Question 1: Which of the following statements are true?

\displaystyle \text{ i)  }  27:36=4.5:6       \displaystyle \text{ ii)  }  \frac{3}{4} : \frac{15}{16}= \frac{2}{3} : \frac{5}{6}  

\displaystyle \text{ iii)  }  \text{ Rs. 14 : Rs 21=2 pens :3 pens}        \displaystyle \text{ iv)    6.5 km :2.6  km=Rs.  60 :Rs  24}  

Answer:

\displaystyle \text{ i)  }  27:36=4.5:6  

\displaystyle  4.5:6 = \frac{4.5\times 6}{6\times 6} = \frac{27}{36}  

Hence True.

\displaystyle \text{ ii)  }  \frac{3}{4} : \frac{15}{16} = \frac{2}{3} : \frac{5}{6}  

\displaystyle  \frac{3}{4} : \frac{15}{16} = \frac{3}{4} \times \frac{16}{15} = \frac{4}{5}  

\displaystyle  \frac{2}{3} : \frac{5}{6} = \frac{2}{3} \times \frac{6}{5} = \frac{4}{5}  

Hence True

\displaystyle \text{ iii)  }  Rs. 14 : \ Rs 21=2 \ pens :3 \ pens  

\displaystyle  \frac{Rs. \ 14}{Rs. \ 21} = \frac{2}{3}  

\displaystyle  \frac{2 \ pens}{3 \ pens} = \frac{2}{3}  

Hence True

\displaystyle \text{ iv)  }  6.5 \ km :2.6 \ km=Rs. \ 60 :Rs \ 24  

\displaystyle  \frac{6.5 \ km}{2.6 \ km} = \frac{5}{2}  

\displaystyle  \frac{Rs. \ 60}{Rs. \ 24} = \frac{5}{2}  

Hence True.

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Q.2. Check whether the following numbers are in proportion or not:

\displaystyle \text{ i)  }  8, 12, 18, 24       \displaystyle \text{ ii)  }  6.4, 3.6, 4.8, 2. 7       \displaystyle \text{ iii)  }  11 \frac{1}{3} , 9 \frac{1}{3} , 8 \frac{1}{2} , 7  

\displaystyle \text{ iv)  }  0.36, 1.8, 6.4, 32       \displaystyle \text{ v)  }  \frac{3}{4} , \frac{5}{6} , \frac{7}{8} , \frac{9}{10}  

Answer:

\displaystyle \text{ i)  }  8, 12, 18, 24  

We have:

\displaystyle  8 : 12 = 2 : 3  

\displaystyle  18 : 24 = 3: 4  

Therefore \displaystyle  8 : 12 \neq 18 : 24  

\displaystyle \text{ Hence }  8, 12, 18, 24 \text{ are not in proportion }

\displaystyle \text{ ii)  }  6.4, 3.6, 4.8, 2. 7  

We have

\displaystyle  6.4:3.6= \frac{64}{36} = \frac{32}{18} = \frac{16}{9}  

\displaystyle  4.8 :2.7= \frac{48}{27} = \frac{16}{9}  

Therefore \displaystyle  6.4 :3.6=4.8 :27  

\displaystyle \text{ Hence }  6.4, 3.6, 4.8, 2.7 \text{ are in proportion } .

\displaystyle \text{ iii)  }  11 \frac{1}{3} , 9 \frac{1}{3} , 8 \frac{1}{2} , 7  

We have

\displaystyle  11 \frac{1}{3} : 9 \frac{1}{3} = \frac{34}{3} : \frac{28}{3} = \frac{34}{28} = \frac{17}{14}  

\displaystyle  8 \frac{1}{2} : 7= \frac{17}{2} :7= \frac{17}{14}  

Therefore \displaystyle  11 \frac{1}{3} :9 \frac{1}{3} = 8 \frac{1}{2} : 7  

\displaystyle \text{ Hence }  11 \frac{1}{3} , 9 \frac{1}{3} , 8 \frac{1}{2} , 7  \text{ are in proportion }

\displaystyle \text{ iv)  }  0.36, 1.8, 6.4, 32  

We have:

\displaystyle  0.36 :1.8= \frac{36}{180} = \frac{1}{5}  

\displaystyle  6.4 :32= \frac{64}{320} = \frac{1}{5}  

Therefore \displaystyle  0.36 : 1.8 = 6.4 : 32  

\displaystyle \text{ Hence }  0.36, 1.8, 6.4, 32 \text{ are in proportion }

\displaystyle \text{ v)  }  \frac{3}{4} , \frac{5}{6} , \frac{7}{8} , \frac{9}{10}  

We have:

\displaystyle  \frac{3}{4} :\frac{5}{6} = \frac{3}{4}\times \frac{6}{5} = \frac{9}{10}  

\displaystyle  \frac{7}{8}:\frac{9}{10} = \frac{7}{8}\times \frac{10}{9} = \frac{70}{72}  

