{"id":8227,"date":"2024-01-30T02:32:58","date_gmt":"2024-01-29T21:02:58","guid":{"rendered":"https:\/\/tsboardsolutions.in\/?p=8227"},"modified":"2024-02-03T09:32:48","modified_gmt":"2024-02-03T04:02:48","slug":"ts-10th-class-maths-solutions-chapter-1-ex-1-3","status":"publish","type":"post","link":"https:\/\/tsboardsolutions.in\/ts-10th-class-maths-solutions-chapter-1-ex-1-3\/","title":{"rendered":"TS 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.3"},"content":{"rendered":"

Students can practice TS 10th Class Maths Solutions<\/a> Chapter 1 Real Numbers Ex 1.3 to get the best methods of solving problems.<\/p>\n

TS 10th Class Maths Solutions Chapter 1 Real Numbers Exercise 1.3<\/h2>\n

Question 1.
\nWrite the following rational numbers in their decimal form and also state which are terminating and which are non-terminating repeating decimal form.<\/p>\n

(i) \\(\\frac{3}{8}\\)
\nSolution:
\n\\(\\frac{3}{8}\\) = \\(\\frac{3}{2.2 .2}\\) = \\(\\frac{3}{2^3}\\) (\u2235 Denominator consists of only 2’s. terminating decimal)
\n\\(\\frac{3 \\times 5^3}{2^3 \\times 5^3}\\) = \\(\\frac{3 \\times 125}{10^3}\\) = \\(\\frac{375}{1000}\\) = 0.375<\/p>\n

(ii) \\(\\frac{229}{400}\\)
\nSolution:
\n\\(\\frac{229}{400}\\) = \\(\\frac{229}{2.2 .2 .2 \\times 5 \\times 5}\\) (\u2235 Denominator consists of only 2 \u2018s.)
\n= \\(\\frac{229}{2^4 \\cdot 5^2}\\) (terminating decimal)
\n= \\(\\frac{229 \\times 5^2}{2^4 \\times 5^2 \\times 5^2}\\) = \\(\\frac{229 \\times 5^2}{2^4 \\times 5^4}\\)
\n= \\(\\frac{229 \\times 25}{(2 \\times 5)^4}\\) = \\(\\frac{5725}{10^4}\\)
\n= \\(\\frac{5725}{10000}\\) = 0.5725 (\u2235 Denominator consists of only 2’s. terminating decimal)<\/p>\n

\"TS<\/p>\n

(iii) 4\\(\\frac{1}{5}\\)
\nSolution:
\n4\\(\\frac{1}{5}\\) = \\(\\frac{21}{5}\\) (terminating decimal)
\n\\(\\frac{21}{5}\\) = 4.2 (\u2235 Denominator consists of only 2\u2019s.)<\/p>\n

(iv) \\(\\frac{2}{11}\\)
\nSolution:
\n\\(\\frac{2}{11}\\) (\u2235 Denominator doesn’t contain 2s or 5’s or both. Hence it is an non-terminating, repeating decimal) (non-terminating, re-peating decimal because the denomination does not contain power of 2 or power of 5 of both 2 and 5)
\n\\(\\frac{2}{11}\\) = 0.181818……..
\n= \\(0 . \\overline{18}\\)<\/p>\n

(v) \\(\\frac{8}{125}\\)
\nSolution:
\n\\(\\frac{8}{125}\\) = \\(\\frac{8}{5^3}\\) (\u2235 Denominator does not consists of only 2\u2019s.) (terminating decimal)
\n= \\(\\frac{8}{5^3}\\) \u00d7 \\(\\frac{2^3}{2^3}\\) = \\(\\frac{8 \\times 8}{(5 \\times 2)^3}\\) = \\(\\frac{64}{1000}\\) = 0.064<\/p>\n

Question 2.
\nWithout performing division, state whether the following rational numbers will have a terminating decimal form or a non-terminating repeating decimal form.<\/p>\n

(i) \\(\\frac{13}{3125}\\)
\nSolution:
\n\\(\\frac{13}{3125}\\). It is of the form \\(\\frac{p}{q}\\)
\n\\(\\frac{13}{3125}\\) = \\(\\frac{13}{5 \\times 5 \\times 5 \\times 5 \\times 5}\\) = \\(\\frac{13}{5^5}\\)
\n\u2235 q = 55<\/sup> which is of the form 2n<\/sup>5m<\/sup> (n = 0; m = 5)
\nGiven rational number has a terminating decimal expansion.<\/p>\n

