Question 1:  Which of the following statements are true?

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

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

Answer:

i) 27:36=4.5:6

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

Hence True.

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

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

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     ii) 6.4, 3.6, 4.8, 2. 7     iii) 11 \frac{1}{3} , 9 \frac{1}{3} , 8 \frac{1}{2} , 7

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

Answer:

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

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.

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

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

v) \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      ii) 5.6 : 3.5 : :x : 1.25      iii) 1 \frac{4}{5} : 2 \frac{4}{5}  ::  x :3 \frac{1}{2}      iv) \frac{2}{3} :  \frac{4}{7}  ::1 \frac{5}{6}  :x

Answer:

i) 8 :x :: 6:27 

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

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 

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} 

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     ii) 3 \frac{1}{3} , 1 \frac{2}{3} , 2 \frac{1}{2}      iii) 1 \frac{5}{7} , 2 \frac{3}{14} , 3 \frac{3}{5}     iv) 1 \frac{1}{5} , 1 \frac{3}{5} , 2.1

Answer:

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}

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}

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}  

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

Answer:

i) 12, 16 

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

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

ii) 4.5, 6 

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

Then 4.5 :6 ::6 :x \ or\ x=8 

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}

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:

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

Answer:

i) 8 \ and\  18  

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

ii) 0.3 \ and\  2. 7

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

iii) 66 \frac{2}{3} \ and\  6

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

iv) 1.25 \ and\  0.45

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

v) \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 x

Answer:

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

Answer:

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.

Answer:

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.

Answer:

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?

Answer:

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.

Answer:

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}