Question 1:  Which of the following statements are true?

i) 27:36=4.5:6

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

Hence True.

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

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

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

Hence True

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iii) Rs. 14 : \ Rs 21=2 \ pens :3 \ pens

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

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

Hence True

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iv) 6.5 \ km :2.6 \ km=Rs. \ 60 :Rs \ 24

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

\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:

i) 8, 12, 18, 24

We have:

8 : 12 = 2 : 3

18 : 24 = 3: 4

Therefore \ 8 : 12 \neq 18 : 24

Hence 8, 12, 18, 24 are not in proportion

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ii) 6.4, 3.6, 4.8, 2. 7

We have

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

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

Therefore 6.4 :3.6=4.8 :27

Hence 6.4, 3.6, 4.8, 2.7 are in proportion.

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

We have

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

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

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

Hence 11 \frac{1}{3}, 9 \frac{1}{3}, 8 \frac{1}{2}, 7 are in proportional

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iv) 0.36, 1.8, 6.4, 32

We have:

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

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

Therefore  0.36 : 1.8 = 6.4 : 32

Hence 0.36, 1.8, 6.4, 32 are in proportion

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vi) \frac{3}{4}, \frac{5}{6}, \frac{7}{8}, \frac{9}{10} 

We have:

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

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

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

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

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

i) 8 :x :: 6:27 

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

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ii) 5.6 : 3.5 : :x : 1.25 

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

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

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

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

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

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

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Q.4. Find the fourth proportional to:

i) 2.8, 14 \ and\ 3.5 

Let the fourth proportional term be x 

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

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

Let the fourth proportional term be x 

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

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

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

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

Let the fourth proportional term be x 

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

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

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1 \frac{1}{5}, 1 \frac{3}{5}, 2.1 

Let the fourth proportional term be x 

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

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

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Q.5. Find the third proportional to:

i) 12, 16 

Let the third proportional to 12 \ and\ 16 \ be \ x 

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

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ii) 4.5, 6 

Let the third proportional to 4.5 \ and\ 6 \ be\ x 

Then 4.5 :6 ::6 :x \ or\ x=8 

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iii) 5 \frac{1}{2}, 16 \frac{1}{2} 

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

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

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iv) 3 \frac{1}{2},8 \frac{3}{4} 

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

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

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Q.6. Find the mean proportional between:

8 \ and\  18  

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

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0.3 \ and\  2. 7 

Mean proportional between 0.3 \ and\  2.7 =  \frac{\sqrt{3\times 27=81}}{10}=0.9 

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66 \frac{2}{3} \ and\  6

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

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1.25 \ and\  0.45

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

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\frac{1}{7} \ and \  \frac{4}{63}

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

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

7:x ::x :28 \ or\  x^2=7 \times 28 \ or\  x=14 

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

x= \sqrt{18\times 50}=30 

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

We have 7 :5 ::x :45.5 \ or\  x=63.7 

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

We have 5 :9 ::x :2331 \ or\  x=1295

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

Let the number to be subtracted = x 

Therefore

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

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

solving for x \ we \ get \ x=2  

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

Let the quantity of zinc be x

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

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