Olympiad Mathematics - Rational Numbers
Exam Duration: 45 Mins Total Questions : 30
Which of the following is not a rational number(s)?
- (a)
\(\frac { -2 }{ 9 } \)
- (b)
\(\frac { 4 }{ -7 } \)
- (c)
\(\frac { -3 }{ -17 } \)
- (d)
\(\frac { \sqrt { 2 } }{ 3 } \)
\(\frac { \sqrt { 2 } }{ 3 } \) is not a rational number
What type of a number is\(\frac { -3 }{ 0 } \) ?
- (a)
A positive rational number
- (b)
A negative rational number.
- (c)
Either a positive or a negative rational number
- (d)
Neither a positive nor a negative rational number.
Since, denominator is 0, it is not a rational number.
Which among the following is a rational number equivalent to\(\frac { 5 }{ 3 } \) ?
- (a)
\(\frac { -20 }{ 15 } \)
- (b)
\(\frac { 25 }{ -15 } \)
- (c)
\(\frac { 25 }{ 15 } \)
- (d)
\(\frac { 15 }{ 25 } \)
What type of a numerator does \(\frac { 0 }{ 7 } \) have?
- (a)
Positive
- (b)
Negative
- (c)
Either positive or negative
- (d)
Neither positive nor negative
Which is the greatest?
- (a)
\(\frac { -5 }{ 11 } \)
- (b)
\(\frac { 5 }{ 12 } \)
- (c)
\(\frac { -5 }{ 17 } \)
- (d)
\(\frac { -5 }{ 13 } \)
Which is the correct descending order of -2,\(\frac { 4 }{ -5 } \) ,\(\frac { -11 }{ 20 } \),\(\frac { 3 }{ 4 } \)?
- (a)
\(\frac { 3 }{ 4 } >-2>\frac { -11 }{ 20 } >\frac { 4 }{ -5 } \)
- (b)
\(\frac { 3 }{ 4 } >\frac { -11 }{ 20 } >\frac { 4 }{ -5 } >-2\)
- (c)
\(\frac { 3 }{ 4 } >\frac { 4 }{ -5 } >-2>\frac { -11 }{ 20 } \)
- (d)
\(\frac { 3 }{ 4 } >\frac { 4 }{ -5 } >\frac { -11 }{ 20 } >-2\)
L.C.M. of 5,4 and 20 is 20.
\(\frac { 4 }{ -5 } \times \frac { 4 }{ 4 } =\frac { 16 }{ -20 } ;\frac { 3 }{ 4 } =\frac { 3\times 5 }{ 4\times 5 } =\frac { 15 }{ 20 } \)
\(-2<\frac { 4 }{ -5 } <\frac { -11 }{ 20 } <\frac { 3 }{ 4 } \) or
\(\frac { 3 }{ 4 } >\frac { -11 }{ 20 } >\frac { 4 }{ -5 } -2\) -2 is the required descending order.
Rewrite equivalent fractions of the given fractions and then compare then. Arrange them in descending order.
P: The quotient of two integers is always a rational number.
Q:\(\frac { 1 }{ 0 } \) is not rational.Which of the following statements is true?
- (a)
P is true and Q is the correct explanation of P.
- (b)
P is false and Q is the correct explanation of P.
- (c)
P is true and Q is false.
- (d)
Both P and Q are false.
Since, \(\frac { 1 }{ 0 } \) is not rational, the quotient of two integers is not always rational.
What is the result of \(2-\frac { 11 }{ 39 } +\frac { 5 }{ 26 } \) ?
- (a)
\(\frac { 149 }{ 39 } \)
- (b)
\(\frac { 149 }{ 78 } \)
- (c)
\(\frac { 149 }{ 76 } \)
- (d)
\(\frac { 149 }{ 98 } \)
Which is the equivalent of\(\frac { -143 }{ 21 } \) ?
- (a)
\(-6+\frac { 17 }{ 21 } \)
- (b)
\(6+\left( \frac { -17 }{ 21 } \right) \)
- (c)
(-6) + \(\left( \frac { -17 }{ 21 } \right) \)
- (d)
-6
Which of the following statements is correct?
- (a)
0 is called the additive identity for rational numbers.
- (b)
1 is called the multiplicative identity for rational numbers
- (c)
The additive inverse of 0 is zero itself
- (d)
All the above.
Consider \(\frac { 4 }{ 5 } \)
(A) Since\(\frac { 4 }{ 5 } +0=\frac { 4 }{ 5 } \) , is the additive identity of rational numbers.
(B) Since \(\frac { 4 }{ 5 } \times 1=\frac { 4 }{ 5 } \),1 is the multiplicative identity of rational numbers
(C) Since 0 + 0 = 0, 0 is the additive inverse of 0
What number should be added to\(\frac { -5 }{ 6 } \) to get\(\frac { 3 }{ 2 } \) ?
- (a)
\(\frac { -7 }{ 3 } \)
- (b)
2\(\frac { 1 }{ 3 } \)
- (c)
\(\frac { 8 }{ 3 } \)
- (d)
\(\frac { -8 }{ 3 } \)
Which of the following statements is true?
