Chapter 1 of Class 8 Maths focuses on Rational Numbers, which are numbers that can be expressed as the ratio of two integers, pq\frac{p}{q}qp, where ppp and qqq are integers, and q≠0q \neq 0q=0. This chapter covers operations, properties, and representation of rational numbers.
Exercise 1.1: Understanding Rational Numbers
Q1.
Is 75\frac{7}{5}57 a rational number?
Solution:
Yes, 75\frac{7}{5}57 is a rational number because it can be expressed as a ratio of two integers, where the denominator is non-zero.
Q2.
Identify the rational numbers from the following:
(a) 43,\frac{4}{3},34,
(b) 0,0,0,
(c) 2,\sqrt{2},2,
(d) −5-5−5
Solution:
(a) 43\frac{4}{3}34 is a rational number.
(b) 000 is a rational number because it can be expressed as 01\frac{0}{1}10.
(c) 2\sqrt{2}2 is not a rational number (it’s irrational).
(d) −5-5−5 is a rational number because it can be expressed as −51\frac{-5}{1}1−5.
Exercise 1.2: Operations with Rational Numbers
Q1.
Add: 35+25\frac{3}{5} + \frac{2}{5}53+52
Solution:
Since the denominators are the same, we add the numerators:35+25=3+25=55=1\frac{3}{5} + \frac{2}{5} = \frac{3 + 2}{5} = \frac{5}{5} = 153+52=53+2=55=1
Q2.
Subtract: 78−48\frac{7}{8} – \frac{4}{8}87−84
Solution:
Since the denominators are the same, we subtract the numerators:78−48=7−48=38\frac{7}{8} – \frac{4}{8} = \frac{7 – 4}{8} = \frac{3}{8}87−84=87−4=83
Q3.
Multiply: 34×56\frac{3}{4} \times \frac{5}{6}43×65
Solution:
Multiply the numerators and denominators:34×56=3×54×6=1524=58\frac{3}{4} \times \frac{5}{6} = \frac{3 \times 5}{4 \times 6} = \frac{15}{24} = \frac{5}{8}43×65=4×63×5=2415=85
Q4.
Divide: 710÷25\frac{7}{10} \div \frac{2}{5}107÷52
Solution:
To divide by a fraction, multiply by its reciprocal:710÷25=710×52=3520=74\frac{7}{10} \div \frac{2}{5} = \frac{7}{10} \times \frac{5}{2} = \frac{35}{20} = \frac{7}{4}107÷52=107×25=2035=47
Exercise 1.3: Properties of Rational Numbers
Q1.
What is the additive inverse of 37\frac{3}{7}73?
Solution:
The additive inverse of 37\frac{3}{7}73 is −37-\frac{3}{7}−73, because37+(−37)=0\frac{3}{7} + \left( -\frac{3}{7} \right) = 073+(−73)=0
Q2.
What is the multiplicative inverse of 45\frac{4}{5}54?
Solution:
The multiplicative inverse of 45\frac{4}{5}54 is 54\frac{5}{4}45, because45×54=1\frac{4}{5} \times \frac{5}{4} = 154×45=1
Q3.
Is −68\frac{-6}{8}8−6 a rational number in its simplest form?
Solution:
The fraction −68\frac{-6}{8}8−6 simplifies to −34\frac{-3}{4}4−3, which is a rational number in its simplest form.
Exercise 1.4: Representation of Rational Numbers on the Number Line
Q1.
Represent 23\frac{2}{3}32 on the number line.
Solution:
To represent 23\frac{2}{3}32, divide the line segment between 0 and 1 into 3 equal parts. Count 2 parts from 0 to mark 23\frac{2}{3}32.
Q2.
Represent −54-\frac{5}{4}−45 on the number line.
Solution:
To represent −54-\frac{5}{4}−45, first mark -1 on the number line, then move 1 unit to the left to mark −54-\frac{5}{4}−45.
40 Additional Practice Questions with Solutions
1.
Add: 56+26\frac{5}{6} + \frac{2}{6}65+62
Solution:56+26=5+26=76\frac{5}{6} + \frac{2}{6} = \frac{5 + 2}{6} = \frac{7}{6}65+62=65+2=67
2.
Subtract: 79−49\frac{7}{9} – \frac{4}{9}97−94
Solution:79−49=7−49=39=13\frac{7}{9} – \frac{4}{9} = \frac{7 – 4}{9} = \frac{3}{9} = \frac{1}{3}97−94=97−4=93=31
3.
Multiply: 37×58\frac{3}{7} \times \frac{5}{8}73×85
Solution:37×58=1556\frac{3}{7} \times \frac{5}{8} = \frac{15}{56}73×85=5615
4.