(ii) \\(\\frac{11}{12}\\)
\nSolution:
\n\\(\\frac{11}{12}\\) it is of the form \\(\\frac{p}{q}\\).
\n\\(\\frac{11}{12}\\) = \\(\\frac{11}{2 \\times 2 \\times 3}\\) = \\(\\frac{11}{2^2 \\times 3}\\) = 0.916666.
\n\u2235 Here q = 22<\/sup> \u00d7 3 which is not of the form 2n<\/sup> \u00d7 5m<\/sup>.
\n\u2234 Given rational number has a non-termi-nating repeating decimal expansion.<\/p>\n

\"TS<\/p>\n

(iii) \\(\\frac{64}{455}\\)
\nSolution:
\n\\(\\frac{64}{455}\\) It is of the form \\(\\frac{p}{q}\\).
\n\\(\\frac{64}{455}\\) = \\(\\frac{64}{5 \\times 7 \\times 13}\\)
\n\u2235 q = 5 \u00d7 7 \u00d7 13 which is not of the form 2n<\/sup>.5m<\/sup>.
\n\u2235 Given rational number has a non-terminating, repeating decimal expansion.<\/p>\n

(iv) \\(\\frac{15}{1600}\\)
\nSolution:
\n\\(\\frac{15}{1600}\\) It is of the form \\(\\frac{p}{q}\\).
\n\\(\\frac{15}{1600}\\) = \\(\\frac{3}{320}\\) = \\(\\frac{3}{2^6 \\times 5}\\)
\n\u2235 Here q = 26<\/sup> \u00d7 51<\/sup> which is of the form 2n<\/sup>.5m<\/sup> (m = 1, n = 6)
\n\u2234 Given rational number has a terminating decimal expansion.<\/p>\n

(v) \\(\\frac{29}{343}\\)
\nSolution:
\n\\(\\frac{29}{343}\\) It is of the form \\(\\frac{p}{q}\\).
\n\\(\\frac{29}{343}\\) = \\(\\frac{29}{7 \\times 7 \\times 7}\\) = \\(\\frac{29}{7^3}\\)
\n\u2235 Here q = 73<\/sup> which is not of the form 2n5m.
\n\u2234 Given rational number has a non-terminating, repeating decimal expansion.<\/p>\n

(vi) \\(\\frac{23}{2^3 \\cdot 5^2}\\)
\nSolution:
\n\\(\\frac{23}{2^3 \\times 5^2}\\) It is of the form \\(\\frac{p}{q}\\).
\n\u2235 Here q = 23<\/sup> \u00d7 52<\/sup> which is of the form of 2n<\/sup>5m<\/sup> (n = 3, m = 2).
\n\u2234 Given rational number has a terminating decimal expansion.<\/p>\n

(vii) \\(\\frac{129}{2^2 \\cdot 5^7 \\cdot 7^5}\\)
\nSolution:
\n\\(\\frac{129}{2^2 \\times 5^7 \\times 7^5}\\) It is of the form \\(\\frac{p}{q}\\).
\n\u2235 Here q = 22<\/sup> \u00d7 57<\/sup> \u00d7 75<\/sup> which is not of the form 2n<\/sup>5m<\/sup>.
\n\u2234 Given rational number has a non-terminating, repeating decimal expansion.<\/p>\n

(viii) \\(\\frac{9}{15}\\) = \\(\\frac{3}{5}\\) It is of the form \\(\\frac{p}{q}\\).
\nSolution:
\n\\(\\frac{9}{15}\\) = \\(\\frac{3}{5}\\) It is of the form \\(\\frac{p}{q}\\).
\nHere q = 51<\/sup> which is of the form 2n<\/sup>5m<\/sup> (n = 0; m = 1). .
\n\u2234 Given rational number has a terminating decimal expansion.<\/p>\n

(ix) \\(\\frac{36}{100}\\)
\nSolution:
\n\\(\\frac{36}{100}\\) It is of the form \\(\\frac{p}{q}\\).
\n\\(\\frac{36}{100}\\) = \\(\\frac{36}{2 \\times 2 \\times 5 \\times 5}\\) = \\(\\frac{36}{2^2 5^2}\\)
\n\u2235 Here q = 22<\/sup>. 52<\/sup> which is of the form 2n<\/sup>5m<\/sup> (n = 2, m = 2).
\n\u2234 Given rational number has a terminating decimal expansion.<\/p>\n

