The sum of all five numbers is 30.

Question

The fifth number plus the third number equals fourteen. The fourth number is one more than the second number. The first number is one less than twice the second number. The second number plus the third number equals ten.

Answer 1

Let the numbers be:  a, b, c, d, and e

 

Translate:

  "The sum of all five numbers is 30."   means   a+b+c+d+e = 30     [eq1]

  "The fifth number plus the third number equals fourteen."   means  

              e             +              c                   = 14       [eq2]

 

  "The fourth number is one more than the second number."  means 

                    d         =      1      +                   b       [eq3]

 

  "The first number is one less than twice the second number."  means 

                  a        = - 1         +            2b        [eq4]

 

  "The second number plus the third number equals ten."    means

                b                +              c                =   10       [eq5]

 

The math:

  Notice that values are given in terms of b, so substitute for all but b:

        a       +         b         +      c        + d      +    e    = 30      [eq1]

  (2b-1)      +        b         +      c        +  d      +    e    =  30     [use eq4]

  (2b-1)      +       b          +      c       +  (1+b)  +  e    =  30    [use eq3]

  (2b-1)    +       b            +    (c+e)   +  (1+b)           = 30      [re-arrange]

  (2b)-1)   +      b             +   14        +  (1+b)           = 30      [use eq2]

           4b + 14 = 30             [collect terms]

           4b =  16             [subtract 14 from both sides]

            b = 4                    [divide both sides by 4]

 

   a = 2b - 1         [from eq4]

   a = 2(4) - 1

   a = 7

   

   b+c = 10           [eq5]

   4+c = 10

      c  = 6

 

    e + c = 14    [eq2]

    e + 6 = 14

    e = 8

 

    d = 1 + b    [eq3]

    d = 1 + 4

    d = 5

 

[Note: use eq1-eq5 in any convenient order.]

 

The numbers are:   7, 4, 6, 5, 8

 

Check:

  Is a+b+c+d+e = 30   ?     [eq1]

     7+4+6+5+8 = 30   ?

      30=30   ?yes    

  Is  e+c=14   ?   [eq2]

       8+6=14   ?

        14=14   ?yes

  Is  d=1+b   ?   [eq3]

      5=1+4   ?

      5 = 5   ?yes

 

... check eq4 and eq5 also.