}}\].
Solution: Multiply both 16 and 21 by a same natural number.
\[\frac{16\times 2}{21\times 2}=\frac{32}{42}\]
Thus\[\frac{16}{21}\] and\[\frac{32}{42}\] are equivalent fractions.
Unit Fraction
The fractions in the form\[\frac{p}{q}\] (when p = 1 and \[q\ne 0\]) are called unit fractions.
\[\frac{1}{4},\frac{1}{5},\frac{1}{6}\]are unit fractions.
\[\frac{p}{14}\]is a unit fraction. Find the value of P. more... 
Introduction
When a figure is divided in equal number of point. The point of the figure a represented by the fraction. In a fraction, numerator represents the required number of parts of figure and denominator represents the total number of points in which the whole figure is divider into. 
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