Positive Rational Number as Exponents
Let us consider 'a' be a positive rational number and \[x\] (which is \[\frac{m}{n}\]) be a positive rational exponent then it can also be written as \[{{a}^{x\left( i.e.{{a}^{\frac{m}{n}}} \right)}}\]or the root of \[{{a}^{m}}.\]
For example, \[{{4}^{\frac{3}{2}}}={{\left( {{4}^{3}} \right)}^{\frac{1}{2}}}={{\left( 4\times 4\times 4 \right)}^{\frac{1}{2}}}={{\left( 64 \right)}^{\frac{1}{2}}}=8\] 
Law of Exponents
For any two rational numbers a and b and for any integer's m and n we have:
Introduction
For any natural number p, the exponent is defined as: \[{{p}^{n}}=p\times p\times p.........\](up to n times), where \[{{p}^{n}}\] is called \[{{n}^{th}}\] power of p and read as "p raised to the power n".
Write 2401 in exponential form.
Solution: \[7\times 7\times 7\times 7={{7}^{4}}\]and \[\left( -7 \right)\times \left( -7 \right)\times \left( -7 \right)\times \left( -7 \right)=\left( -7 \right)4.\] Here \[{{7}^{4}}\]and \[{{\left( -7 \right)}^{4}}\]are said to be exponential form of 2401.
Note: \[a{}^\circ =1\]for all rational number 'a', but in the case of negative exponent it is expressed as \[{{(a)}^{-n}}\]and is called the exponent of base "a" is \[-n\]and it can also be defined as \[{{a}^{-n}}=\frac{1}{{{a}^{n}}}.\] 
Introduction
We know that a mathematical statement of equality which involves one or more than one variables is called an equation.
An equation in which variables are of one degree is called linear equation. If there Is only one variable, then the equation is said to be linear equation in one variable.
Linear Equation
The general form of linear equation in one variable is \[ax+b=0,\] where a and b are constant.
The general form of linear equation in two variable is \[ax+by+c=0,\] where a, b, c are constants, for example \[4x+5=5\] is a linear equation in one variable.
Solutions of the Linear Equations
The real number which satisfies the given equation is called the solution of the equation, for example 6 is the root of the equation \[3x+4=5x-8,\] because when we put x = 6 then L.H.S = 22, R.H.S = 22. That is why, to find the solution of an equation means that you have to find the value of variable for which L.H.S = R.H.S The following are the methods to solve a linear equation:
Trial and Error Method
In this method we just guess the roots of the equation.
The value of variable for which L.H.S = R.H.S is the solution of the equation. v+3
e.g. \[\frac{y+3}{3}+8=11,\] we guess different values of y, suppose 3, 6, 9 etc.
If we put y = 3 then L.H.S. = 10 and R.H.S = 11.
Therefore, L.H.S\[\ne \] R.H.S. hence y = 3 is not a solution. Put y = 6 then L.H.S. = 11 and R.H.S. = 11. Therefore, L.H.S. = R.H.S. hence y = 6 is the solution of given equation
Systematic Method
Solve \[19x+2=40\]
Solution:
\[19x+2=40\] \[\Rightarrow 19x+2-2=40-2\](subtracting 2 from both sides)
\[\Rightarrow 19x+0=38\Rightarrow 19x=38\]
\[\Rightarrow \frac{19x}{19}=\frac{38}{19}(dividing\text{ }both\text{ }sides\text{ }by\text{ }19)\]
\[\Rightarrow x=2.\]Therefore, \[x=2\]is the solution of given equation.
Transposition Method
In this method the following steps are involved:
Step 1: Identify the unknown quantity
Step 2: Transfer all the term containing variable on the left hand side and constant term on the right hand side.
Step 3: Simplify left hand side and right hand side in such a way that each side containing only one term.
Step 4: Solve the equation obtained above by systematic method.
Solve \[3(y+3)-2\text{ (}y-1\text{)}=5(y-5)\]
Solution:
\[3(y+3)-2(y-1)=5(y-5)\]
\[\Rightarrow 3y+9-2y+2=5y-25\](Expanding the bracket)
\[\Rightarrow y+11=5y-25\]
\[\Rightarrow y-5y=-25-11\] (Transposing 5y on LHS and 11 on RHS)
\[\Rightarrow 4y=-36\Rightarrow \frac{-4y}{-4}=\frac{-36}{-4}=9\]
Therefore, y = 9 is the solution of given equation.
more... 
Application of Linear Equation
When you are solving the word problem you should follow the following steps:
Step 1: Read the problem carefully and specify the given and required parameters.
Step 2: Represent the unknown quantity by variables like x, y w....etc.
