Ratio, Proportion

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  1. If P =
    4xy
    		 , find the value of
    x + y

    P + 2x
    		 +
    P + 2y
    		 .
    P − 2xP − 2y








  1. View Hint View Answer Discuss in Forum

    We have

    P =
    4xy
    		 =
    2x × 2y
    x + yx + y

    ⇒ 
    P
    		 =
    2y
    2xx + y

    and,  
    P
    		 =
    2x
    2yx + y

    Now ,  
    P
    		 =
    2y
    2xx + y

    On applying componendo and dividendo , we have
    P + 2x
    		 =
    2y + x + y
    P − 2x2y − x − y

    =
    x + 3y
    		       ...(i)
    y − x

    Again ,  
    P
    		 =
    2x
    2yx + y

    ⇒ 
    P + 2y
    		 =
    2x + x + y
    P − 2y2x − x − y

    =
    3x + y
    		       ...(ii)
    x − y

    (i) and (ii)
    P + 2x
    		 +
    P + 2y
    P − 2xP − 2y

    =
    x + 3y
    		 +
    3x + y
    y − xx − y

    =
    3x + y
    		 +
    x + 3y
    x − yy − x

    =
    3x + y
    		 −
    x + 3y
    x − yx − y

    =
    3x + y − x −3y
    x − y

    =
    2x − 2y
    		 = 2
    x − y

    Correct Option: C

    We have

    P =
    4xy
    		 =
    2x × 2y
    x + yx + y

    ⇒ 
    P
    		 =
    2y
    2xx + y

    and,  
    P
    		 =
    2x
    2yx + y

    Now ,  
    P
    		 =
    2y
    2xx + y

    On applying componendo and dividendo , we have
    P + 2x
    		 =
    2y + x + y
    P − 2x2y − x − y

    =
    x + 3y
    		       ...(i)
    y − x

    Again ,  
    P
    		 =
    2x
    2yx + y

    ⇒ 
    P + 2y
    		 =
    2x + x + y
    P − 2y2x − x − y

    =
    3x + y
    		       ...(ii)
    x − y

    (i) and (ii)
    P + 2x
    		 +
    P + 2y
    P − 2xP − 2y

    =
    x + 3y
    		 +
    3x + y
    y − xx − y

    =
    3x + y
    		 +
    x + 3y
    x − yy − x

    =
    3x + y
    		 −
    x + 3y
    x − yx − y

    =
    3x + y − x −3y
    x − y

    =
    2x − 2y
    		 = 2
    x − y

  1. 4a + 9b
    		 =
    4c + 9d
    		 then
    a
    		 = ?
    4a − 9b4c − 9db








  1. View Hint View Answer Discuss in Forum

    Here,  
    4a + 9b
    		 =
    4c + 9d
    4a − 9b4c − 9d

    On applying componendoand dividendo , we have
    4a + 9b + 4a − 9b
    4a + 9b − 4a + 9b

    =
    4c + 9d + 4c − 9d
    4c + 9d − 4c + 9d

    ⇒ 
    8a
    		 =
    8c
    		 ⇒
    a
    		 =
    c
    18b18dbd

    Correct Option: B

    Here,  
    4a + 9b
    		 =
    4c + 9d
    4a − 9b4c − 9d

    On applying componendoand dividendo , we have
    4a + 9b + 4a − 9b
    4a + 9b − 4a + 9b

    =
    4c + 9d + 4c − 9d
    4c + 9d − 4c + 9d

    ⇒ 
    8a
    		 =
    8c
    		 ⇒
    a
    		 =
    c
    18b18dbd

  1. If b is the mean proportional between a and c , then
    a2 − b2 + c2
    		 = ?
    a−2 − b−2 + c−2








  1. View Hint View Answer Discuss in Forum

    Here b is the mean proportional between a and c .
    ∴  a : b : : b : c

    ⇒ 
    a
    		 =
    b
    		 ⇒ b2 = ac
    bc

    Now,  
    a2 − b2 + c2
    a−2 − b−2 + c−2

    =
    a2 − b2 + c2
    1
    		 −
    1
    		 +
    1
    a2b2c2

    =
    a2 − ac + c2
    1
    		 −
    1
    		 +
    1
    a2acc2

    =
    a2 − ac + c2
    c2 − ac + a2
    a2c2

    =
    (a2 − ac + c2) a2c2
    		 = a2c2
    c2 − ac + a2

    (ac)2 = (b2)2 = b4

    Correct Option: A

    Here b is the mean proportional between a and c .
    ∴  a : b : : b : c

    ⇒ 
    a
    		 =
    b
    		 ⇒ b2 = ac
    bc

    Now,  
    a2 − b2 + c2
    a−2 − b−2 + c−2

    =
    a2 − b2 + c2
    1
    		 −
    1
    		 +
    1
    a2b2c2

    =
    a2 − ac + c2
    1
    		 −
    1
    		 +
    1
    a2acc2

    =
    a2 − ac + c2
    c2 − ac + a2
    a2c2

    =
    (a2 − ac + c2) a2c2
    		 = a2c2
    c2 − ac + a2

    (ac)2 = (b2)2 = b4

  1. The ratio between two numbers is 3 : 4 . If their LCM is 120, find the numbers .








  1. View Hint View Answer Discuss in Forum

    Let the numbers be 3x and 4x .
    Then , LCM of 3x and 4x
    = 3 × 4 × x = 12x
    ∴  12x = 120
    ⇒  x = 10
    So the numbers are 3x
    = 3 × 10 = 30 and,
    4x = 4 × 10 = 40

    Correct Option: D

    Let the numbers be 3x and 4x .
    Then , LCM of 3x and 4x
    = 3 × 4 × x = 12x
    ∴  12x = 120
    ⇒  x = 10
    So the numbers are 3x
    = 3 × 10 = 30 and,
    4x = 4 × 10 = 40

  1. What must be added to the numbers 10, 20, 30 and 50 so that the sums are proportional ?








  1. View Hint View Answer Discuss in Forum

    Let x be added in each number to make them proportional.
    ∴  10 + x : 20 + x :: 30 + x : 50 + x
    ⇒  (10 + x ) (50 + x )
    = (20 + x ) (30 + x )
    ⇒  500 + 50 x + 10x + x2
    = 600 + 20x + 30x + x2
    ⇒  500 + 60x + x2
    = 600 + 50x + x2
    ⇒  60x – 50x = 600 – 500 = 100
    ⇒  10x = 100

    ⇒  x =
    100
    		 = 10
    10

    Correct Option: C

    Let x be added in each number to make them proportional.
    ∴  10 + x : 20 + x :: 30 + x : 50 + x
    ⇒  (10 + x ) (50 + x )
    = (20 + x ) (30 + x )
    ⇒  500 + 50 x + 10x + x2
    = 600 + 20x + 30x + x2
    ⇒  500 + 60x + x2
    = 600 + 50x + x2
    ⇒  60x – 50x = 600 – 500 = 100
    ⇒  10x = 100

    ⇒  x =
    100
    		 = 10
    10