This is a generalization of
Stopping for lunch
Abe and Ben are on a road trip, and stop at a roadside picnic table for lunch. Abe has P sandwiches. Bert has Q sandwiches. Cal comes along and asks if they would share the sandwiches between the three of them. And they did, each got (P+Q)/3 sandwiches. When Cal is finished he leaves P+Q dollars to pay for the sandwiches.
Abe and Ben thereafter divide the money according to the ratio of the sandwiches each of them gave to Cal.
(For example, if (P, Q) = (5, 3), then Abe gives 5-8/3=7/3 sandwiches and, Ben gives 3 -8/3=1/3 sandwiches to Cal. Therefore, the 8 dollars which has been contributed by Cal should be divided in the ratio 7:1, so that Abe receives 7 dollars and, Ben receives 1 dollar.)
It is observed that Abe’s share of the money is precisely R times that of Ben.
Given that each of P, Q and R is a
positive integer with Q < P < 2Q:
(i) Express P in terms of Q whenever, gcd(P, Q) = 1 and R = 1 (mod 3)
(ii) If (P, R) =(Q+m, m+1), then determine Q in terms of m.
DECLARE FUNCTION gcd# (a#, b#)
DEFDBL A-Z
CLS
FOR tot = 3 TO 100
FOR q = -INT(-tot / 3) TO INT(tot / 2)
p = tot - q
IF p > q AND p < 2 * q THEN
dollars = p + q
abegave = p - (p + q) / 3
bengave = q - (p + q) / 3
r = abegave / bengave
rr = INT(r + .5)
IF ABS(rr - r) < .000001# THEN
r = rr
IF gcd(p, q) = 1 AND r MOD 3 = 1 THEN
PRINT p; q,
PRINT USING "#####.####"; abegave; bengave;
PRINT r,
PRINT USING "####.##"; (p + q) * r / (r + 1); (p + q) / (r + 1); dollars
END IF
END IF
END IF
NEXT q
NEXT tot
STOP
PRINT
FOR tot = 3 TO 100
FOR q = -INT(-tot / 3) TO INT(tot / 2)
p = tot - q
IF p > q AND p < 2 * q THEN
dollars = p + q
abegave = p - (p + q) / 3
bengave = q - (p + q) / 3
r = abegave / bengave
rr = INT(r + .5)
m = p - q
r = rr
IF r = m + 1 THEN
PRINT p; q; m,
PRINT USING "#####.####"; abegave; bengave;
PRINT r,
PRINT USING "####.##"; (p + q) * r / (r + 1); (p + q) / (r + 1); dollars
END IF
END IF
NEXT q
NEXT tot
FUNCTION gcd (a, b)
x = a: y = b
DO
q = INT(x / y)
r = x - y * q
x = y: y = r
LOOP UNTIL r = 0
gcd = x
END FUNCTION
finds
for part (i):
Abe Ben Abe's Ben's Total
p q gave Gave r share share Dollars
3 2 1.3333 0.3333 4 4.00 1.00 5.00
5 3 2.3333 0.3333 7 7.00 1.00 8.00
7 4 3.3333 0.3333 10 10.00 1.00 11.00
9 5 4.3333 0.3333 13 13.00 1.00 14.00
11 6 5.3333 0.3333 16 16.00 1.00 17.00
13 7 6.3333 0.3333 19 19.00 1.00 20.00
15 8 7.3333 0.3333 22 22.00 1.00 23.00
17 9 8.3333 0.3333 25 25.00 1.00 26.00
19 10 9.3333 0.3333 28 28.00 1.00 29.00
21 11 10.3333 0.3333 31 31.00 1.00 32.00
23 12 11.3333 0.3333 34 34.00 1.00 35.00
25 13 12.3333 0.3333 37 37.00 1.00 38.00
27 14 13.3333 0.3333 40 40.00 1.00 41.00
29 15 14.3333 0.3333 43 43.00 1.00 44.00
31 16 15.3333 0.3333 46 46.00 1.00 47.00
33 17 16.3333 0.3333 49 49.00 1.00 50.00
35 18 17.3333 0.3333 52 52.00 1.00 53.00
37 19 18.3333 0.3333 55 55.00 1.00 56.00
39 20 19.3333 0.3333 58 58.00 1.00 59.00
41 21 20.3333 0.3333 61 61.00 1.00 62.00
43 22 21.3333 0.3333 64 64.00 1.00 65.00
45 23 22.3333 0.3333 67 67.00 1.00 68.00
47 24 23.3333 0.3333 70 70.00 1.00 71.00
49 25 24.3333 0.3333 73 73.00 1.00 74.00
51 26 25.3333 0.3333 76 76.00 1.00 77.00
53 27 26.3333 0.3333 79 79.00 1.00 80.00
55 28 27.3333 0.3333 82 82.00 1.00 83.00
57 29 28.3333 0.3333 85 85.00 1.00 86.00
59 30 29.3333 0.3333 88 88.00 1.00 89.00
61 31 30.3333 0.3333 91 91.00 1.00 92.00
63 32 31.3333 0.3333 94 94.00 1.00 95.00
65 33 32.3333 0.3333 97 97.00 1.00 98.00
where p = 2*q - 1 in these instances.
