Linear Equation


  1. The difference between the squares of two numbers is 256000 and the sum of the numbers is 1000. The numbers are









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    Let us assume the numbers are a and b.
    According to question,
    a2 - b2 = 256000 .......................(1)
    and a + b = 1000 .........................(2)
    Solve the above equation.

    Correct Option: B

    Let us assume the numbers are a and b.
    According to question,
    a2 - b2 = 256000 .......................(1)
    and a + b = 1000 .........................(2)
    On dividing the Equation (1) with Equation (2),
    (a2 - b2)/(a + b) = 256000/1000
    (a2 - b2)/(a + b) = 256
    (a + b)(a - b)/(a + b) = 256
    a - b = 256...............................(3)
    Add the equation (2) and (3), we will get
    a + b + a - b = 1000 + 256
    2a = 1256
    a = 628
    Put the value of a in equation (2), we will get
    658 + b = 1000
    b = 1000 - 628
    b = 372
    So answer is 628 , 372.


  1. Two-fifths of one -fourth of three -seventh of a number is 15. What is half of that number?









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    Let us assume the number is P.
    According the question,
    three -seventh of a number = 3/7 x P
    one -fourth of three -seventh of a number = 1/4 x 3/7 x P
    Two-fifths of one -fourth of three -seventh of a number = 2/5 x 1/4 x 3/7 x P
    According to question,
    Two-fifths of one -fourth of three -seventh of a number = 15
    2/5 x 1/4 x 3/7 x P = 15
    Solve the above equation.

    Correct Option: D

    Let us assume the number is P.
    According the question,
    three -seventh of a number = 3/7 x P
    one -fourth of three -seventh of a number = 1/4 x 3/7 x P
    Two-fifths of one -fourth of three -seventh of a number = 2/5 x 1/4 x 3/7 x P
    ⇒ 2/5 x 1/4 x 3/7 x P = 15
    ⇒ 2 x 1 x 3 x P = 15 x 7 x 4 x 5
    P = 15 x 7 x 4 x 5 / 2 x 3
    P = 5 x 7 x 2 x 5
    P = 350
    P /2 = 350/2
    P /2 = 175



  1. The numbers are such that the square of one is 224 less than 8 times the square of the other. If the numbers be in the ratio 3 : 4, the numbers are









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    Let us assume the ratio term is r.
    then the number will be 3r and 4r.
    According to question,
    (4r)2 + 224 = 8 x (3r)2
    Solve the equation and find the answer.

    Correct Option: C

    Let us assume the ratio term is r.
    then the number will be 3r and 4r.
    According to question,
    (4r)2 + 224 = 8 x (3r)2
    16r2 + 224 = 8 x 9r2
    16r2 + 224 = 72r2
    72r2 - 16r2 = 224
    56r2 = 224
    r2 = 224/56
    r2 = 4
    r = 2

    First number = 3r = 3 x 2 = 6
    Second number = 4r = 4 x 2 = 8


  1. A person on tour has total 360 for his daily expenses. He decides to extend his tour programme by 4 days which leads to cutting down daily expenses by 3 a day . The numbers of days of tour are :









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    Method 1 to solve this question.
    Let us assume the numbers of days of tour before extending the tour are D and daily expenses is E.
    According to question,
    Total expenses on tour = per day expenses x total number of days
    Total expenses on tour = E x D = ED
    360 = ED
    Again according to question,
    After extending the tour by 4 days, Number of days of tour = D + 4
    Expenses will be reduced by 3 rupees, then everyday expenses = E - 3
    so total expenses = (D + 4) x (E - 3)
    Solve the equation and get the answer.

    Method 2 to solve this question.
    Let us assume the original day of tour is d days.
    Given that his tour is extended for 4 days
    Hence daily expenses per days = 360/(d + 4)
    Therefore, According to question,
    360/d - 360/(d + 4) = 3
    Solve the equation and get the answer.

