1. A rational number is defined as a number that can be expressed in the form \( \dfrac{p}{q} \), where \(p\) and \(q\) are integers and
(a) \(q = 0\)
(b) \(q = 1\)
(c) \(q \neq 1\)
(d) \(q \neq 0\)
Which of the following rational numbers is positive?
(a) \(-\dfrac{8}{7}\)
(b) \(\dfrac{19}{-13}\)
(c) \(\dfrac{-3}{-4}\)
(d) \(\dfrac{-21}{13}\)
Which of the following rational numbers is negative?
(a) \(-\dfrac{3}{7}\)
(b) \(\dfrac{-5}{-8}\)
(c) \(\dfrac{9}{8}\)
(d) \(\dfrac{3}{-7}\)
In the standard form of a rational number, the common factor of numerator and denominator is always:
(a) 0
(b) 1
(c) -2
(d) 2
Which of the following rational numbers is equal to its reciprocal?
(a) 1
(b) 2
(c) \(\dfrac{1}{2}\)
(d) 0
The reciprocal of \(\dfrac{1}{2}\) is
(a) 3
(b) 2
(c) -1
(d) 0
The standard form of \(-\dfrac{48}{60}\) is
(a) \(\dfrac{48}{60}\)
(b) \(\dfrac{-60}{48}\)
(c) \(-\dfrac{4}{5}\)
(d) \(\dfrac{-4}{-5}\)
Which of the following is equivalent to \(\dfrac{4}{5}\)?
(a) \(\dfrac{5}{4}\)
(b) \(\dfrac{16}{25}\)
(c) \(\dfrac{16}{20}\)
(d) \(\dfrac{15}{25}\)
How many rational numbers are there between two rational numbers?
(a) 1
(b) 0
(c) unlimited
(d) 100
In the standard form of a rational number, the denominator is always a
(a) 0
(b) negative integer
(c) positive integer
(d) 1
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
Which is greater number in the following:
(a) \(-\dfrac{1}{2}\)
(b) 0
(c) \(\dfrac{1}{2}\)
(d) -2
\(-\dfrac{3}{8}\) is a _____ rational number.
\(-\dfrac{3}{8}\) is a negative rational number.
1 is a _____ rational number.
1 is a positive rational number.
The standard form of \(-\dfrac{8}{-36}\) is _____.
The standard form of \(-\dfrac{8}{-36}\) is \(\dfrac{2}{7}\).
The standard form of \(\dfrac{18}{-24}\) is _____.
The standard form of \(\dfrac{18}{-24}\) is \(-\dfrac{3}{4}\).
On a number line, \(-\dfrac{1}{2}\) is to the _____ of zero (0).
On a number line, \(-\dfrac{1}{2}\) is to the left of zero (0).
On a number line, \(\dfrac{4}{3}\) is to the _____ of zero (0).
On a number line, \(\dfrac{4}{3}\) is to the right of zero (0).
\(-\dfrac{1}{2}\) is _____ than \(\dfrac{1}{5}\).
\(-\dfrac{1}{2}\) is smaller than \(\dfrac{1}{5}\).
\(-\dfrac{3}{5}\) is _____ than 0.
\(-\dfrac{3}{5}\) is smaller than 0.
\(-\dfrac{16}{24}\) and \(\dfrac{20}{-16}\) represent _____ rational numbers.
\(-\dfrac{16}{24}\) and \(\dfrac{20}{-16}\) represent different rational numbers.
\(-\dfrac{27}{45}\) and \(-\dfrac{3}{5}\) represent _____ rational numbers.
\(-\dfrac{27}{45}\) and \(-\dfrac{3}{5}\) represent same rational numbers.
Additive inverse of \(\dfrac{2}{3}\) is _____.
Additive inverse of \(\dfrac{2}{3}\) is -\dfrac{2}{3}.
\(-\dfrac{3}{5} + \dfrac{2}{5} = \) _____.
\(-\dfrac{3}{5} + \dfrac{2}{5} = -\dfrac{1}{5}\).
\(-\dfrac{5}{6} + -\dfrac{1}{6} = \) _____.
