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ABCD - Math Puzzle

by Andrew
(New York)











































The four numbers a < b < c < d can be paired in six different ways. If each pair has a different sum, and if the four smallest sums are 1, 2, 3, and 4,
what are all possible values of d?

Comments for ABCD - Math Puzzle

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Nov 01, 2011
ABCD - Math Puzzle
by: Staff


Question:

by Andrew
(New York)



The four numbers a < b < c < d can be paired in six different ways. If each pair has a different sum, and if the four smallest sums are 1, 2, 3, and 4, what are all possible values of d?



Answer:


Order is not important
No repetition of the same combination

Permutations possible = n!/[r!(n-r)!]

n = 4

r = 2


Permutations possible = 4!/[2!(4-2)!]

= 4!/[2!(4-2)!]

= (4*3*2*1)/(2*1*2*1)

= 6

{a,b} {a,c} {a,d} {b,c} {b,d} {c,d}


The six possibilities for the sum the pairs:

a + b
a + c
a + d
b + c
b + d
c + d


since a < b < c < d, the combinations with the four smallest sums are:

a + b
a + c
a + d
b + c


Therefore, the four smallest sums are:

a + b = 1
a + c = 2
a + d = 3
b + c = 4


This is a linear system of four equations


Solve for c

b + c = 4
b - b + c = 4 - b
0 + c = 4 - b
c = 4 - b

a + c = 2


solve for a

substitute 4 - b for c

a + (4 - b) = 2
a + 4 - b = 2
a + 4 - 4 - b = 2 - 4
a + 0 - b = -2
a - b = -2

add the following two equations:

a + b = 1
a - b = -2
--------------
a + a + b - b = 1 - 2

2a + 0 = -1
2a = -1

divide each side of the equation by 2

2a/2 = -½
a*(2/2) = -½
a*(1) = -½
a = -½


substitute -½ for a in the following equation

a + d = 3
-½ + d = 3
-½ + ½ + d = 3 + ½
0 + d = 3 + ½
d = 3 + ½
d = 3.5


the final answer is: d = 3.5





Thanks for writing.

Staff
www.solving-math-problems.com


Nov 02, 2011
only answer?
by: Anonymous

is 3.5 the only answer?

Nov 02, 2011
Math Puzzle
by: Staff


3.5 is the only answer because it is the only value of d which will satisfy the linear system of four equations:

a + b = 1
a + c = 2
a + d = 3
b + c = 4

although the numerical solutions for all four letters were not included with the answer (all the question asked for was the value of d), they are:

(a=-1/2, b=3/2, c=5/2, d=7/2)



Thanks for writing.

Staff
www.solving-math-problems.com



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