- (a)
The reciprocals of 1 and -1 are themselves.
- (b)
Zero has no reciprocal
- (c)
The product of two rational numbers is a rational number
- (d)
All the above
What is the product of a rational number and its reciprocal?
- (a)
0
- (b)
1
- (c)
-1
- (d)
2
Evaluate p + q given p = \(\left( -2\frac { 1 }{ 5 } \right) \)and q =\(\left( -1\frac { 1 }{ 3 } \right) \)
- (a)
1\(\frac { 8 }{ 15 } \)
- (b)
\(\left( -3\frac { 8 }{ 15 } \right) \)
- (c)
\(\left( -2\frac { 8 }{ 15 } \right) \)
- (d)
\(3\frac { 8 }{ 15 } \)
Given P=\(\left( -2\frac { 1 }{ 5 } \right) \) and q =\(\left( -1\frac { 1 }{ 3 } \right) \)
p+q =\(\left( -2\frac { 1 }{ 5 } \right) +\left( -1\frac { 1 }{ 3 } \right) =-\frac { 11 }{ 5 } -\frac { 4 }{ 3 } \)
= \(-\left( \frac { 53 }{ 15 } \right) =-3\frac { 8 }{ 15 } \)
Compute \(1\frac { 1 }{ 4 } +\left( \frac { -8 }{ 3 } \right) -\left( \frac { -5 }{ 9 } \right) \)
- (a)
\(\frac { -31 }{ 36 } \)
- (b)
\(\frac { 13 }{ 36 } \)
- (c)
\(\frac { 15 }{ 36 } \)
- (d)
\(\frac { -29 }{ 36 } \)
Find the value of \(-6\frac { 2 }{ 3 } \times \frac { 2 }{ 5 } \)
- (a)
\(2\frac { 2 }{ 3 } \)
- (b)
\(2\frac { 1 }{ 5 } \)
- (c)
\(-2\frac { 2 }{ 3 } \)
- (d)
\(-2\frac { 1 }{ 3 } \)
\(-6\frac { 2 }{ 3 } \times \frac { 2 }{ 5 } =-\frac { 20 }{ 3 } \times \frac { 2 }{ 5 } =\left( -2\frac { 2 }{ 3 } \right) \)
For what value of 'x' is\(\frac { 7 }{ 8 } -\left( -\frac { 11 }{ 4 } \right) +x=3\frac { 7 }{ 24 } \) ?
- (a)
\(\frac { -1 }{ 3 } \)
- (b)
\(\frac { 1 }{ 3 } \)
- (c)
\(\frac { 2 }{ 3 } \)
- (d)
-3
\(\frac { 7 }{ 8 } -\left( -\frac { 11 }{ 4 } \right) +x=3\frac { 7 }{ 24 } \)
⇒ x= \(\frac { 79 }{ 24 } -\frac { 29 }{ 8 } =\left( \frac { -8 }{ 24 } \right) =\left( \frac { -1 }{ 3 } \right) \)
What should be added to\(\frac { -7 }{ 8 } \) to get \(\frac { 4 }{ 9 } \) ?
- (a)
\(\frac { 72 }{ 95 } \)
- (b)
\(\frac { 95 }{ 72 } \)
- (c)
\(\frac { -72 }{ 95 } \)
- (d)
\(\frac { -95 }{ 72 } \)
Let the number to be added be x
Then, \(\frac { -7 }{ 8 } +x=\frac { 4 }{ 9 } \Rightarrow x=\frac { 4 }{ 9 } -\left( \frac { -7 }{ 8 } \right) \)
= \(\frac { 4 }{ 9 } +\frac { 7 }{ 8 } \) \(\left[ since\quad -\left( \frac { -7 }{ 8 } \right) =\frac { 7 }{ 8 } \right] \)
= \(\frac { (32+63) }{ 72 } =\frac { 95 }{ 72 } \)
Hence, the required number is \(\frac { 95 }{ 72 } \)
What is the reciprocal of -8?
- (a)
8
- (b)
\(\frac{1}{8}\)
- (c)
\(\frac{-1}{8}\)
- (d)
-8
Given a =1\(\frac { 5 }{ 7 } \) ,b =\(\frac { 1 }{ 4 } \) .c =\(\frac { 1 }{ 9 } \) and d=\(\left( -1\frac { 1 }{ 4 } \right) \) evaluate a(b-c) ÷ d.
- (a)
\(\frac { -4 }{ 21 } \)
- (b)
\(\frac { -6 }{ 23 } \)
- (c)
\(\frac { -5 }{ 27 } \)
- (d)
\(\frac { 4 }{ 21 } \)
Which of the following statements is true?
- (a)
1 and -1 are reciprocal of themselves.
- (b)
Zero has no reciprocal.
- (c)
The product of two rational numbers is a rational number.
- (d)
All of these
The value of x such that \(-{3\over8}and {x\over-24}\) are equivalent rational numbers is_____________.
- (a)
64
- (b)
-64
- (c)
-9
- (d)
9
\({-3\over8}={x\over-24}\Rightarrow {-3\over8}\times{3\over3}={-x\over24}\)
\(\Rightarrow {-9\over24}={-x\over24}\Rightarrow x=9\)
A rational number\({-2\over3}\)
- (a)
Lies to the left side of 0 on the number line.
