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Get Real, Find The Triplets (Posted on 2007-05-21)

Determine all possible triplets (a, b, c) of real numbers satisfying the following system of equations, where [n] denotes the greatest integer ≤n, and {n}=n–[n].

{a}	+	b	+	[c]	=	5.5
a	+	[b]	+	{c}	=	6.7
[a]	+	{b}	+	c	=	8.4

Submitted by K Sengupta

Rating: 3.5000 (2 votes)

Solution: (Hide)

(i) {a}+ b + [c] = 5.5
(ii) a + [b] + {c} = 6.7
(iii) [a] + {b}+ c = 8.4

Adding (i), (ii) and (iii), we obtain:
2(a+b+c) = 20.6 (since a = [a]+{a}; b =[b]+[b], and so on)

Separately subtracting (i), (ii) and (iii) in turn from (iv), we obtain:

[a] + {c} = 4.8; {b}+ [c] = 3.6; {a}+ [b] =1.9

Now, we know that [n] must be an integer, and accordingly:

[a] + {c} = 4.8, gives [a] = 4 and {c}= 0.8.
{b}+ [c] = 3.6, gives [c] = 3 and {b} = 0.6
{a}+ [b] =1.9 gives [b] = 1 and {a} = 0.9

Consequently:

a = [a]+{a} = 4+0.9 = 4.9

b = [b] + {b} = 1.6

c = [c] + {c} = 3.8

Comments: ( You must be logged in to post comments.)

	Subject	Author	Date
	Solutions	Old Original Oskar!	2007-05-21 10:06:45

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