Answer:
Given, present age of father = 42 yr and present age of son = 14 yr (a) Required ratio \[\text{=}\frac{\text{Present}\,\text{age}\,\text{of}\,\text{father}}{\text{Present}\,\text{age}\,\text{of}\,\text{son}}=\frac{42yr}{14yr}=\frac{42}{14}\] \[=\frac{42\div 14}{14\div 14}=\frac{3}{1}=3:1\] [\[\because \]HCF of 42 and 14 = 14] (b) When son's age was 12 yr i.e. 2 yr back (because son's present age is 14 yr), then father's age = (42 ? 2) = 40 yr \[\therefore \] Required ratio \[=\frac{2yr\,back\,father's\,age}{2yr\,back\,son's\,age}=\frac{40yr}{12yr}=\frac{40}{12}=\frac{40\div 4}{12\div 4}\] \[\because \]\[40=2\times 2\times 2\times 5\] and \[12=2\times 2\times 3\] \[\therefore \] HCF of 40 and \[12=2\times 2=4=\frac{10}{3}=10:3\] (c) After 10 yr father and son's age will be (42 + 10) yr and (14 + 10) yr respectively i.e. 52 yr and 24 yr. \[\therefore \] Required ratio \[\text{=}\frac{\text{After}\,\text{10yr}\,\text{father }\!\!'\!\!\text{ sage}}{\text{After}\,\text{10yr}\,\text{son }\!\!'\!\!\text{ s}\,\text{age}}=\frac{52yr}{24yr}=\frac{52\div 4}{24\div 4}\] \[\because \] \[52=2\times 2\times 13\] and \[24=2\times 2\times 2\times 3\] \[\therefore \] HCF of 52 and \[24=4=\frac{13}{6}=13:6\] (d) When father's age was 30 yr i.e. 12 yr back (because father's present age is 42 yr and (42 ? 12) = 30 yr. Then, son's age = (14 ? 12) = 2 yr Required ratio \[=\frac{\text{12 yr back father }\!\!'\!\!\text{ s age }}{\text{12 yr back son s age}}\] \[=\frac{30yr}{2yr}=\frac{30}{2}\]\[=\frac{30\div 2}{2\div 2}=\frac{15}{1}=15:1\] [\[\because \]HCF of 30 and 2 = 2]
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