A rational number is defined as a number that can be expressed in the form where p and q are integers and
A. q = 0
B. q = 1
C. q ≠ 1
D. q ≠ 0
Division by 0 is not defined i.e. you cannot divide any number by 0.
Addition of 0 is defined. p + 0 = p;
Subtraction of 0 is defined p-0 = p
Multiplication by 0 is defined p×0 = 0.
Which of the following rational numbers is positive?
A.
B.
C.
D.
A rational number is positive if
1.both numerator and denominator are greater than 0, i.e. positive.
2.both numerator and denominator are less than 0 , i.e. negative.
In option (C) both numerator and denominator are negative.
Which of the following rational numbers is negative?
A.
B.
C.
D.
Option(A) :
Both numerator and denominator is positive,
∴ number is positive.
Option(B): Both numerator and denominator is negative,
∴ number is positive.
Option(C): Both numerator and denominator is positive,
∴ number is positive.
Option(D): Numerator is positive but denominator is negative
∴ number is negative
In the standard form of a rational number, the common factor ofnumerator and denominator is always:
A. 0
B. 1
C. – 2
D. 2
Standard form is the simplest form.
If there is any common factor other than 1, we will divide numerator and denominator both by that factor to convert it into standard form .
Which of the following rational numbers is equal to its reciprocal?
A. 1
B. 2
C.
D. 0
1 =
To find reciprocal of fraction,
Interchange its numerator and denominator .
Recipracal of 1 = = 1
2 =
∴ Reciprocal of 2 =
Reciprocal of = 2
Reciprocal of 0 = not defined
The reciproal ofis
A. 3
B. 2
C. – 1
D. 0
To find reciprocal of fraction,
Interchange its numerator and denominator .
∴ Reciprocal of = 2
The standard form ofis
A.
B.
C.
D.
Common factor of 4 and 5 is only 1.
Which of the following is equivalent to
A.
B.
C.
D.
Common factor of 4 and 5 is only 1
∴ is in standard form
Common factor of 5 and 4 is only 1
∴ is in standard form
Common factor of 16 and 25 is only 1
∴ is in standard form
Hence, is equivalent to
How many rational numbers are there between two rational numbers?
A. 1
B. 0
C. unlimited
D. 100
We can find any number of rational numbers between two rational numbers by making their denominator and increasing the numerator of the smaller number.
For example, we have to find rational numbers between = and
So, the rational numbers between and will be:
Similarly, we can find any number of rational numbers between
And
In the standard form of a rational number, the denominator is always a
A. 0
B. negative integer
C. positive integer
D. 1
Standard form is such that 1 is the only common factor of numerator and denominator.
And denominator should be positive.
To reduce a rational number to its standard form, we divide its numerator and denominator by their
A. LCM
B. HCF
C. product
D. multiple
Standard form is such that 1 is the only common factor of numerator and denominator.
Since, HCF is the greatest number that is a factor of both numerator and denominator , if we will divide both by HCF, there will be no common factor except 1.
Which is greater number in the following:
A.
B. 0
C.
D. -2
Positive number>zero>negative number.
Fill in the blanks to make the statements true.
is a ______ rational number.
negative
A rational number is negative if either of numerator and denominator is negative but not both.
Fill in the blanks to make the statements true.
1 is a ______ rational number.
Positive rational number.
Example, -1 is a negative rational number.
Fill in the blanks to make the statements true.
The standard form ofis ______.
HCF of 8 and 36 is 4.
Now, dividing numerator and denominator by 4
Fill in the blanks to make the statements true.
The standard form ofis ______.
To convert into standard form ,
Divide numerator and denominator by HCF and make denominator positive.
HCF of 18, 24 = 6
Dividing numerator and denominator by 6
Multiplying -1 to both numerator and denominator (to make denominator positive)
Fill in the blanks to make the statements true.
On a number line, is to the ______ of zero (0).
left
Numerator is negative and denominator is positive.
∴ is a negative rational number.
On a number line,
Negative numbers are to the left side of zero(0).
Fill in the blanks to make the statements true.
On a number line, is to the ______ of zero (0).
Right
Both numerator and denominator are positive.