\(-\dfrac{5}{6} + -\dfrac{1}{6} = -1\).
\(\dfrac{3}{4} \times ( -\dfrac{2}{3} ) = \) _____.
\(\dfrac{3}{4} \times ( -\dfrac{2}{3} ) = -\dfrac{1}{2}\).
\(-\dfrac{5}{3} \times ( -\dfrac{3}{5} ) = \) _____.
\(-\dfrac{5}{3} \times ( -\dfrac{3}{5} ) = 1\).
\(-\dfrac{6}{7} = \dfrac{__}{42}\)
\(-\dfrac{6}{7} = \dfrac{-36}{42}\)
\(\dfrac{1}{2} = \dfrac{6}{__}\)
\(\dfrac{1}{2} = \dfrac{6}{12}\)
\(-\dfrac{2}{9} - \dfrac{7}{9} = \) _____.
\(-\dfrac{2}{9} - \dfrac{7}{9} = -1\).
\(-\dfrac{7}{8} \; \Box \; \dfrac{8}{9}\)
\(-\dfrac{7}{8} < \dfrac{8}{9}\)
\(\dfrac{3}{7} \; \Box \; \dfrac{-5}{6}\)
\(\dfrac{3}{7} > -\dfrac{5}{6}\)
\(\dfrac{5}{6} \; \Box \; \dfrac{8}{4}\)
\(\dfrac{5}{6} < 2\)
\(-\dfrac{9}{7} \; \Box \; \dfrac{4}{-7}\)
\(-\dfrac{9}{7} < -\dfrac{4}{7}\)
\(\dfrac{8}{8} \; \Box \; \dfrac{2}{2}\)
\(\dfrac{8}{8} = \dfrac{2}{2}\)
The reciprocal of _____ does not exist.
The reciprocal of zero does not exist.
The reciprocal of 1 is _____.
The reciprocal of 1 is 1.
\(-\dfrac{3}{7} \div ( -\dfrac{7}{3} ) = \) _____
\(-\dfrac{3}{7} \div ( -\dfrac{7}{3} ) = \dfrac{9}{49}\)
\(0 \div ( -\dfrac{5}{6} ) = \) _____
\(0 \div ( -\dfrac{5}{6} ) = 0\)
\(0 \times ( -\dfrac{5}{6} ) = \) _____
\(0 \times ( -\dfrac{5}{6} ) = 0\)
_____ \(\times ( -\dfrac{2}{5} ) = 1\)
-\dfrac{5}{2} \(\times ( -\dfrac{2}{5} ) = 1\)
The standard form of rational number -1 is _____.
The standard form of rational number -1 is -1.
If m is a common divisor of a and b, then \(\dfrac{a}{b} = \dfrac{a \div m}{\_\_\_}\)
If m is a common divisor of a and b, then \(\dfrac{a}{b} = \dfrac{a \div m}{b \div m}\).
If p and q are positive integers, then \(\dfrac{p}{q}\) is a _____ rational number and \(\dfrac{p}{-q}\) is a _____ rational number.
If p and q are positive integers, then \(\dfrac{p}{q}\) is a positive rational number and \(\dfrac{p}{-q}\) is a negative rational number.
Two rational numbers are said to be equivalent or equal, if they have the same _____ form.
Two rational numbers are said to be equivalent or equal, if they have the same simplest form.
If \(\dfrac{p}{q}\) is a rational number, then q cannot be _____.
If \(\dfrac{p}{q}\) is a rational number, then q cannot be zero.
Every natural number is a rational number but every rational number need not be a natural number.
Every integer is a rational number but every rational number need not be an integer.
Every negative integer is not a negative rational number.
If \(\dfrac{p}{q}\) is a rational number and m is a non-zero integer, then \(\dfrac{p}{q} = \dfrac{p \times m}{q \times m}\).
If \(\dfrac{p}{q}\) is a rational number and m is a non-zero common divisor of p and q, then \(\dfrac{p}{q} = \dfrac{p \div m}{q \div m}\).
In a rational number, denominator always has to be a non-zero integer.