- (b)
Lies to the right side of 0 on the number line
- (c)
Is not possible to represent on the number line
- (d)
None of these
\(5\over8\) is the rational number between \(3\over 4\) and\(1\over2\) . Which of the following is not a rational number between \(3\over 4\) and \(1\over2\)?
- (a)
\(9\over16\)
- (b)
\(13\over16\)
- (c)
\(10\over16\)
- (d)
\(11\over16\)
Which of the following sum is in the simplest form?
- (a)
\({4\over9}+{-5\over9}\)
- (b)
\({-2\over5}+{13\over20}\)
- (c)
\({-5\over12}+{11\over-12}\)
- (d)
\({-7\over8}+{1\over12}+{2\over3}\)
\({4\over9}+{-5\over9}={4-5\over9}={-1\over9}\)
\({-2\over5}+{13\over20}={-8+13\over20}={5\over20}\)
\({-5\over12}+{11\over-12}={-5\over12}-{11\over12}={-5-11\over12}={-16\over12}\)
\({-7\over8}+{1\over12}+{2\over3}={-12+2+16\over24}={-3\over24}\)
\(\therefore {-1\over9}\)is in the simplest form.
Simplify :\(\left( \frac { \left( -18\frac { 1 }{ 3 } \times 2\frac { 8 }{ 11 } \right) -\left( 4\frac { 5 }{ 7 } \times 2\frac { 1 }{ 3 } \right) }{ \left| \frac { 3 }{ 5 } +\left( \frac { -9 }{ 10 } \right) \right| +\left| -\left( \frac { -3 }{ 5 } \right) \right| } \right) \)
- (a)
\(63{4\over81}\)
- (b)
\(-23{7\over9}\)
- (c)
\(-67{7\over9}\)
- (d)
\(12{6\over17}\)
We have,\(\left( \frac { \left( -18\frac { 1 }{ 3 } \times 2\frac { 8 }{ 11 } \right) -\left( 4\frac { 5 }{ 7 } \times 2\frac { 1 }{ 3 } \right) }{ \left| \frac { 3 }{ 5 } +\left( \frac { -9 }{ 10 } \right) \right| +\left| -\left( \frac { -3 }{ 5 } \right) \right| } \right) \)
\(=\frac { \left( \frac { -55 }{ 3 } \times \frac { 30 }{ 11 } \right) -\left( \frac { 33 }{ 7 } \times \frac { 7 }{ 3 } \right) }{ \left| \frac { 3 }{ 5 } -\frac { 9 }{ 10 } \right| +\left| \frac { 3 }{ 5 } \right| } =\frac { -50-11 }{ \left| \frac { 6-9 }{ 10 } \right| +\frac { 3 }{ 5 } } \)
\(={-61\over{3\over10}+{3\over5}}={-61\over{3+6\over10}}\)
\(={-61\times 10\over9}={-610\over9}=-67{7\over9}\)
A farmer grows vegetables in his field. In \({2\over3}\) of the field, he grows potatoes, in \(1\over4\) of the field, he grows onions and in the rest of the field, he grows tomatoes. In what part of the field does he grow tomatoes?
- (a)
\(1\over12\)
- (b)
\(11\over12\)
- (c)
\(3\over4\)
- (d)
\(1\over6\)
Let the total area of the field be 1.
Part of the field in which the farmer grows tomatoes
\(=1-({2\over3}+{1\over4})=1-({8+3\over12})=1-{11\over12}={1\over12}\)
From his home, Rahul walks \(6\over7\) km towards school and then returns\(5\over6\) km on the same way towards his home to reach a landmark. At what distance will he be now from his home?
- (a)
\({1\over42}km\)
- (b)
\({1\over43}km\)
- (c)
\({30\over42}km\)
- (d)
\({11\over42}km\)
Distance between home and school = \({6\over7}km\)
And, distance between school and landmark=\({5\over6}km\)
\(\therefore\) Distance between home and landmark \({6\over7}-{5\over6}={36-35\over42}={1\over42}km\)
Which of the following options hold?
(1) Every integer is a rational number and every fraction is a rational number.
(2) A rational number \(p\over q\) is positive if p and q are either both positive or both negative.
(3) A rational number \(p\over q\) is negative if one of p and q is positive and other is negative.
(4) If there are two rational numbers with common denominator then the one with the larger numerator is larger than the other.
- (a)
Both 1 and 4 are correct
- (b)
Both 2 and 3 are incorrect
- (c)
Only 1 is correct
- (d)
All are correct
Fill in the blanks.
(i) The number P is neither positive nor negative rational number.
(ii)There are Q number of rational numbers between two rational numbers.
(iii) A rational number is defined as a number which can be expressed in the form of \(p\over q\) where p and q are R and q is not equal to S
- (a)
P Q R S 1 limited whole numbers 0 - (b)
P Q R S 0 limited integers 1 - (c)
P Q R S 1 unlimited whole numbers 0 - (d)
P Q R S 0 unlimited integers 0