∴ is a positive number.
On a number line,
Positive numbers are to the right side of zero(0).
Fill in the blanks to make the statements true.
is ______ than
less
is a negative rational number.
is a positive rational number.
Negative number<zero(0)<positive number.
Fill in the blanks to make the statements true.
is ______ than 0.
less
is a negative number.
Negative number<zero(0)<positive number.
Fill in the blanks to make the statements true.
andrepresent ______ rational numbers.
Negative
If either numerator or denominator is negative, but not both,
The rational number is negative.
Fill in the blanks to make the statements true.
andrepresent ______ rational numbers.
equivalent
Two rational numbers are said to be equivalent if they are equal.
HCF of 3 and 5 is 1.
∴ is in standard form.
HCF of 27 and 45 is 9.
Dividing numerator and denominator of by 9
∴
Hence, both are equivalent rational numbers.
Fill in the blanks to make the statements true.
Additive inverse ofis ______.
Additive inverse of a number is the number adding which, the result is zero.
To find additive inverse,
Reverse the sign of the rational number.
Fill in the blanks to make the statements true.
Fill in the blanks to make the statements true.
LCM of 5 and 6 is 30.
Multiplying numerator and denominator of by 6.
We get,
Multiplying numerator and denominator of by 5.
We get,
∴
LCM of 35 and 30 = 5
Dividing numerator and denominator by 5
Fill in the blanks to make the statements true.
HCF of 6, 12 = 6
Dividing numerator and denominator by 6
Fill in the blanks to make the statements true.
HCF of 15, 15 = 15
Dividing numerator and denominator by 15
Fill in the blanks to make the statements true.
-36
To make denominator 42,
We have to multiply numerator and denominator by 42÷7 = 6
Multiplying numerator and denominator by 6
∴
Fill in the blanks to make the statements true.
12
To make numerator 6,
We have to multiply numerator and denominator by 6÷1 = 6
Multiplying numerator and denominator by 6
∴
Fill in the blanks to make the statements true.
-1
LCM of 9, 9 = 9
Dividing Numerator and denominator by 9
Fill in the blanks to make the statements true.
Fill in the boxes with the correct symbol >, < or = .
<
In
Numerator is positive, Denominator is negative
∴ It is negative rational number.
In
Both numerator and denominator are positive
∴ it is positive rational number .
Negative number < Positive Number
Fill in the blanks to make the statements true.
Fill in the boxes with the correct symbol >, < or = .
>
In
Numerator is negative, Denominator is positive
∴ It is negative rational number.
In
Both numerator and denominator are positive
∴ it is positive rational number .
Positive number < Negative Number
Fill in the blanks to make the statements true.
Fill in the boxes with the correct symbol >, < or = .
<
LCM of 6 and 4 = 12
To make denominator 12,
Multiplying numerator and denominator of by 2
Multiplying numerator and denominator of by 3
10<24
∴
∴
Fill in the blanks to make the statements true.
Fill in the boxes with the correct symbol >, < or = .
<
To compare 2 negative numbers ,
First compare them without negative sign
And then reverse the sign of greater than/less than
Comparing and
9>4
∴
Reversing the sign
.
Fill in the blanks to make the statements true.
Fill in the boxes with the correct symbol >, < or = .
=
LCM of 8, 2 = 8
To make denominator of , 8
Multiply numerator and denominator by 8÷2 = 4
8 = 8
∴
∴
Fill in the blanks to make the statements true. The reciprocal of ______ does not exist.
Zero
For Reciprocal, numerator should be made denominator and denominator should be made numerator.
Division by zero is not defined
∴ zero cannot be in denominator
Hence, reciprocal of zero does not exist.
Fill in the blanks to make the statements true. The reciprocal of 1 is ______.
1
1 =
Interchanging numerator by denominator ,
Reciprocal = 1.
Fill in the blanks to make the statements true.
For division by a rational number,
Multiply by its reciprocal.
Interchanging numerator and denominator
Reciprocal of
∴
Multiplying numerator and denominator by -1
Fill in the blanks to make the statements true.
0
Division of zero by any number is zero.
Fill in the blanks to make the statements true.
