The DAV Class 7 Maths Book Solutions Pdf and DAV Class 7 Maths Chapter 1 Worksheet 2 Solutions of Rational Numbers offer comprehensive answers to textbook questions.
DAV Class 7 Maths Ch 1 WS 2 Solutions
Question 1.
In each of the following cases, show that the rational numbers are equivalent.
(i) \(\frac{4}{9}\) and \(\frac{44}{99}\)
Answer:
\(\frac{4}{9}\) and \(\frac{44}{99}\)
\(\frac{4}{9}\) = \(\frac{4}{9}\) hence they are equivalent rational numbers.
(ii) \(\frac{7}{-3}\) and \(\frac{35}{-15}\)
Answer:
\(\frac{7}{-3}\) and \(\frac{35}{-15}\)
\(\frac{7}{-3}\) = \(\frac{7}{-3}\), So, they are equivalent rational numbers.
(iii) \(\frac{-3}{5}\) and \(\frac{-12}{20}\)
Answer:
\(\frac{-3}{5}\) and \(\frac{-12}{20}\)
\(\frac{-3}{5}\) = \(\frac{-3}{5}\) So, they are not equivalent rational numbers.
Question 2.
In each of the following cases, show that the rational numbers are not equivalent.
(i) \(\frac{4}{9}\) and \(\frac{16}{27}\)
Answer:
\(\frac{4}{9}\) and \(\frac{16}{27}\)
⇒ 4 × 27 and 16 × 9
⇒ 108 ≠ 144
(ii) \(\frac{-100}{3}\) and \(\frac{300}{9}\)
Answer:
\(\frac{-100}{3}\) and \(\frac{300}{9}\)
⇒ -100 × 9 and 300 × 3
⇒ -900 ≠ 900
(iii) \(\frac{3}{-17}\) and \(\frac{8}{-51}\)
Answer:
\(\frac{3}{-17}\) and \(\frac{8}{-51}\)
⇒ 3 × (-51) and 8 × (-17)
⇒ -153 ≠ -136
So, they are not equivalent rational numbers.
Question 3.
Write three rational numbers equivalent to each of the following:
(i) \(\frac{4}{7}\)
Answer:
\(\frac{4}{7}=\frac{4 \times 2}{7 \times 2}=\frac{8}{14}\)
and \(\frac{4}{7}=\frac{4 \times 3}{7 \times 3}=\frac{12}{21}\) and \(\frac{4 \times 4}{7 \times 4}=\frac{16}{28}\)
Hence, the three equivalent fractions are
\(\frac{8}{14}=\frac{12}{21}=\frac{16}{28}\)
(ii) \(\frac{36}{108}\)
Answer:
\(\frac{36}{108}=\frac{36 \div 2}{108 \div 2}=\frac{18}{54}\)
and \(\frac{36 \div 3}{108 \div 3}=\frac{12}{36}\) and \(\frac{36 \div 4}{108 \div 4}=\frac{9}{27}\)
Hence, the equivalent fractions are
\(\frac{18}{54}, \frac{12}{36}\) and \(\)
(iii) \(\frac{-5}{-7}\)
Answer:
\(\frac{-5}{-7}=\frac{-5 \times 2}{-7 \times 2}=\frac{-10}{-14}\)
and \(\frac{-5 x-2}{-7 x-2}=\frac{10}{14}\) and \(-\frac{5 \times 3}{-7 \times 3}=\frac{-15}{-21}\)
Hence, the equivalent fractions are
\(\frac{-10}{-14}, \frac{10}{14}\) and \(\frac{-15}{-21}\)
(iv) \(\frac{-72}{180}\)
Answer:
\(\frac{-72}{180}=\frac{-72 \div 2}{180 \div 2}=\frac{-36}{90}\)
and \(\frac{-72 \div 3}{180 \div 3}=\frac{-24}{60}\) and \(\frac{-72 \div 4}{180 \div 4}=\frac{-18}{45}\)
Hence the equivalent rational numbers are
\(\frac{-36}{90}, \frac{-24}{60}\) and \(\frac{-18}{45}\)
Question 4.
Express \(\frac{3}{5}\) as rational number with numerator,
(i) -21
Answer:
\(\frac{3}{5}=\frac{3 \times-7}{5 \times-7}\)
= \(\frac{-21}{-35}\)
(ii) 150
Answer:
\(\frac{3}{5}=\frac{3 \times 50}{5 \times 50}\)
= \(\frac{150}{250}\)
Question 5.
Express \(\frac{4}{-7}\) as a rational number with denominator
(i) 84
Answer:
\(\frac{4}{-7}=\frac{4 \times-12}{-7 x-12}\)
= \(\frac{-48}{84}\)
(ii) -28
Answer:
\(\frac{4}{-7}=\frac{4 \times 4}{-7 \times 4}\)
= \(\frac{16}{-28}\)
Question 6.
Express \(\frac{90}{216}\) as a rational number with numerator 5.
Answer:
\(\frac{90}{216}=\frac{90 \div 18}{216 \div 18}\)
= \(\frac{5}{12}\)
Question 7.
Express \(\frac{-64}{256}\) as a rational number with denominator 8.
Answer:
\(\frac{-64}{256}=\frac{-64 \div 32}{256 \div 32}\)
= \(\frac{-2}{8}\)
Question 8.
Find the equivalent forms of the rational numbers having a common denominator in each of the following collections of rational numbers:
(i) \(\frac{2}{5}, \frac{6}{13}\)
Answer:
The L.C.M of 5 and 13 = 5 × 13 = 65
= \(\frac{2 \times 13}{5 \times 13}, \frac{6 \times 5}{13 \times 5}\)
⇒ \(\frac{26}{65}, \frac{30}{65}\)
(ii) \(\frac{1}{7}, \frac{2}{8}, \frac{3}{14}\)
Answer:
The L.C.M of 7, 8, 14 = 56
= \(\frac{1}{7} \times \frac{8}{8}, \frac{2 \times 7}{8 \times 7}, \frac{3 \times 4}{14 \times 4}\)
⇒ \(\frac{8}{56}, \frac{14}{56}, \frac{12}{56}\)
(iii) \(\frac{5}{12}, \frac{7}{4}, \frac{9}{60}, \frac{11}{3}\)
Answer:
The L.C.M of 12, 4, 60 and 3 = 60
= \(\frac{5}{12} \times \frac{5}{5}, \frac{7}{4} \times \frac{15}{15}, \frac{9}{60} \times \frac{1}{1}, \frac{11}{3} \times \frac{20}{20}\)
= \(\frac{25}{60}, \frac{105}{60}, \frac{9}{60}, \frac{220}{60}\)