If \(\dfrac{p}{q}\) is a rational number and m is a non-zero integer, then \(\dfrac{p \times m}{q \times m}\) is a rational number not equivalent to \(\dfrac{p}{q}\).
Sum of two rational numbers is always a rational number.
All decimal numbers are also rational numbers.
The quotient of two rationals is always a rational number.
Two rationals with different numerators can never be equal.
8 can be written as a rational number with any integer as denominator.
\(\dfrac{4}{6}\) is equivalent to \(\dfrac{2}{3}\).
The rational number \(-\dfrac{3}{4}\) lies to the right of zero on the number line.
The rational numbers \(-\dfrac{12}{-5}\) and \(-\dfrac{7}{17}\) are on the opposite sides of zero on the number line.
66. Match the following:
| Column I | Column II |
|---|---|
| (i) \(\dfrac{a}{b} \div \dfrac{a}{b}\) | (a) \(-\dfrac{a}{b}\) |
| (ii) \(\dfrac{a}{b} \div \dfrac{c}{d}\) | (b) \(-1\) |
| (iii) \(\dfrac{a}{b} \div (-1)\) | (c) 1 |
| (iv) \(\dfrac{a}{b} \div \dfrac{-a}{b}\) | (d) \(\dfrac{bc}{ad}\) |
| (v) \(\dfrac{b}{a} \div (\dfrac{d}{c})\) | (e) \(\dfrac{ad}{bc}\) |
(i) ↔ (c)
(ii) ↔ (e)
(iii) ↔ (a)
(iv) ↔ (b)
(v) ↔ (d)
Write each of the following rational numbers with positive denominators: \(\dfrac{5}{-8}, \dfrac{15}{-28}, \dfrac{-17}{-13}\).
\(\dfrac{5}{-8} = -\dfrac{5}{8}\)
\(\dfrac{15}{-28} = -\dfrac{15}{28}\)
\(\dfrac{-17}{-13} = \dfrac{17}{13}\)
Express \(\dfrac{3}{4}\) as a rational number with denominator:
(i) 36
(ii) -80
(i) \(\dfrac{27}{36}\)
(ii) \(\dfrac{-60}{-80}\)
Reduce each of the following rational numbers in its lowest form:
(i) \(\dfrac{-60}{72}\)
(ii) \(\dfrac{91}{-364}\)
(i) \(-\dfrac{5}{6}\)
(ii) \(-\dfrac{1}{4}\)
Express each of the following rational numbers in its standard form:
(i) \(\dfrac{-12}{-30}\)
(ii) \(\dfrac{14}{-49}\)
(iii) \(\dfrac{-15}{35}\)
(iv) \(\dfrac{299}{-161}\)
(i) \(\dfrac{2}{5}\)
(ii) \(-\dfrac{2}{7}\)
(iii) \(-\dfrac{3}{7}\)
(iv) \(-\dfrac{13}{7}\)
Are the rational numbers \(\dfrac{-8}{28}\) and \(\dfrac{32}{-112}\) equivalent? Give reason.
Yes, both simplify to \(-\dfrac{2}{7}\).
Arrange the rational numbers \(-\dfrac{7}{10}, \dfrac{5}{-8}, \dfrac{2}{-3}, -\dfrac{1}{4}, -\dfrac{3}{5}\) in ascending order.
-\dfrac{7}{10}, -\dfrac{2}{3}, -\dfrac{5}{8}, -\dfrac{3}{5}, -\dfrac{1}{4}
Represent the following rational numbers on a number line: \(\dfrac{3}{8}, -\dfrac{7}{3}, \dfrac{22}{-6}\).
Points marked at \(\dfrac{3}{8}\), -\dfrac{7}{3}, -\dfrac{11}{3}\) on number line.
If \(-\dfrac{5}{7} = \dfrac{x}{28}\), find the value of x.