0
Multiplying any number by 0 gives 0.
Fill in the blanks to make the statements true. ______ ×
Multiplying a rational number by its reciprocal gives 1
Numerator of -2
Denominator of 5
Interchanging numerator and denominator
Reciprocal of
Fill in the blanks to make the statements true. The standard form of rational number –1 is ______.
In standard form
HCF of numerator and Denominator should be 1
And
Denominator should be positive.
Fill in the blanks to make the statements true. If m is a common divisor of a and b, then
b÷m
A fraction is not changed if both numerator and denominator is
1. Multiplied by same non zero number.
2. Divided by same non zero number.
Fill in the blanks to make the statements true. If p and q are positive integers, thenis a ______ rational number andis a ______ rational number.
positive, negative
If both numerator and denominator are positive,
Rational number is positive.
If either of numerator or denominator is negative but not both,
Rational number is negative.
Fill in the blanks to make the statements true. Two rational numbers are said to be equivalent or equal, if they havethe same ______ form.
Standard
If two numbers are equal, they will be reduced to same fraction after dividing numerator and denominator by their HCF.
Fill in the blanks to make the statements true.
Ifis a rational number, then q cannot be ______.
Zero
Division by zero is not defined.
State whether the statements are True or False.
Every natural number is a rational number but every rational number need not be a natural number.
True
Every natural number N can be represented int the form of as
State whether the statements are True or False.
Zero is a rational number.
True
Zero can be represented in the form as or etc.
State whether the statements are True or False.
Every integer is a rational number but every rational number neednot be an integer.
True
Every Integer I can be represented in the form as
∴ Every integer is a rational number
Every rational number cannot be an integer, as it can have irremovable fractional part . Example :
State whether the statements are True or False.
Every negative integer is not a negative rational number.
False
Every Integer I can be represented in the form as
∴ Every integer is a rational number
While representing the integer in the form
Its value will not be changed .
i.e. if integer is less than 0
corresponding rational number will also be less than 0
∴ every negative integer is a negative rational number.
State whether the statements are True or False.
Ifis a rational number and m is a non-zero integer, then
True
A fraction is not changed if both numerator and denominator is
1. Multiplied by same non zero number.
2. Divided by same non zero number.
State whether the statements are True or False.
Ifis a rational number and m is a non-zero common divisor of p and q, then
True
A fraction is not changed if both numerator and denominator is
1. Multiplied by same non zero number.
2. Divided by same non zero number.
State whether the statements are True or False.
In a rational number, denominator always has to be a non-zero integer.
True
It is true that, in a rational number of the form, , where q ≠ 0.
State whether the statements are True or False.
Ifis a rational number and m is a non-zero integer, thenis a rational number not equivalent to
False
A rational number is not changed if both numerator and denominator is
1. Multiplied by same non zero number.
2. Divided by same non zero number.
State whether the statements are True or False.
Sum of two rational numbers is always a rational number.
True
To add two rational numbers,
We will make their denominators same.
Then, we will add numerator
So, the result is again in the form
Hence, Sum of two rational numbers is always a rational number
State whether the statements are True or False.
All decimal numbers are also rational numbers.
True
All decimal numbers are also rational numbers because every decimal number can be expressed in the form of , where p and q are integers and q ≠ 0
For example,
1. .88 is a decimal number, and it can be expressed as .
Converting it into its standard form,
⇒ =
⇒ , which is a rational number
Since 22 and 25 both are integers and q ≠ 0
State whether the statements are True or False.
The quotient of two rationals is always a rational number.
False
Let’s take 1 and 0 as two numbers,
Both of these are rational numbers but their division is not defined.
⇒ = not defined
Also, is not a rational number as q = 0.
Hence, The quotient of two rationals is not always a rational number.
State whether the statements are True or False.
Every fraction is a rational number.
True
Every fraction can be expressed in the form , where p and q are integers and q ≠ 0
For, example:
one – fourth part
Since both 1 and 4 are integers and 4 ≠ 0
Hence, is a rational number.
Note: All integers and fractions are rational numbers.
State whether the statements are True or False.
Two rationals with different numerators can never be equal.
