question_answer 17)

Determine, if the following are in proportion. (a) 15, 45, 40, 120             (b) 33, 121, 9, 96               (c) 24, 28, 36, 48                (d) 32, 48, 70, 210 (e) 4, 6, 8, 12      (f) 33, 44, 75, 100 TIPS Firstly, find the ratio between two terms of each pair. If the ratio of both pairs is same, then they are in proportion, otherwise not.   

Answer:

                (a) We have, 15, 45, 40, 120 \[\therefore \]Ratio of 15 to 45 \[=\frac{15}{45}=\frac{15\div 15}{45\div 15}\]     [\[\because \]HCF of 15 and 45 = 15] \[=\frac{1}{3}=1:3\] and ratio of 40 to 120 \[=\frac{40}{120}=\frac{40\div 40}{120\div 40}=\frac{1}{3}=1:3\] [\[\because \]HCF of 40 and \[120=\frac{1}{3}=1:3\]] Here, 15 : 45 = 40 : 120 = 1 : 3 Therefore, 15, 45, 40, and 120 are in proportion. (b) We have, 33, 121, 9, 96 \[\therefore \] Ratio of 33 to \[121=\frac{33}{121}=\frac{33\div 11}{121\div 11}\]                              [\[\because \] HCF of 33 and 121 = 11] and ratio of 9 to 96 \[=\frac{9}{96}=\frac{9\div 3}{96\div 3}\]                       [\[\because \]HCF of 9 and 96 = 3] \[=\frac{3}{32}=3:32\] Here,\[3:11\ne 3:32\] i.e. \[33:121\ne 9:96\] Therefore, 33, 121, 9 and 96 are not in proportion. (c) We have, 24, 28, 36, 48 \[\therefore \] Ratio of 24 to 28 \[=\frac{24}{28}=\frac{24\div 4}{28\div 4}\]                         [\[\because \]HCF of 24 and\[28=2\times 2=4\]] \[=\frac{6}{7}=6:7\] and ratio of 36 to 48 \[=\frac{39}{48}=\frac{36\div 12}{48\div 12}=\frac{3}{4}=3:4\] [\[\because \]HCF of 36 and \[48=2\times 2\times 3=12\]] Here,\[6:7\ne 3:4\] i.e.\[24:28\ne 36:48\] Therefore, 24, 28, 36 and 48 are not in proportion. (d) We have, 32, 48, 70, 210 \[\therefore \]Ratio of 32 to \[48=\frac{32}{48}=\frac{32\div 16}{48\div 16}=\frac{2}{3}=2:3\] [\[\therefore \]HCF of 32 and \[48=2\times 2\times 2\times 2=16\]] and ratio of 70 to \[210=\frac{70}{210}=\frac{70\div 70}{210\div 70}=\frac{1}{3}=1:3\] [\[\because \]HCF of 70 and\[210=2\times 5\times 7=70\]] Here, \[2:3\ne 1:3\]i.e.\[32:48\ne 70:210\] Therefore, 32, 48, 70 and 210 are not in proportion. (e) We have; 4, 6, 8, 12 \[\therefore \] Ratio of 4 to \[6=\frac{4}{6}=\frac{4\div 2}{6\div 2}=\frac{2}{3}=2:3\] [dividing numerator and denominator both by 2] and ratio of 8 to\[12=\frac{8}{6}=\frac{8\div 4}{12\div 4}=\frac{2}{3}=2:3\] [dividing numerator and denominator both by 4] Here, 4 : 6 = 8 : 12 = 2 : 3 Therefore, 4, 6, 8 and 12 are in proportion. (f) We have 33, 44, 75, 100 \[\therefore \] Ratio of 33 to \[44=\frac{33}{44}=\frac{33\div 11}{44\div 11}\] [\[\because \]HCF of 33 and 44 = 11] \[=\frac{3}{4}=3:4\] and ratio of 75 to \[100=\frac{75}{100}=\frac{75\div 25}{100\div 25}=\frac{3}{4}=3:4\] [\[\because \]HCF of 75 and\[100=5\times 5=25\]] Here, 33 : 44 = 75 : 100 = 3 : 4 Therefore, 33, 44, 75 and 100 are in proportion.