\(x = -20\)
Give three rational numbers equivalent to:
(i) \(-\dfrac{3}{4}\)
(ii) \(\dfrac{7}{11}\)
(i) -\dfrac{6}{8}, -\dfrac{9}{12}, -\dfrac{12}{16}
(ii) \dfrac{14}{22}, \dfrac{21}{33}, \dfrac{28}{44}
Write the next three rational numbers to complete the pattern:
(i) \(\dfrac{4}{-5}, \dfrac{8}{-10}, \dfrac{12}{-15}, \dfrac{16}{-20}, \_\_, \_\_, \_\_\)
(ii) \(\dfrac{-8}{7}, \dfrac{-16}{14}, \dfrac{-24}{21}, \dfrac{-32}{28}, \_\_, \_\_, \_\_\)
(i) -\dfrac{20}{25}, -\dfrac{24}{30}, -\dfrac{28}{35}
(ii) -\dfrac{40}{35}, -\dfrac{48}{42}, -\dfrac{56}{49}
List four rational numbers between \(\dfrac{5}{7}\) and \(\dfrac{7}{8}\).
\(\dfrac{42}{56}, \dfrac{44}{56}, \dfrac{46}{56}, \dfrac{48}{56}\)
Find the sum of:
(i) \(\dfrac{8}{13} + \dfrac{3}{11}\)
(ii) \(\dfrac{7}{3} + \dfrac{-4}{3}\)
(i) \(\dfrac{127}{143}\)
(ii) 1
Solve:
(i) \(\dfrac{29}{4} - \dfrac{30}{7}\)
(ii) \(\dfrac{5}{13} - \dfrac{-8}{26}\)
(i) \(\dfrac{83}{28}\)
(ii) \(\dfrac{9}{13}\)
Find the product of:
(i) \(\dfrac{-4}{5} \times \dfrac{-5}{12}\)
(ii) \(\dfrac{-22}{11} \times \dfrac{-21}{11}\)
(i) \(\dfrac{1}{3}\)
(ii) \(\dfrac{42}{11}\)
Simplify:
(i) \(\dfrac{13}{11} \times \dfrac{-14}{5} + \dfrac{13}{11} \times \dfrac{-7}{5} + \dfrac{-13}{11} \times \dfrac{34}{5}\)
(ii) \(\dfrac{6}{5} \times \dfrac{3}{7} - \dfrac{1}{5} \times \dfrac{3}{7}\)
(i) -13
(ii) \(\dfrac{3}{7}\)
Simplify:
(i) \(\dfrac{3}{7} \div (\dfrac{21}{-55})\)
(ii) \(1 \div ( -\dfrac{1}{2} )\)
(i) \(-\dfrac{55}{49}\)
(ii) -2
Which is greater in the following?
(i) \(\dfrac{3}{4}, \dfrac{7}{8}\)
(ii) -\(\dfrac{3}{5}, \dfrac{1}{9}\)
(i) \(\dfrac{7}{8}\)
(ii) \(\dfrac{1}{9}\)
Write a rational number in which the numerator is less than \(-7×11\) and the denominator is greater than \(12+4\).
Examples: -\dfrac{78}{17}, -\dfrac{79}{18}
If \(x=\dfrac{1}{10}\) and \(y=\dfrac{-3}{8}\), then evaluate \(x+y, x-y, x×y, x÷y\).
(i) \(x+y = -\dfrac{11}{40}\)
(ii) \(x-y = \dfrac{19}{40}\)
(iii) \(x×y = -\dfrac{3}{80}\)
(iv) \(x÷y = -\dfrac{4}{15}\)
Find the reciprocal of the following:
(i) \((\dfrac{1}{2} \times \dfrac{1}{4}) + (\dfrac{1}{2} \times 6)\)
(ii) \(\dfrac{20}{51} \times \dfrac{4}{91}\)
(iii) \(\dfrac{3}{13} \div \dfrac{-4}{65}\)
(iv) \((-5 \times \dfrac{12}{15}) - (-3 \times \dfrac{2}{9})\)
(i) \(\dfrac{8}{25}\)
(ii) \(\dfrac{4641}{80}\)
(iii) \(-\dfrac{4}{15}\)
(iv) \(-\dfrac{3}{10}\)
Complete the following table by finding the sums:
| + | -1/9 | 4/11 | -5/6 |
|---|---|---|---|
| 2/3 | 5/9 | 34/33 | -1/6 |
| -5/4 | -49/36 | -39/44 | -25/12 |
| -1/3 | -4/9 | 1/33 | -7/6 |
Write each of the following numbers in the form \(\dfrac{p}{q}\):
(a) six-eighths
(b) three and half
(c) opposite of 1
(d) one-fourth
(e) zero
(f) opposite of three-fifths
(a) \(\dfrac{6}{8}\)
(b) \(\dfrac{7}{2}\)
(c) \(-1\)
(d) \(\dfrac{1}{4}\)
(e) \(0\)
(f) \(-\dfrac{3}{5}\)
If \(p = m \times t\) and \(q = n \times t\), then \(\dfrac{p}{q} = \_\_\).