False
Let’s take (first rational) and (second rational)
As given both the rationals have different numerators.
Since, 15 and 18 have 3 in common hence,
⇒ =
⇒ (second rational) = (first rational )
Hence, two rational with different numerators can be equal.
State whether the statements are True or False.
8 can be written as a rational number with any integer as the denominator.
True,
8 can be written as , where ‘m’ is any integer.
For example:
State whether the statements are True or False.
is equivalent to
True
First rational number =
4 = 2 × 2
6 = 2 × 3
H.C.F. = 2
So, to convert into its standard form, divide the numerator and denominator by their H.C.F.
⇒ =
⇒
Now, consider the second rational number,
=
= First rational number
Hence, is equivalent to .
State whether the statements are True or False.
The rational numberlies to the right of zero on the number line.
False
is a negative rational number, and all negative numbers lie left of zero on the number line.
State whether the statements are True or False.
The rational numbers and are on the opposite sides of zero on the number line.
True
The rational number have ‘-‘ common in both numerator and denominator so,
⇒ =
⇒ A positive rational number.
Hence, it lies right of zero on the number line
is a negative rational number.
So, it lies left of zero on the number line.
⇒ and lies on the opposite side of zero on the number line.
State whether the statements are True or False.
Every rational number is a whole number.
False
is a rational number but it is not a whole number because,
The whole number is natural number including 0.
0,1,2,3,…….
State whether the statements are True or False.
Zero is the smallest rational number.
False
A rational number can be negative for example,
-1,-8,……
Which are less than zero.
Match the following:
(i) (c)
⇒
⇒
= 1
Note: To divide one rational number by other non zero rational number, We multiply the first rational number by reciprocal of the other.
(ii) (e)
⇒
⇒
Note: To divide one rational number by other non-zero rational number, We multiply the first rational number by reciprocal of the other.
(iii) (a)
⇒
⇒
Reciprocal of (-1) is itself (-1).
(iv) (b)
⇒
= -1
Note: To divide one rational number by other non-zero rational number, We multiply the first rational number by reciprocal of the other.
(v) (d)
⇒
⇒
Write each of the following rational numbers with positive denominators:
Multiplying by -1 in both numerator and denominator we get,
⇒
⇒
Now, for
Multiplying by -1 in both numerator and denominator we get,
⇒
⇒
Now, for
Multiplying by -1 in both numerator and denominator we get,
⇒
⇒
Expressas a rational number with denominator:
(i) 36 (ii) – 80
(i) as a rational number with denominator = 36
Multiplying by 9 in both numerator and denominator we get,
⇒
⇒
(ii) as a rational number with denominator = -80
Multiplying by -20 in both numerator and denominator we get,
⇒
⇒
Reduce each of the following rational numbers in its lowest form:
(i)
(ii)
(i) Considering the rational number =
-60 = 2 × 2 × 3 × 5 × -1
72 = 2 × 2 × 2 × 3 × 3
H.C.F. = 12
To convert into it’s lowest form,
Divide the numerator and denominator by H.C.F.
⇒
⇒
(ii) considering the rational number =
91 = 7 × 13
-364 = 2 × 2 × 7 × 13 × -1
H.C.F. = 91
To convert into it’s lowest form,
Divide the numerator and denominator by H.C.F.
⇒
⇒
=
Express each of the following rational numbers in its standard form:
A.
B.
C.
D.
A. Considering the rational number =
-12 = 2 × 2 × 3 × -1
72 = 2 × 2 × 2 × 3 × 3
H.C.F. = 12
To convert into it’s standard form,
Divide the numerator and denominator by H.C.F.
⇒
⇒
B. Considering the rational number =
14 = 2 × 7
-49 = 7 × 7 × -1
H.C.F. = 7
To convert into it’s standard form,
Divide the numerator and denominator by H.C.F.
⇒
⇒
C. Considering the rational number =
-15 = 3 × 5 × -1
35 = 5 × 7
H.C.F. = 5
To convert into it’s standard form,
Divide the numerator and denominator by H.C.F.
⇒
⇒
D. Considering the rational number =
299 = 13 × 23
161 = 7 × 23
H.C.F. = 23
To convert into standard form,
Divide the numerator and denominator by H.C.F.