Step 3: Convert the mathematical statements into mathematical problem.
Step 4: Use the conditions to form an equation.
Step 5: Solve the equation for the unknown and check whether the solution satisfies the equating or not.
The sum of three consecutive multiples of 8 is 888. Which one of following options is the group of those numbers?
(a) 504, 342 and 342
(b) 234, 567 and 604
(c) 234, 564 and 905
(d) 288, 296 and 304
(e) None of these
Answer: (d)
Explanation
Let the first multiple of 8 be 8x then the next two multiples of 8 will be \[8(x+1)\And 8(x+2)\]
It is given that the sum of these three consecutive multiples is 888.
\[\therefore 8x+8(x+1)+8(x+2)\text{ }=888\]
\[\Rightarrow 8x+8x+8+8x+16\]\[=888\text{ }\Rightarrow 24x+24=888\]
\[\Rightarrow 24x=888-24\Rightarrow 24x=864\Rightarrow x=\frac{864}{24}=36\]
Therefore, three consecutive multiples of 8 are, \[8\times 36,8\times 37\And 8\times 38.\]i.e., 288, 296 and 304
The denominator of a rational number is greater than its numerator by 6. If enumerator is increased by 5 and the denominator is decreased by 3 then the number obtained is \[\frac{5}{4},\] find the rational number.
(a) \[\frac{5}{11}\]
(b) \[\frac{11}{5}\]
(c) \[\frac{12}{3}\]
(d) \[\frac{9}{8}\]
(e) None of these
Answer: (a)
Explanation
Let the numerator of the rational number be x.
Then the denominator of the rational number will be \[x+6\]
It is given that the numerator and denominator of the number are increased and decreased by 5 and 3 respectively then the number obtained is \[\frac{5}{4}\]
\[\therefore \] Numerator of the new rational number\[~=x\text{+}5\]
Denominator of the new rational number \[=(x+6)-3=x+3\]
\[\therefore \] New rational number \[=\frac{x+5}{x+3}\]
But the new rational number is given as \[\frac{5}{4}\]
\[~\therefore \frac{x+5}{x+3}=\frac{5}{4}\Rightarrow 4(x+5)=5(x+3)\](By cross multiplication)
\[\Rightarrow 4x+20=5x+15\] \[4x-5x=15-20\][transposing 5x to L.H.S. and 20 to R.H.S.]
\[\Rightarrow -x=-5\Rightarrow \] or \[x=5\]
\[\therefore \] Numerator of the rational number \[=5\]
Denominator of the rational number \[=5+6=11\]
\[\therefore \] The required rational number \[=\frac{5}{11}\]
A steamer goes downstream from one port to another in 6 hours. It covers the same distance up stream in 7 hours. If the speed of the stream is 2 km/hours then find the speed of the steamer in still water.
(a) 20km/h
(b) 30 Km/h
(c) 26 Km/h
(d) 48Km/h
(e) None of these
Answer: (c)
Explanation
Let the speed of the steamer in still water be \[x\text{ }Km/h\]
It is given that while going down stream the steamer takes 6 hours to cover the distance between two ports.
\[\therefore \] more... 
Introduction
In this chapter we will study about the comparison of two or more quantities. When we compare only two quantities of same kind, it is called ratio and more than two quantities is called proportion.
Ratio
A ratio is a relation between two quantities of same kind. Comparison is made between the two quantities by considering what part of one quantity is that of the other quantity. The two quantities are called the terms of ratio. If \[x\]and y are two quantities of same kind then the ratio of \[x\] to \[y\]is \[x/y\]or \[x:y.\]It is represented by \[x:y.\]
Important Points Related to Ratio
Jennifer mixes 600 ml of orange juice with 2L of apple juice to make a fruit drink. Find the ratio of orange juice to apple juice in its simplest from.
(a) 1:3
(b) 300:1
(b) 3:10
(d) 3:2
(e) None of these
Answer: (c)
Explanation
\[600:2000=\frac{600}{2000}=\times \frac{6\times 100}{20\times 100}=\frac{6}{20}=\frac{3\times 2}{2\times 10}=\frac{3}{10}=3:10\]
If \[4x+3y:6x+5y=\frac{11}{17}\]then find\[~x:y.\]
(a) 0:1
(b) 2:1
(b) 1:0
(d) 5:0
(e) None of these
Answer: (b)
Explanation
\[\frac{4x+3y}{6x+5y}=\frac{11}{17}\Rightarrow 17(4x+3y)=11(6x+5y)\]
\[\Rightarrow 68x+51y=66x+55y\Rightarrow 68x-66x\text{ }=55y-51y\]
\[\Rightarrow 2x=4y\Rightarrow \frac{x}{y}=\frac{4}{2}\Rightarrow x:y=2:1\]
If A, B, C, D are quantities of same kind and the ratio of A to B is 3 : 4, B to C is 5 : 7 and C to D is 8 : 9 then the ratio of
1. A to C is 15 : 28
2. B to D is 40 : 63
3. A to D is 10 : 21
Which one of the following options is correct?