for part (ii)
Abe Ben Abe's Ben's Total
p q m gave Gave r share share Dollars
5 4 1 2.0000 1.0000 2 6.00 3.00 9.00
6 5 1 2.3333 1.3333 2 7.33 3.67 11.00
7 5 2 3.0000 1.0000 3 9.00 3.00 12.00
7 6 1 2.6667 1.6667 2 8.67 4.33 13.00
8 6 2 3.3333 1.3333 3 10.50 3.50 14.00
9 6 3 4.0000 1.0000 4 12.00 3.00 15.00
8 7 1 3.0000 2.0000 2 10.00 5.00 15.00
11 7 4 5.0000 1.0000 5 15.00 3.00 18.00
13 8 5 6.0000 1.0000 6 18.00 3.00 21.00
15 9 6 7.0000 1.0000 7 21.00 3.00 24.00
17 10 7 8.0000 1.0000 8 24.00 3.00 27.00
19 11 8 9.0000 1.0000 9 27.00 3.00 30.00
21 12 9 10.0000 1.0000 10 30.00 3.00 33.00
23 13 10 11.0000 1.0000 11 33.00 3.00 36.00
25 14 11 12.0000 1.0000 12 36.00 3.00 39.00
27 15 12 13.0000 1.0000 13 39.00 3.00 42.00
29 16 13 14.0000 1.0000 14 42.00 3.00 45.00
31 17 14 15.0000 1.0000 15 45.00 3.00 48.00
33 18 15 16.0000 1.0000 16 48.00 3.00 51.00
35 19 16 17.0000 1.0000 17 51.00 3.00 54.00
37 20 17 18.0000 1.0000 18 54.00 3.00 57.00
39 21 18 19.0000 1.0000 19 57.00 3.00 60.00
41 22 19 20.0000 1.0000 20 60.00 3.00 63.00
43 23 20 21.0000 1.0000 21 63.00 3.00 66.00
45 24 21 22.0000 1.0000 22 66.00 3.00 69.00
47 25 22 23.0000 1.0000 23 69.00 3.00 72.00
49 26 23 24.0000 1.0000 24 72.00 3.00 75.00
51 27 24 25.0000 1.0000 25 75.00 3.00 78.00
53 28 25 26.0000 1.0000 26 78.00 3.00 81.00
55 29 26 27.0000 1.0000 27 81.00 3.00 84.00
57 30 27 28.0000 1.0000 28 84.00 3.00 87.00
59 31 28 29.0000 1.0000 29 87.00 3.00 90.00
61 32 29 30.0000 1.0000 30 90.00 3.00 93.00
63 33 30 31.0000 1.0000 31 93.00 3.00 96.00
65 34 31 32.0000 1.0000 32 96.00 3.00 99.00
In most of these instances, Q = m + 3.
However, for some (P,Q), this is not the case. Two of these cases, (6,5) and (7,6) involve dollar amounts that do not exactly follow the agreed-upon division of dollars as the cents to not come out even.
But the case of (P,Q) = (8,7) also has a Q that's not 3 more than m, as Q = m + 6, even though this instance isn't unusual in other respects.
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Posted by Charlie
on 2012-09-28 23:44:25 |
computer exploration
