In this odd-man-out question on number triples, four ordered groups of three numbers are given: (13, 26, 39), (15, 30, 45), (17, 34, 51) and (19, 38, 59). In three of these groups, the second and third numbers are exactly 2 times and 3 times the first number. Which triple is the odd one out because it does not follow this 1 : 2 : 3 ratio pattern?

Introduction / Context:
This question is a numerical odd-man-out problem involving sets of three numbers. The groups given are (13, 26, 39), (15, 30, 45), (17, 34, 51) and (19, 38, 59). In aptitude tests, such triples often follow a simple ratio or multiplier pattern, such as the second number being 2 times the first and the third number being 3 times the first. The task is to recognize the common pattern shared by most triples and then find the one that fails to satisfy it.

Given Data / Assumptions:
Four groups of three numbers are given: (13, 26, 39), (15, 30, 45), (17, 34, 51) and (19, 38, 59).
In three of the groups, the second number is 2 times the first number and the third number is 3 times the first number.
In one group, this ratio pattern is broken.
Only basic multiplication is needed to test the relationship.

Concept / Approach:
The key idea is to check the simple ratio 1 : 2 : 3. For a triple (a, b, c) to follow this pattern, b should be equal to 2a and c should be equal to 3a. This makes the triple effectively (a, 2a, 3a). By verifying this for each group, we can see which triple does not respect the relationship. That triple will be the odd one out.

Step-by-Step Solution:
Step 1: For (13, 26, 39), check: 2 * 13 = 26 and 3 * 13 = 39. So this triple follows the 1 : 2 : 3 pattern. Step 2: For (15, 30, 45), check: 2 * 15 = 30 and 3 * 15 = 45. This triple also follows the same 1 : 2 : 3 pattern. Step 3: For (17, 34, 51), check: 2 * 17 = 34 and 3 * 17 = 51. This triple again follows the desired pattern. Step 4: For (19, 38, 59), check: 2 * 19 = 38, which matches the second number, but 3 * 19 = 57, not 59. Step 5: Conclude that only the last triple fails the 1 : 2 : 3 ratio for the third number, making (19, 38, 59) the odd one out.
Verification / Alternative check:
You can verify by computing the ratios b/a and c/a for each group. For the first three groups, b/a = 2 and c/a = 3 exactly, but in the fourth group, b/a = 2 is correct, while c/a = 59 / 19 is not equal to 3. This mismatch clearly indicates that the fourth triple does not share the same simple ratio structure as the others. Another quick mental check is to see that 59 is only 2 units larger than 57, which would have been the correct multiple for 3 * 19.

Why Other Options Are Wrong:
(13, 26, 39) precisely follows the pattern (a, 2a, 3a) with a = 13, so it is not the odd one out.
(15, 30, 45) fits the same pattern with a = 15 and matches both 2a and 3a exactly.
(17, 34, 51) also respects the 1 : 2 : 3 relationship and belongs to the main group of correctly constructed triples.
(19, 38, 59) fails because the third number should have been 57 instead of 59 to conform to the 3 times rule.

Common Pitfalls:
A common error is to check only whether the second number is twice the first and overlook the relation with the third number. Because the second number in the last triple fits the rule, some candidates may mistakenly assume the whole triple is correct. Always confirm both conditions when dealing with patterns that involve multiple related numbers. Another pitfall is to look for more complicated patterns when a simple multiplicative relationship already explains three of the four options perfectly.

Final Answer:
The number triple that does not follow the 1 : 2 : 3 multiple pattern is (19, 38, 59).