⇒
⇒
Are the rational numbersandequivalent? Give reason.
Yes
Considering the rational number
-8 = 2 × 2 × 2 × -1
28 = 2 × 2 × 7
H.C.F. = 4
To convert into standard form,
Divide the numerator and denominator by H.C.F.
⇒
⇒
Now, considering the rational number =
32 = 2 × 2 × 2 × 2 × 2
-112 = 2 × 2 × 2 × 2 × 7 × -1
H.C.F. = 16
To convert into standard form,
Divide the numerator and denominator by H.C.F.
⇒
⇒
=
Since, and have same standard form.
Hence, they are equivalent.
Arrange the rational numbersin ascending order.
For comparing the rational numbers we should make the denominator equal of all rationals.
We have to compare so,
L.C.M. of 10, 8 , 3 , 4 , 5
= 120
⇒
⇒
As we know,
-84 < -80 < -75 < -72 < -30
So, order of ascending will be:
⇒
Represent the following rational numbers on a number line:
Since,
, we will divide the 0-1 in 8 parts, and will mark
Since,
, we will divide the (-3) - (-2) into three parts.
Since,
, we will divide (-4) – (-3) into six parts.
If find the value of x.
According to the question,
Using cross multiplication we get,
⇒ -5 × 28 = 7 × x
⇒ 7 × x = -5 × 28
⇒ x =
⇒ x = -5 × 4
⇒ x = -20
Hence, the value of x = -20.
Give three rational numbers equivalent to:
i.
ii.
i. NOTE: If the numerator and denominator of a rational number is multiplied or divided by a non – zero integer, we get a rational number which is said to be equivalent of given number.
First number equivalent to
⇒
⇒
Second number equivalent to
⇒
⇒
Third number equivalent to
⇒
⇒
ii. NOTE: If the numerator and denominator of a rational number is multiplied or divided by a non – zero integer, we get a rational number which is said to be equivalent of given number.
First number equivalent to
⇒
⇒
Second number equivalent to
⇒
⇒
Third number equivalent to
⇒
⇒
Write the next three rational numbers to complete the pattern:
i.
ii.
(i) As we can see the pattern is,
So, the next three rational numbers are:
(ii) As we can see the pattern is,
So, the next three rational numbers are:
List four rational numbers betweenand
A rational number between and
⇒
⇒
⇒
Second rational number between and
⇒
⇒
⇒
Third rational number between and
⇒
⇒
⇒
Fourth rational number between and
⇒
⇒ ()
⇒
Find the sum of
i.
ii.
i.
⇒
⇒
⇒
ii.
⇒
⇒
⇒ = 1.
Solve:
i.
ii.
i.
⇒
⇒
⇒
ii.
⇒
⇒
⇒ =
Find the product of:
i.
ii.
i.
Product of two rational numbers =
⇒
⇒
⇒
ii.
Product of two rational numbers =
⇒
⇒
⇒
Simplify:
i.
ii.
i. Product of two rational numbers =
⇒
⇒
⇒
⇒
⇒ =
⇒ 13
ii. Product of two rational numbers =
⇒
⇒
⇒ =
Simplify:
i.
ii.
i. To divide one rational by other non – zero rational number, We multiply the first rational number by the reciprocal of the other.
⇒
⇒
⇒
ii. To divide one rational by other non – zero rational number, We multiply the first rational number by the reciprocal of the other.
⇒
⇒ -2
Which is greater in the following?
i.
ii.
(i) L.C.M. of 4 and 8 = 8
⇒ ,
⇒
Hence, is greater than .
(ii) Simplifying the above fractions we get,
=
L.C.M. of 7 and 9 = 63
⇒
⇒
Since, first number is ‘negative’ so,
⇒ is greater than .
Write a rational number in which the numerator is less than‘–7 × 11’ and the denominator is greater than ‘12 + 4’.