Therefore \displaystyle  \frac{3}{4} : \frac{5}{6} \neq \frac{7}{8} : \frac{9}{10}  

\displaystyle \text{ Hence }  \frac{3}{4} , \frac{5}{6} , \frac{7}{8} , \frac{9}{10} \text{ are not in proportion }

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Q.3. Find the value of x in each of the following:

\displaystyle \text{ i)  }  8 :x :: 6:27       \displaystyle \text{ ii)  }  5.6 : 3.5 : :x : 1.25       \displaystyle \text{ iii)  }  1 \frac{4}{5} : 2 \frac{4}{5} :: x :3 \frac{1}{2}       \displaystyle \text{ iv)  }  \frac{2}{3} : \frac{4}{7} ::1 \frac{5}{6} :x  

Answer:

\displaystyle \text{ i)  }  8 :x :: 6:27  

\displaystyle \text{ We have  }  \frac{8}{x} = \frac{6}{27} \text{ or  }x=  \frac{ 8 \times 27}{6} =36  

\displaystyle \text{ ii)  }  5.6 : 3.5 : :x : 1.25  

\displaystyle \text{ We have  }  \frac{56}{35} = \frac{x}{1.25} \text{ or  }x=  \frac{56\times 1.25}{35} = 2  

\displaystyle \text{ iii)  }  1 \frac{4}{5} : 2 \frac{4}{5} :: x :3 \frac{1}{2}  

\displaystyle \text{ We have  }  \frac{9}{5} : \frac{14}{5} = x : \frac{7}{2}  

\displaystyle  \frac{9}{14} = \frac{2x}{7} \text{ or  }x=  \frac{9\times 7}{14\times 2} = \frac{9}{4}  

\displaystyle \text{ iv)  }  \frac{2}{3} : \frac{4}{7} ::1 \frac{5}{6} :x 

\displaystyle \text{ We have  }  \frac{2}{3} \times \frac{7}{4}= \frac{11}{6x} \text{ or  }x=  \frac{11}{7}  

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Q.4. Find the fourth proportional to:

\displaystyle \text{ i)  }  2.8, 14 \text{ and }3.5       \displaystyle \text{ ii)  }  3 \frac{1}{3} , 1 \frac{2}{3} , 2 \frac{1}{2}       \displaystyle \text{ iii)  }  1 \frac{5}{7} , 2 \frac{3}{14} , 3 \frac{3}{5}       \displaystyle \text{ iv)  }  1 \frac{1}{5} , 1 \frac{3}{5} , 2.1  

Answer:

\displaystyle \text{ i)  }  2.8, 14 \text{ and }3.5  

Let the fourth proportional term be \displaystyle  x  

\displaystyle \text{ We have  }  \frac{2.8}{14} = \frac{3.5}{x} \text{ or  }x= \frac{35}{2}  

\displaystyle \text{ ii)  }  3 \frac{1}{3} , 1 \frac{2}{3} , 2 \frac{1}{2}  

Let the fourth proportional term be \displaystyle  x  

\displaystyle  3 \frac{1}{3} :1 \frac{2}{3} = 2 \frac{1}{2} :x  

\displaystyle  \frac{10}{3} : \frac{5}{3} = \frac{5}{2} :x  

\displaystyle  \frac{10}{3} \times \frac{3}{5} = \frac{5}{2x} \text{ or  }x= \frac{5}{4}  

\displaystyle \text{ iii)  }  1 \frac{5}{7} , 2 \frac{3}{14} , 3 \frac{3}{5}  

Let the fourth proportional term be \displaystyle  x  

\displaystyle  1 \frac{5}{7} : 2 \frac{3}{14} = 3 \frac{3}{5} :x  

\displaystyle  \frac{12}{7} : \frac{31}{14} = \frac{18}{5} :x \text{ or  }x= \frac{93}{20}  

\displaystyle \text{ iv)  }  1 \frac{1}{5} , 1 \frac{3}{5} , 2.1  

Let the fourth proportional term be \displaystyle  x  

\displaystyle  1 \frac{1}{5} :1 \frac{3}{5} :: 2.1 :x  

\displaystyle  \frac{6}{5} : \frac{8}{5} :: 2.1 :\text{ or  }x=2.8  

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Q.5. Find the third proportional to:

\displaystyle \text{ i)  }  12, 16       \displaystyle \text{ ii)  }  4.5, 6       \displaystyle \text{ iii)  }  5 \frac{1}{2} , 16 \frac{1}{2}       \displaystyle \text{ iv)  }  3 \frac{1}{2} ,8 \frac{3}{4}  

Answer:

\displaystyle \text{ i)  }  12, 16  

\displaystyle \text{ Let the third proportional to  }   12 \text{ and }16 \text{ be }x  