Divide: 610÷35\frac{6}{10} \div \frac{3}{5}106÷53
Solution:610÷35=610×53=3030=1\frac{6}{10} \div \frac{3}{5} = \frac{6}{10} \times \frac{5}{3} = \frac{30}{30} = 1106÷53=106×35=3030=1
5.
Find the additive inverse of 78\frac{7}{8}87.
Solution:
The additive inverse of 78\frac{7}{8}87 is −78-\frac{7}{8}−87.
6.
Find the multiplicative inverse of −59\frac{-5}{9}9−5.
Solution:
The multiplicative inverse of −59\frac{-5}{9}9−5 is −95\frac{-9}{5}5−9.
7.
Express 1525\frac{15}{25}2515 in its simplest form.
Solution:
1525=35\frac{15}{25} = \frac{3}{5}2515=53 (dividing both numerator and denominator by 5).
8.
Represent 54\frac{5}{4}45 on the number line.
Solution:
First mark 1 and then move 1 unit to the right to mark 54\frac{5}{4}45.
9.
Add 12+23\frac{1}{2} + \frac{2}{3}21+32.
Solution:12+23=36+46=76\frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{7}{6}21+32=63+64=67
10.
Subtract 38−78\frac{3}{8} – \frac{7}{8}83−87.
Solution:38−78=3−78=−48=−12\frac{3}{8} – \frac{7}{8} = \frac{3 – 7}{8} = \frac{-4}{8} = \frac{-1}{2}83−87=83−7=8−4=2−1
11.
Multiply 25×−34\frac{2}{5} \times \frac{-3}{4}52×4−3.
Solution:25×−34=−620=−310\frac{2}{5} \times \frac{-3}{4} = \frac{-6}{20} = \frac{-3}{10}52×4−3=20−6=10−3
12.
Divide 56÷49\frac{5}{6} \div \frac{4}{9}65÷94.
Solution:56÷49=56×94=4524=158\frac{5}{6} \div \frac{4}{9} = \frac{5}{6} \times \frac{9}{4} = \frac{45}{24} = \frac{15}{8}65÷94=65×49=2445=815
13.
Add −34+12-\frac{3}{4} + \frac{1}{2}−43+21.
Solution:−34+12=−34+24=−3+24=−14-\frac{3}{4} + \frac{1}{2} = -\frac{3}{4} + \frac{2}{4} = \frac{-3 + 2}{4} = \frac{-1}{4}−43+21=−43+42=4−3+2=4−1
14.
Find the additive inverse of −115\frac{-11}{5}5−11.
Solution:
The additive inverse of −115\frac{-11}{5}5−11 is 115\frac{11}{5}511.
15.
Simplify the expression: 49+39\frac{4}{9} + \frac{3}{9}94+93.
Solution:49+39=4+39=79\frac{4}{9} + \frac{3}{9} = \frac{4 + 3}{9} = \frac{7}{9}94+93=94+3=97
16.
Multiply −78×25-\frac{7}{8} \times \frac{2}{5}−87×52.
Solution:−78×25=−1440=−720-\frac{7}{8} \times \frac{2}{5} = \frac{-14}{40} = \frac{-7}{20}−87×52=40−14=20−7
17.
Divide 910÷32\frac{9}{10} \div \frac{3}{2}109÷23.
Solution:910÷32=910×23=1830=35\frac{9}{10} \div \frac{3}{2} = \frac{9}{10} \times \frac{2}{3} = \frac{18}{30} = \frac{3}{5}109÷23=109×32=3018=53
18.
Write −68\frac{-6}{8}8−6 as a rational number in its simplest form.
Solution:
−68=−34\frac{-6}{8} = \frac{-3}{4}8−6=4−3.
19.
Represent 35\frac{3}{5}53 on the number line.
Solution:
Divide the segment between 0 and 1 into 5 equal parts, and mark 3 parts from 0 to represent 35\frac{3}{5}53.
20.
Find the reciprocal of 56\frac{5}{6}65.
Solution:
The reciprocal of 56\frac{5}{6}65 is 65\frac{6}{5}56.
21.
Subtract −47-\frac{4}{7}−74 from 67\frac{6}{7}76.
Solution:67−(−47)=67+47=107\frac{6}{7} – \left(-\frac{4}{7}\right) = \frac{6}{7} + \frac{4}{7} = \frac{10}{7}76−(−74)=76+74=710
22.
Simplify 69+−49\frac{6}{9} + \frac{-4}{9}96+9−4.
Solution:69+−49=6−49=29\frac{6}{9} + \frac{-4}{9} = \frac{6 – 4}{9} = \frac{2}{9}96+9−4=96−4=92
23.
Find the multiplicative inverse of 97\frac{9}{7}79.
Solution:
The multiplicative inverse of 97\frac{9}{7}79 is 79\frac{7}{9}97.