Numerator is less than ‘-7 × 11’
⇒ Numerator is less than ‘-77’
⇒ Numerator = -78, -79,………
Denominator is greater than ’12 + 4’
⇒ Denominator is greater than ‘16’
⇒ Denominator = 17, 18,……
So, Rational number =
If x =and y =then
evaluate x + y, x – y, x × y and x ÷ y.
x + y = ()
⇒
⇒
⇒
x – y = ()
⇒
⇒
⇒ =
x × y = ()
⇒
x ÷ y = ()
⇒
⇒ =
Find the reciprocal of the following:
i.
ii.
iii.
iv.
i. () + ()
⇒ + 3
⇒
ii. ()
⇒
iii.
⇒
⇒
iv. () – ()
⇒ - ()
⇒ -4 +
⇒
⇒
Complete the following table by finding the sums:
I.
⇒
⇒
II.
⇒
⇒
III.
⇒
⇒
IV.
⇒
⇒
⇒
⇒
V.
⇒
⇒
VI.
⇒
⇒
VII.
⇒
⇒
VIII.
⇒
⇒
Write each of the following numbers in the formwhere p and q are integers:
A. six-eighths
B. three and half
C. opposite of 1
D. one-fourth
E. zero
F. opposite of three-fifths
A.
B. =
C.
D.
E.
F.
If p = m × t and q = n × t, then
P = m × t
q = n × t
⇒
⇒ =
Given thatandare two rational numbers with different denominators and both of them are in standard form. To compare these rational numbers we say that:
(a) if p × s < r × q
(b) if _______ = ______
(c) if p × s > r × q
(a) If p × s < r × q
By cross multiplication,
Since, and are two rational numbers
⇒
(b) Using cross multiplication we get,
⇒ p × s = q × r
(c) If p × s > r × q
Since, and are two rational numbers
⇒
In each of the following cases, write the rational number whose numerator and denominator are respectively as under:
(a) 5 – 39 and 54 – 6
(b) (–4) × 6 and 8 ÷ 2
(c) 35 ÷ (–7) and 35 –18
(d) 25 + 15 and 81 ÷ 40
(a) Numerator = 5 – 39
⇒ Numerator = -34
Denominator = 54 – 6
⇒ Denominator = 48
⇒ Rational number =
(b) Numerator = -4 × 6
Numerator = -24
Denominator = 8 ÷ 2
Denominator = 4
⇒ Rational number =
=
(c) Numerator =
⇒ Numerator = -5
Denominator = 35 – 18
⇒ Denominator = 17
⇒ Rational number =
(d) Numerator = 25 + 15
⇒ Numerator = 40
Denominator =
⇒ Rational number =
⇒ Rational number =
Write the following as rational numbers in their standard forms:
A. 35%
B. 1.2
C.
D. 240 ÷ (– 840)
E. 115 ÷ 207
A. 35% =
⇒
B. 1.2 =
⇒
C. =
⇒
D. 240 ÷ (-840)
⇒ =
⇒
E. 115 ÷ 207
⇒
Find a rational number exactly halfway between:
A.
B.
C.
D.
A.
⇒ [ Adding additive inverse to
= 0
B.
⇒
⇒
⇒ =
C.
⇒
⇒
⇒
⇒
D.
⇒
⇒
⇒
Taking and find :
(a) the rational number which when added to x gives y.
(b) the rational number which subtracted from y gives z.
(c) the rational number which when added to z gives us x.
(d) the rational number which when multiplied by y to get x.
(e) the reciprocal of x + y.
(f) the sum of reciprocals of x and y.
(g) (x ÷ y) × z
(h) (x – y) + z
(i) x + (y + z)
(j) x ÷ (y ÷ z)
(k) x – (y + z)
(a) Let’s the rational number is p.
⇒ x + p = y
⇒ p = y – x
⇒ p =
⇒ p =
⇒ p =
⇒ p =
(b) Let’s the rational number is p.