\(\dfrac{p}{q} = \dfrac{m}{n}\)
Given \(\dfrac{p}{q}\) and \(\dfrac{r}{s}\) are rationals in standard form:
(a) \(\dfrac{p}{q} < \dfrac{r}{s}\), if \(p \times s < r \times q\)
(b) \(\dfrac{p}{q} = \dfrac{r}{s}\), if \(p \times s = r \times q\)
(c) \(\dfrac{p}{q} > \dfrac{r}{s}\), if \(p \times s > r \times q\)
(a) \(\dfrac{p}{q} < \dfrac{r}{s}\)
(b) \(p \times s = r \times q\)
(c) \(\dfrac{p}{q} > \dfrac{r}{s}\)
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) -34/48
(b) -24/4
(c) -5/17
(d) 1600/81
Write the following as rational numbers in their standard forms:
(a) 35%
(b) 1.2
(c) -6 3/7
(d) 240 ÷ (-840)
(e) 115 ÷ 207
(a) 7/20
(b) 6/5
(c) -45/7
(d) -2/7
(e) 5/9
Find a rational number exactly halfway between:
(a) -1/3 and 1/3
(b) 1/6 and 1/9
(c) 5/-13 and -7/9
(d) 1/15 and 1/12
(a) 0
(b) 5/36
(c) -136/234
(d) 3/40
Taking x=-4/9, y=5/12, z=7/18, find:
(a) rational number added to x gives y
(b) rational number subtracted from y gives z
(c) rational number added to z gives x
(d) rational number multiplied by y gives x
(e) reciprocal of x+y
(f) 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) 31/36
(b) 1/36
(c) -5/6
(d) -48/45
(e) -36
(f) 3/20
(g) -56/135
(h) -17/36
(i) 13/36
(j) -56/135
(k) -5/4
What should be added to -1/2 to obtain the nearest natural number?
3/2
What should be subtracted from -2/3 to obtain the nearest integer?
1/3
What should be multiplied with -5/8 to obtain the nearest integer?
8/5
What should be divided by 1/2 to obtain the greatest negative integer?
-1/2
A rope 68m long is cut into pieces of length 4 1/4 m. Find number of such pieces.
16
If 12 shirts of equal size are prepared from 27m cloth, what is length required for each shirt?
2.25 m
Insert 3 equivalent rational numbers between:
(i) -1/2 and 1/5
(ii) 0 and -10
(i) -3/20, -6/40, -9/60
(ii) -5, -10/2, -15/3
Put (√) wherever applicable:
| Number | Natural | Whole | Integer | Fraction | Rational |
|---|---|---|---|---|---|
| -114 | √ | √ | |||
| 19/27 | √ | √ | |||
| 623/1 | √ | √ | √ | √ | √ |
| -19 3/4 | √ | √ | |||
| 73/71 | √ | √ | |||
| 0 | √ | √ | √ | √ |
As shown in the table above.
a and b are different numbers from 1–50. What is largest value of (a–b)/(a+b) and (a+b)/(a–b)?
49/51 and 99
150 students study English, Maths or both. 62% study English, 68% study Maths. How many study both?
45
A body floats 2/9 above surface. What is ratio of submerged volume to exposed volume?
7:2 or 7/2
Find the odd one out:
(a) 4/3×3/4
(b) -3/2×-2/3
(c) 2×1/2
(d) -1/3×3/1
(d)
Find the odd one out:
(a) 4/-9
(b) -16/36
(c) -20/-45
(d) 28/-63
(c)