    Correct Option: B

    Method 1 to solve this question.
    Let us assume the numbers of days of tour before extending the tour are D and daily expenses is E.
    According to question,
    Total expenses on tour = per day expenses x total number of days
    Total expenses on tour = E x D = ED
    360 = ED
    ED = 360
    E = 360/D .................(1)
    Again according to question,
    After extending the tour by 4 days, Number of days of tour = D + 4
    Expenses will be reduced by 3 rupees, then everyday expenses = E - 3
    so total expenses = (D + 4) x (E - 3)
    360 = (D + 4) x (E - 3)
    (D + 4) x (E - 3) = 360 .............................................(2)
    Put the value of E in Equation (2). we will get,
    (D + 4) x (360/D - 3) = 360
    ⇒ (D + 4) x ( (360 - 3D)/D ) = 360
    ⇒ (D + 4) x ( (360 - 3D) ) = 360 x D
    ⇒ (D + 4) x (360 - 3D) = 360 x D
    After multiplication by algebra law,
    ⇒ 360 x D - 3D x D + 4 x 360 - 4 x 3D = 360D
    ⇒ 360 x D - 3D + 1440 - 12D = 360D
    ⇒ - 3D + 1440 - 12D = 360D - 360 x D
    ⇒ - 3D + 1440 - 12D = 0
    ⇒ 3D - 1440 + 12D = 0
    D - 480 + 4D = 0
    D + 4D - 480 = 0
    D + 24D - 20D - 480 = 0
    D(D + 24) - 20(D + 24) = 0
    ⇒ (D + 24) (D - 20) = 0
    Either (D + 24) = 0 or (D - 20) = 0
    So D = - 24 or D = 20
    But days cannot be negative so D = 20 days.

    Method 2 to solve this question.
    Let us assume the original day of tour is d days.
    Given that his tour is extended for 4 days
    Hence daily expenses per days = 360/(d + 4)
    Therefore, According to question,
    360/d - 360/(d + 4) = 3
    ⇒ (360(d + 4) - 360d)/d x (d + 4) = 3
    ⇒ (360(d + 4) - 360d)/d x (d + 4) = 3
    ⇒ 360(d + 4) - 360d = 3d x (d + 4)
    ⇒ 360d + 1440 – 360d = 3(d2 + 4d)
    ⇒ 1440 = 3d2 + 12d
    ⇒ 3d2 + 12d – 1440 = 0
    ⇒ d2 + 4d – 480 = 0
    ⇒ d2 + 24d – 20d – 480 = 0
    ⇒ d(d + 24) – 20(d + 24) = 0
    ⇒ (d + 24)(d – 20) = 0
    ⇒ (d + 24) = 0 or (d – 20) = 0
    ? d = –24 or d = 20
    Since days cannot be negative, d = 20
    Hence his original duration of the tour is 20 days.



  1. The sum of three consecutive odd numbers is 20 more than the first of these numbers.What is the middle number ?









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    Let us assume the first Odd number is a.
    then 2nd odd number = a + 2
    and 3rd odd number = a + 4
    According to question,
    Sum of three consecutive odd numbers is 20 more than the first of these numbers;
    a + (a + 2) + (a + 4) = a + 20
    Solve the equation and get the answer.

    Correct Option: B

    Let us assume the first Odd number is a.
    then 2nd odd number = a + 2
    and 3rd odd number = a + 4
    According to question,
    Sum of three consecutive odd numbers is 20 more than the first of these numbers;
    a + (a + 2) + (a + 4) = a + 20
    ⇒ a + a + 2 + a + 4 - a = 20
    ⇒ 2a + 6 = 20
    ⇒ 2a + 6 = 20
    ⇒ 2a = 20 - 6
    ⇒ 2a = 14
    ⇒ a = 7
    The First odd number a = 7
    The second odd number = a + 2 = 7 + 2 = 9
    Second odd number is middle number = 9