(a) 1, 2 and 3
(b) 1 and 2
(b) 2 and 3
(d) 1 and 5
(e) None of These
Answer: (a)
Explanation
\[\frac{A}{B}=\frac{3}{4},\frac{B}{C}=\frac{5}{7}\And \frac{C}{D}=\frac{8}{9}\Rightarrow A=\frac{3}{4}B=\frac{5}{7}C\Rightarrow A=\frac{3}{4}\times \frac{5}{7}C\] Or \[\frac{A}{C}=\frac{15}{28}\] \[\Rightarrow C=\frac{8}{9}D\Rightarrow B=\frac{5}{7}\times \frac{8}{9}D\Rightarrow B\frac{40}{63}D\]Or \[\frac{B}{D}=\frac{40}{63}\]
\[\Rightarrow A=\frac{3}{4}B=\frac{3}{4}\times \frac{40}{63}D\frac{10}{21}D\Rightarrow \frac{A}{D}=\frac{10}{21}\]
Two numbers are in the ratio of 2 : 3. If sum of their squares is 468 then Find the numbers.
(a) (12, 16)
(b) (12, 18)
(c) (14, 20)
(d) (12, 22)
(e) None of these
Answer: (b)
Explanation
Let one number be \[2x\] and other number will be 3x then more... 
Proportion
It is the equality of two ratios i.e. if a: b = c: d, then ad = cd that implies product of extremes = product of means. Four quantities p, q, r, s are in proportion if ps = qr
Important Points Related to Proportion
If \[\frac{a}{b}=\frac{c}{d}\]then
The Ratio between two quantities is 7: 9. If the first quantity is 511 then find the other quantity.
(a) 655
(b) 555
(c) 65
(d) 656
(e) None of these
Answer: (c)
Explanation
Let the other quantity be x then
\[7:9=511:x\Rightarrow x=\frac{511\times 9}{7}=657\]
Find two numbers so that their mean proportional is 14 and third proportional is 112.
(a) 6 and 27
(b) 7 and 28
(b) 9 and 29
(d) 10 and 30
(e) None of these
Answer: (b)
Explanation
Let the number be \[x\]and \[y,\]then according to question
\[\sqrt{xy}=14\Rightarrow xy=196.........(i)\]
\[\Rightarrow \frac{x}{y}=\frac{y}{112}\Rightarrow {{y}^{2}}=112x.....(ii)\]
From (i) \[y\frac{196}{x}\Rightarrow \frac{{{(196)}^{2}}}{{{x}^{2}}}=112x\Rightarrow {{x}^{3}}=343\Rightarrow x=7\]and \[y=\frac{196}{7}=28\]
Hence, the required numbers are 7 and 28
The ration of the volume of three buckets is 3 : 4 : 5. Buckets contains the mixture of water and alcohol. If the mixture contains water and alcohol in the ratio 1 : 4, 1 : 3 and 2 : 5 respectively then find the ratio of water and alcohol when the mixture in all containers are poured in fourth container.
(a) 35:57
(b) 53:157
(c) 157:53
(d) 35:157
(e) None of these
Answer: (b)
Find the ratio among the time taken by three buses to travel the same distances if the ratios of their speed are 5:4:6.
(a) 10 :12 :15
(b) 10 :15 : 12
(b) 15:12:10
(d) 12:15:10
(e) None of these
Answer: (d)
Rational Number
Important Point Related to Rational Numbers
Types of Rational Number
Rational numbers are of two types.
Properties of Rational Number
Equivalent Rational Numbers
If we multiply or divide the numerator or denominator of a rational by the same none zero integers then we get equivalent rational number.
Find the equivalent rational numbers of \[\frac{p}{q}\].
Solution:
\[\frac{2\times P}{2\times q},\frac{3\times p}{3\times q},\frac{4\times p}{4\times q},\frac{0.235\times p}{0.235\times q}.\]
We can write infinite equivalent rational numbers of a rational number.
Lowest Form
Divide the numerator and denominator by the HCF of (Numerator, Denominator) by ignoring the sign of it, so that we get the new numerator and denominator which are co-prime.
Find the lowest form of \[\frac{-60}{96}.\]
Solution:
The HCF of 60 and 96 is 12. Therefore, divide the numerator and more... 