24.
Divide −89÷23\frac{-8}{9} \div \frac{2}{3}9−8÷32.
Solution:−89÷23=−89×32=−2418=−43\frac{-8}{9} \div \frac{2}{3} = \frac{-8}{9} \times \frac{3}{2} = \frac{-24}{18} = \frac{-4}{3}9−8÷32=9−8×23=18−24=3−4
25.
Add −58+78-\frac{5}{8} + \frac{7}{8}−85+87.
Solution:−58+78=−5+78=28=14-\frac{5}{8} + \frac{7}{8} = \frac{-5 + 7}{8} = \frac{2}{8} = \frac{1}{4}−85+87=8−5+7=82=41
26.
Represent −34\frac{-3}{4}4−3 on the number line.
Solution:
First mark −1-1−1, then move 14\frac{1}{4}41 units to the left to mark −34\frac{-3}{4}4−3.
27.
Simplify the expression: 85+35\frac{8}{5} + \frac{3}{5}58+53.
Solution:85+35=8+35=115\frac{8}{5} + \frac{3}{5} = \frac{8 + 3}{5} = \frac{11}{5}58+53=58+3=511
28.
Subtract 34\frac{3}{4}43 from 54\frac{5}{4}45.
Solution:54−34=5−34=24=12\frac{5}{4} – \frac{3}{4} = \frac{5 – 3}{4} = \frac{2}{4} = \frac{1}{2}45−43=45−3=42=21
29.
Multiply 45×56\frac{4}{5} \times \frac{5}{6}54×65.
Solution:45×56=2030=23\frac{4}{5} \times \frac{5}{6} = \frac{20}{30} = \frac{2}{3}54×65=3020=32
30.
Find the reciprocal of 67\frac{6}{7}76.
Solution:
The reciprocal of 67\frac{6}{7}76 is 76\frac{7}{6}67.
31.
Add −25+75-\frac{2}{5} + \frac{7}{5}−52+57.
Solution:−25+75=−2+75=55=1-\frac{2}{5} + \frac{7}{5} = \frac{-2 + 7}{5} = \frac{5}{5} = 1−52+57=5−2+7=55=1
32.
Find the multiplicative inverse of 43\frac{4}{3}34.
Solution:
The multiplicative inverse of 43\frac{4}{3}34 is 34\frac{3}{4}43.
33.
Simplify 1015\frac{10}{15}1510.
Solution:
1015=23\frac{10}{15} = \frac{2}{3}1510=32 (dividing both numerator and denominator by 5).
34.
Multiply 32×−74\frac{3}{2} \times \frac{-7}{4}23×4−7.
Solution:32×−74=−218\frac{3}{2} \times \frac{-7}{4} = \frac{-21}{8}23×4−7=8−21
35.
Subtract −25−35\frac{-2}{5} – \frac{3}{5}5−2−53.
Solution:−25−35=−2−35=−55=−1\frac{-2}{5} – \frac{3}{5} = \frac{-2 – 3}{5} = \frac{-5}{5} = -15−2−53=5−2−3=5−5=−1
36.
Divide 23÷45\frac{2}{3} \div \frac{4}{5}32÷54.
Solution:23÷45=23×54=1012=56\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}32÷54=32×45=1210=65
37.
Simplify 59−29\frac{5}{9} – \frac{2}{9}95−92.
Solution:59−29=5−29=39=13\frac{5}{9} – \frac{2}{9} = \frac{5 – 2}{9} = \frac{3}{9} = \frac{1}{3}95−92=95−2=93=31
38.
Find the reciprocal of 78\frac{7}{8}87.
Solution:
The reciprocal of 78\frac{7}{8}87 is 87\frac{8}{7}78.
39.
Represent −32-\frac{3}{2}−23 on the number line.
Solution:
First mark −1-1−1, then move 12\frac{1}{2}21 units to the left to mark −32-\frac{3}{2}−23.
40.
Subtract 49−19\frac{4}{9} – \frac{1}{9}94−91.
Solution:49−19=4−19=39=13\frac{4}{9} – \frac{1}{9} = \frac{4 – 1}{9} = \frac{3}{9} = \frac{1}{3}94−91=94−1=93=31
Practice Questions
1.
Add: 79+29\frac{7}{9} + \frac{2}{9}97+92
Solution:79+29=7+29=99=1\frac{7}{9} + \frac{2}{9} = \frac{7 + 2}{9} = \frac{9}{9} = 197+92=97+2=99=1
2.
Subtract: 56−13\frac{5}{6} – \frac{1}{3}65−31
Solution:56−13=56−26=5−26=36=12\frac{5}{6} – \frac{1}{3} = \frac{5}{6} – \frac{2}{6} = \frac{5 – 2}{6} = \frac{3}{6} = \frac{1}{2}65−31=65−62=65−2=63=21
3.