Then, \displaystyle  12 :16 :: 16 :x \text{ or  }x= \frac{64}{3}  

\displaystyle \text{ ii)  }  4.5, 6  

\displaystyle \text{ Let the third proportional to  }   4.5 \text{ and }6 \text{ be } x  

\displaystyle \text{ Then }  4.5 :6 ::6 :x \text{ or  }x=8  

\displaystyle \text{ iii)  }  5 \frac{1}{2} , 16 \frac{1}{2}  

\displaystyle \text{ Let the third proportional to  }   5 \frac{1}{2} , 16 \frac{1}{2} \text{ be } x  

\displaystyle \text{ Then }  5 \frac{1}{2} : 16 \frac{1}{2} :: 16 \frac{1}{2} :x \text{ or  }x= \frac{99}{2}  

\displaystyle \text{ iv)  }  3 \frac{1}{2} ,8 \frac{3}{4}  

\displaystyle \text{ Let the third proportional to  }   3 \frac{1}{2} , 8 \frac{3}{4} \text{ be } x  

\displaystyle \text{ Then }  3 \frac{1}{2} : 8 \frac{3}{4} :: 8 \frac{3}{4} :x \text{ or  }x= 21 \frac{7}{8}  

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Q.6. Find the mean proportional between:

\displaystyle \text{ i)  }  8 \text{ and }18       \displaystyle \text{ ii)  }  0.3 \text{ and }2. 7       \displaystyle \text{ iii)  }  66 \frac{2}{3} \text{ and }6       \displaystyle \text{ iv)  }  1.25 \text{ and } 0.45       \displaystyle \text{ v)  }  \frac{1}{7} \text{ and } \frac{4}{63}  

Answer:

\displaystyle \text{ i)  }  8 \text{ and }18  

\displaystyle \text{ Mean proportional between  }   8 \text{ and }18 = \sqrt{8\times 18} = \sqrt{144} =12  

\displaystyle \text{ ii)  }  0.3 \text{ and }2. 7  

\displaystyle \text{ Mean proportional between  }   0.3 \text{ and }2.7 = \frac{\sqrt{3\times 27}}{10} \frac{9}{10} =0.9 

\displaystyle \text{ iii)  }  66 \frac{2}{3} \text{ and }6  

\displaystyle \text{ Mean proportional between  }   66 \frac{2}{3} \text{ and }6 = \sqrt{\frac{200}{3}\times 6}   \sqrt{400} = 20  

\displaystyle \text{ iv)  }  1.25 \text{ and }0.45  

\displaystyle \text{ Mean proportional between  }   1.25 \text{ and } 0.45 = \sqrt{1.25 \times 0.45}=0.75  

\displaystyle \text{ v)  }  \frac{1}{7} \ and \ \frac{4}{63}  

\displaystyle \text{ Mean proportional between  }   \frac{1}{7} \text{ and }\frac{4}{63} = \frac{\sqrt{1\times 4}}{7\times 63} = \frac{2}{21}  

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Q.7. If \displaystyle  28 is the third proportional to \displaystyle  7 \text{ and }x , find the value of \displaystyle  x

Answer:

\displaystyle  7:x ::x :28 \text{ or  }x^2=7 \times 28 \text{ or  }x=14  

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Q.8. If \displaystyle  18, x, 50 are in continued proportion, find the value of \displaystyle  x

Answer:

\displaystyle  x= \sqrt{18\times 50}=30  

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Q.9. A rod was cut into two pieces in the ratio \displaystyle  7: 5 . If the length of the smaller piece was \displaystyle  45.5 \ cm , then find the length of the longer piece.

Answer:

\displaystyle \text{ We have  }  7 :5 ::x :45.5 \text{ or  }x=63.7  

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Q.10. The areas of two rectangular fields are in the ratio \displaystyle  5: 9 . Find the area of the smaller field if that of the larger field is \displaystyle  2331 sq. meters.

Answer:

\displaystyle \text{ We have  }  5 :9 ::x :2331 \text{ or  }x=1295  

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Q.11. What number must be subtracted from each of the numbers \displaystyle  41, 55, 36, 48 so that the differences are proportional?

Answer:

Let the number to be subtracted \displaystyle  = x  

Therefore

\displaystyle  \frac{41-x}{55-x} = \frac{36-x}{48-x}  

\displaystyle  41 \times 48-48 x-41 x+ x^2=36\times 55-55x-36x+ x^2  

solving for \displaystyle  x \text{ we get } x=2  

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Q.12. An alloy is to contain copper and zinc in the ratio \displaystyle  9 : 4 . Find the quantity of zinc to be melted with \displaystyle  2 \frac{2}{5} kg of copper, to get the desired alloy.

Answer:

Let the quantity of zinc be \displaystyle  x  

\displaystyle \text{ We have  }  9 :4=2 \frac{2}{5} : x \text{ or  } x = \frac{12\times 4}{5\times 9} = \frac{16}{15} \text{ or  } \frac{1 1}{15}