Operation on Rational Numbers
In this topic we will study about addition, subtraction, multiplication and division of rational numbers.
Addition of Rational Numbers
Step 1: Write the rational number in the standard form.
Step 2: Make the denominator same by taking the LCM of denominators.
Step 3: Write all the rational number with the same denominator.
Step 4: Add the numerators
Step 5: Write the numerator getting after addition on the denominator.
Step 6: Reduce the rational number to lowest term.
Add \[2\frac{3}{5},\frac{-15}{13},\frac{-13}{-15},-4\frac{3}{5}\]
Solution:
Step 1: The standard form of given rational numbers are
\[\frac{13}{5},\frac{-15}{13},\frac{13}{15},\frac{-23}{5}\]
Step 2: LCM of 5, 13, 15, 5 is 195
Step3: \[\frac{13}{5}=\frac{13\times 39}{5\times 39}=\frac{507}{195}\]
\[\frac{-15}{13}=\frac{-15\times 15}{13\times 15}=\frac{-225}{195}\]
\[\frac{13}{15}=\frac{13\times 13}{15\times 13}=\frac{169}{195}\]
\[\frac{-23}{5}=\frac{23\times 39}{5\times 39}=\frac{-897}{195}\]
Step 4: The sum of numerators
\[=507+\left( -225 \right)+169+\left( -897 \right)\]\[=-446\]
Step 5: \[\frac{-446}{195}\]
Step 6: \[-2\frac{56}{195}\]
Therefore
\[2\frac{3}{5}+\frac{-15}{13}+\frac{-13}{-15}+\left( -4\frac{3}{5} \right)=-2\frac{56}{195}\]
Subtraction of Two Rational Numbers
Step 1: Write the rational numbers in standard form.
Step 2: Make the denominator same by taking LCM.
Step 3: Subtract numerators.
Step 4: Write the result with denominator and reduce it to lowest terms.
If the sum of two rational numbers is \[\frac{-5}{6}\] and if one of the number is \[-\frac{9}{20}\] then find the other rational number.
Solution:
Sum of Rational number \[=\frac{-5}{16}\]
One number\[=\frac{-9}{20}\], Then the other \[=\frac{-5}{16}-\left( \frac{-9}{20} \right)\]
\[=\frac{-25-\left( -36 \right)}{80}=\frac{-25+36}{80}=\frac{11}{80}\]
Multiplication of Rational Numbers
Product of rational numbers \[=\frac{Product\text{ }of\,the\,numerators}{Product\text{ }of\text{ }the\text{ }denominators}\] and reduce it to the lowest terms.
Reciprocal of Rational Number
For non - zero rational numbers, a rational number is said to be reciprocal of other if the product is 1.
Find the product of \[\frac{13}{6}\times \frac{-18}{91}\times \frac{-5}{9}\times \frac{72}{-125}\]
Solution:
\[\frac{13}{6}\times \frac{-81}{91}\times \frac{-5}{9}\times \frac{72}{-125}=\frac{13\times \left( -18 \right)\times \left( -5 \right)\times 72}{6\times 91\times 9\left( -125 \right)}\]
\[=\frac{-3\times 8}{7\times 25}=\frac{-24}{175}\]
Find the reciprocal of \[\frac{4}{9}\times \frac{-7}{13}\times \frac{-3}{8}\]
Solution:
\[=\frac{4}{9}\times \frac{-7}{13}\times \frac{-3}{8}=\frac{{{\bcancel{4}}^{1}}}{{{\bcancel{9}}_{3}}}\times \frac{-7}{13}\times \frac{-{{\bcancel{3}}^{1}}}{{{\bcancel{8}}_{2}}}=\frac{7}{78}\]
The reciprocal of \[\left( \frac{7}{78} \right)\] is \[\left( \frac{78}{7} \right)\] because \[\frac{7}{78}\times \frac{78}{7}=1\]
Division of Rational Numbers
If a and b are two rational numbers in such a way that \[b\ne 0,\] then \[a\div b\] is same as the product of a and reciprocal of b.
more... Introduction
We have already studied about the integers, the fractions, the decimals and their operations. We know that for a given integer a and b, their addition, subtraction and multiplication are always an integer but the result of division of an integer by a non-negative integer may or may not be an integer. In this chapter we will study about a new number system which is formed by the division of two integers.
A number which is in the form of \[\frac{a}{b}\] where and b are integers and \[b\ne 0\] is called a rational number, for example \[\frac{-1}{2},\frac{3}{-14},\frac{15}{7}\] are rational numbers. 
You need to login to perform this action.
You will be redirected in
3 sec