Multiply: −45×38\frac{-4}{5} \times \frac{3}{8}5−4×83
Solution:−45×38=−1240=−310\frac{-4}{5} \times \frac{3}{8} = \frac{-12}{40} = \frac{-3}{10}5−4×83=40−12=10−3
4.
Divide: 89÷47\frac{8}{9} \div \frac{4}{7}98÷74
Solution:89÷47=89×74=5636=149\frac{8}{9} \div \frac{4}{7} = \frac{8}{9} \times \frac{7}{4} = \frac{56}{36} = \frac{14}{9}98÷74=98×47=3656=914
5.
Find the reciprocal of 56\frac{5}{6}65.
Solution:
The reciprocal of 56\frac{5}{6}65 is 65\frac{6}{5}56.
6.
Express 1824\frac{18}{24}2418 in its simplest form.
Solution:
1824=34\frac{18}{24} = \frac{3}{4}2418=43 (dividing both numerator and denominator by 6).
7.
Subtract 710\frac{7}{10}107 from 910\frac{9}{10}109.
Solution:910−710=9−710=210=15\frac{9}{10} – \frac{7}{10} = \frac{9 – 7}{10} = \frac{2}{10} = \frac{1}{5}109−107=109−7=102=51
8.
Add −34+58\frac{-3}{4} + \frac{5}{8}4−3+85.
Solution:−34+58=−68+58=−6+58=−18\frac{-3}{4} + \frac{5}{8} = \frac{-6}{8} + \frac{5}{8} = \frac{-6 + 5}{8} = \frac{-1}{8}4−3+85=8−6+85=8−6+5=8−1
9.
Find the additive inverse of −67\frac{-6}{7}7−6.
Solution:
The additive inverse of −67\frac{-6}{7}7−6 is 67\frac{6}{7}76.
10.
Multiply 23×45\frac{2}{3} \times \frac{4}{5}32×54.
Solution:23×45=815\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}32×54=158
11.
Divide 1113÷34\frac{11}{13} \div \frac{3}{4}1311÷43.
Solution:1113÷34=1113×43=4439\frac{11}{13} \div \frac{3}{4} = \frac{11}{13} \times \frac{4}{3} = \frac{44}{39}1311÷43=1311×34=3944
12.
Simplify −1218\frac{-12}{18}18−12.
Solution:
−1218=−23\frac{-12}{18} = \frac{-2}{3}18−12=3−2 (dividing both numerator and denominator by 6).
13.
Add 15+415\frac{1}{5} + \frac{4}{15}51+154.
Solution:15+415=315+415=715\frac{1}{5} + \frac{4}{15} = \frac{3}{15} + \frac{4}{15} = \frac{7}{15}51+154=153+154=157
14.
Subtract 911−511\frac{9}{11} – \frac{5}{11}119−115.
Solution:911−511=9−511=411\frac{9}{11} – \frac{5}{11} = \frac{9 – 5}{11} = \frac{4}{11}119−115=119−5=114
15.
Find the multiplicative inverse of −49\frac{-4}{9}9−4.
Solution:
The multiplicative inverse of −49\frac{-4}{9}9−4 is −94\frac{-9}{4}4−9.
16.
Simplify −68+38\frac{-6}{8} + \frac{3}{8}8−6+83.
Solution:−68+38=−6+38=−38\frac{-6}{8} + \frac{3}{8} = \frac{-6 + 3}{8} = \frac{-3}{8}8−6+83=8−6+3=8−3
17.
Multiply −78×23\frac{-7}{8} \times \frac{2}{3}8−7×32.
Solution:−78×23=−1424=−712\frac{-7}{8} \times \frac{2}{3} = \frac{-14}{24} = \frac{-7}{12}8−7×32=24−14=12−7
18.
Divide 1415÷710\frac{14}{15} \div \frac{7}{10}1514÷107.
Solution:1415÷710=1415×107=140105=43\frac{14}{15} \div \frac{7}{10} = \frac{14}{15} \times \frac{10}{7} = \frac{140}{105} = \frac{4}{3}1514÷107=1514×710=105140=34
19.
Express 816\frac{8}{16}168 in its simplest form.
Solution:
816=12\frac{8}{16} = \frac{1}{2}168=21 (dividing both numerator and denominator by 8).
20.
Find the additive inverse of 56\frac{5}{6}65.
Solution:
The additive inverse of 56\frac{5}{6}65 is −56-\frac{5}{6}−65.
Class 7 Maths Chapter 6: Integers Worksheet with Answers
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This worksheet provides a comprehensive set of questions on rational numbers, covering various concepts and operations. If you need further explanation or more practice questions, feel free to ask
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