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| Proportion |
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| Four quantities a, b, c and d are said to be in proportion, if a:b=c:d. |
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| a and d are called extremes of the proportion. |
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| b and c are called means of the proportion. |
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| a:b=c:d |
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| Product of extremes is equal to the product of means in a proportion. In a proportion a:b=c:d, the fourth term 'd' is called the fourth proportional. |
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| The proportion a:b = c:d is often expressed as a:b::c:d and is read as a is to b is the same as c is to d. |
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| Three quantities are said to be in proportion if the ratio of the first to the second is equal to the ratio of the second and third. |
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| a, b and c are said to be in continued proportion if a:b = b:c |
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| If a, b and c are in continued proportion then b (second quantity) is called the mean proportion and c (third quantity) is called the third proportional of a and b. |
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| 1) Find the value of x if 2:5::x:15. |
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| Suggested answer: |
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| 5x=2 x 15 |
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| x = 6 |
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| 2) What must be added to each of the four numbers 10, 18, 22, 38 to make them a proportion? |
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| Suggested answer: |
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| Let x be added. |
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| 380+48x+x2=396+40x+x2 |
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| 8x=16 |
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| x=2 |
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| 3) Find the fourth proportional to 6, 10 and 12. |
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| Suggested answer: |
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| Let the fourth proportional to be x. |
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| 6:10=12:x |
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| 6x=120 (Product of extremes is equal to product of means) |
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| x=20 |
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| 4) Find the mean proportional to 6 and 54. |
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| Suggested answer: |
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| Let x be the mean proportional. |
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| x2= 6 x 54 |
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| =2 x 3 x 3=18 |
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| x=18 |
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| 5) Find the third proportional to 5 and 10. |
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| Suggested answer: |
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| Let x be the third proportional to 5 and 10. |
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| 5:10=10:x. |
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| 5x=10 x 10 (product of extreme is equal to product of means) |
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| 6) Find two numbers such that the mean proportional between them is 24 and the third proportional to them is 192. |
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| Suggested answer: |
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| Let the two numbers be x and y. |
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| 24 is the mean proportional of x and y |
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| x.y=(24)2 …1 |
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| 192 is the third proportional to x and y. |
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| y3= (24)2.24 x 8 |
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| y3= (24)3.23 |
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| y=24 x 2=48. |
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| \ the numbers are 12 and 48. |
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| 7) If a:b::b:c, prove that a:c=a2:b2. |
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| Suggested answer: |
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| a:b::b:c |
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 b2=ac |
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| Suggested answer: |
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| a, b, c are in continued proportion. |
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| Consider the product, |
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| (a+b) (a-b)=a2-b2 |
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| =a2-ac |
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| =a(a-c) |
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| (a+b) (a-b)=a(a-c) |
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