(x) \\(\\frac{77}{210}\\)
\nSolution:
\n\\(\\frac{77}{210}\\) it is of the form \\(\\frac{p}{q}\\).
\n\\(\\frac{77}{210}\\) = \\(\\frac{11}{30}\\) = \\(\\frac{11}{2 \\times 3 \\times 5}\\)
\n\u2235 q = 2 \u00d7 3 \u00d7 5 which is not of the form 2n<\/sup>5m<\/sup>
\n\u2234 Given rational number has non-terminating, repeating decimal expansion.<\/p>\n

Question 3.
\nWrite the following rational numbers in decimal form using Theorem 1.4.<\/p>\n

(i) \\(\\frac{13}{25}\\)
\nSolution:
\n\\(\\frac{13}{25}\\) = \\(\\frac{13}{5 \\times 5}\\) = \\(\\frac{13 \\times 2^2}{5^2 \\times 2^2}\\)
\n= \\(\\frac{13 \\quad 4}{10^2}\\) = \\(\\frac{52}{100}\\) = 0.52<\/p>\n

(ii) \\(\\frac{15}{16}\\)
\nSolution:
\n\\(\\frac{15}{16}\\) = \\(\\frac{15}{2 \\times 2 \\times 2 \\times 2}\\) = \\(\\frac{15 \\times 5^4}{2^4 \\times 5^4}\\)
\n= \\(\\frac{15 \\quad 5^4}{10^4}\\) = \\(\\frac{9375}{10000}\\) = 0.9375<\/p>\n

\"TS<\/p>\n

(iii) \\(\\frac{23}{2^3 \\cdot 5^2}\\)
\nSolution:
\n\\(\\frac{23}{2^3 \\times 5^2}\\) = \\(\\frac{23 \\times 5}{2^3 \\times 5^2 \\times 5}\\)
\n= \\(\\frac{115}{10^3}\\) = \\(\\frac{115}{1000}\\) = 0.115<\/p>\n

(iv) \\(\\frac{7218}{3^2 .5^2}\\)
\nSolution:
\n\\(\\frac{7218}{3^2 \\times 5^2}\\) = \\(\\frac{3 \\times 3 \\times 802}{3^2 \\times 5^2}\\)
\n= \\(\\frac{802 \\times 2^2}{5^2 \\times 2^2}\\) = \\(\\frac{3208}{100}\\) = 32.08<\/p>\n

(v) \\(\\frac{143}{110}\\)
\nSolution:
\n\\(\\frac{143}{110}\\) = \\(\\frac{11 \\times 13}{11 \\times 10}\\) = \\(\\frac{13}{10}\\) = 1.3<\/p>\n

Question 4.
\nExpress the following decimal numbers in the form of q and write the prime factors of q. What do you observe?
\n(i) 43.123
\nSolution:
\n43.123456789
\nGiven decimal expansion terminates. Hence given real number is a rational.
\nIt is in the form of \\(\\frac{p}{q}\\)
\n\"TS
\n\u2235 Here q = 29<\/sup>.59<\/sup> ; q is of the form 2n<\/sup>5m<\/sup> (n = 9; m = 9)<\/p>\n

\"TS<\/p>\n

(ii) 0.120112001120001…
\nSolution:
\n0.120120012000120000 …………
\nGiven decimal expansion is not either ter-minating or non-terminating repeating.
\n\u2235 Hence given real number is not rational.<\/p>\n

(iii) \\(43 . \\overline{12}\\)
\nSolution:
\n\\(43 . \\overline{123456789}\\)
\nGiven decimal expansion is non-terminating, repeating.
\nGiven real number is rational and so of the \\(\\frac{p}{q}\\)
\nLet x = \\(43 . \\overline{123456789}\\)
\nx = \\(43 . \\overline{123456789}\\) …… (2)
\nMultiplying both sides of (1) by 1000000000, we get
\nx = 43123456789.123456789 ……. (2)
\nSubstracting (1) from (2) we get
\n999999999 x = 43123456746
\nx = \\(\\frac{43123456746}{999999999}\\)
\nx = \\(\\frac{14374485582}{333333333}\\)
\nq = 333333333 which is not of the form 2n<\/sup>5m<\/sup>.<\/p>\n","protected":false},"excerpt":{"rendered":"

Students can practice TS 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.3 to get the best methods of solving problems. TS 10th Class Maths Solutions Chapter 1 Real Numbers Exercise 1.3 Question 1. Write the following rational numbers in their decimal form and also state which are terminating and which are non-terminating repeating … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[16],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/8227"}],"collection":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/comments?post=8227"}],"version-history":[{"count":7,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/8227\/revisions"}],"predecessor-version":[{"id":8437,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/8227\/revisions\/8437"}],"wp:attachment":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/media?parent=8227"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/categories?post=8227"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/tags?post=8227"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}