⇒ y – p = z
⇒ p = y – z
⇒ p =
⇒ p =
⇒ p =
(c) ⇒ z + p = x
⇒ p = x – z
⇒ p =
⇒ p =
⇒ p =
⇒ p =
⇒ p × y = x
⇒ p =
⇒ p = =
⇒ p =
(e) ⇒ The reciprocal of x + y =
⇒
⇒ =
(f) ⇒ The reciprocals of x and y are =
⇒ The sum of reciprocals of x and y are =
⇒
⇒ =
⇒ =
⇒
(g) () ×
⇒ ×
⇒
⇒
(h) (- ) +
⇒ () +
⇒
⇒
⇒
(i)
⇒ + ()
⇒ + ()
⇒
⇒
(j)
⇒ ÷ ()
⇒ ÷
⇒
⇒
(k)
⇒ - ()
⇒
⇒ =
⇒
What should be added to to obtain the nearest natural number?
Nearest natural number to is 1.
⇒
⇒ p = 1 +
⇒ p =
What should be subtracted fromto obtain the nearest integer?
Nearest integer to is = -1.
⇒
⇒ p =
⇒ p =
What should be multiplied withto obtain the nearest integer?
Nearest integer to is = -1.
⇒
⇒ Using cross multiplication
⇒ p = =
What should be divided byto obtain the greatest negative integer?
Greatest negative integer = -1.
⇒ p ÷ = -1
⇒ p × 2 = -1
⇒ p =
From a rope 68 m long, pieces of equal size are cut. If length of onepiece isfind the number of such pieces.
Length of one piece = = m
Let’s number of such pieces = p
⇒ p × = 68 m
Using cross multiplication
⇒ p =
⇒ p = 16
Hence, the number of pieces = 16.
If 12 shirts of equal size can be prepared from 27m cloth, what is length of cloth required for each shirt?
Let’s length of cloth required for each shirt = p m
⇒ 12 × p = 27
⇒ p =
⇒ p = m
Insert 3 equivalent rational numbers between
i.
ii. 0 and -10
i. First rational number =
⇒
⇒
Second rational number = =
Third equivalent rational number = =
ii. First rational number =
= = -5
Second rational number =
Third rational number =
Put the (√), wherever applicable
(a) -114
⇒ Integer, Rational number
(b)
⇒ Fraction, Rational number
(c)
⇒ Natural number, Whole number, Integer, Rational number
(d) =
Rational number
(e)
⇒ Fraction, Rational number
(f) 0
⇒ Whole number, Integer, Rational number
‘a’ and ‘b’ are two different numbers taken from the numbers 1 – 50. What is the largest value thatcan have? What is the largest value thatcan have?
For the largest value of .
Let’s a = 50 and b = 1;
⇒ =
⇒
For the largest value of .
Let’s a = 50 and b = 49;
⇒ =
⇒
150 students are studying English, Maths or both. 62 per cent of the students are studying English and 68 per cent are studying Maths. How many students are studying both?
Students studying English, Maths or both are = 150
Student studying English are = = 93
Now, students studying Maths are = = 102
Using the concept of sets,
Students studying both = Students studying Maths + Students Studying English – Students studying both English, Maths or both
⇒ Students studying both = 102 + 93 – 150
⇒ Students studying both = 195 – 150
⇒ Students studying both = 45
A body floatsof its volume above the surface. What is the ratio of the body submerged volume to its exposed volume? Re-write it as a rational number.
Let’s volume of the body = V
So, Exposed volume =
Now, the submerged volume of the body =
=
⇒ Ratio of submerged volume to it’s exposed volume
= = = 7:2
As a Rational number =
Find the odd one out of the following and give reason.
A.
B.
C.
D.
(D)
A. = 1 B. = 1
C. =1 but in option D. = -1.
Find the odd one out of the following and give reason.
A.
B.
C.
D.
(C)
All are standard form of
Except, which is =
Find the odd one out of the following and give reason.
A.
B.
C.
D.
(B)
All Rationals except have even numerator and odd denominator.
Find the odd one out of the following and give reason.
A.
B.
C.
D.
(A)
The Standard form of above rationals are
Since, All rationals have denominator = 5, except which have 7 as a denominator.
What’s the Error? Chhaya simplified a rational number in this mannerWhat error did the student make?
Considering
-25 = 5 × 5 × -1
-30 = 2 × 3 × 5 × -1
H.C.F. = 5
To get standard form,
Dividing by 5 in both numerator and denominator
⇒ =
ERROR: Chhaya did not divide numerator by 5 and denominator by -5.