pkw209 wrote:Hi all,
Couldn't figure out this one. Would really appreciate a brief explanation. thanks!
Q) On a tour, the ratio of the number of women to the number of children was 5 to 2. What was the number of men on the tour?
1) The ratio of the number of children to the number of men was 5 to 11.
2) The number of women on the tour was less than 30.
papgust, one key was this:
we cannot have a fraction of a person so we must get an integer
1) The ratio of the number of children to the number of men was 5 to 11.
Clearly insufficient as there is no info about numbers.
2) The number of women on the tour was less than 30.
Clearly insufficient as there is no info about men.
(1) + (2):
From (1), we know that there are at least 11 men. This is because we cannot further divide 5:11 without cutting at least one kid and one man in half. For example, if we were to divide the ratio 5:11 by 2, we would have: 2.5: 5.5, meaning we would have two and a half children and five and a half men. From (1), we also know that the number of men is a multiple of 11.
And from the stem, we know that the ratio of women to children is 5:2. Because children is the variable that shows up in both ratios, let's think about them. From (1), we know that the number of children is a multiple of 5; from the stem, we know that the number of children is a multiple of 2. Therefore, the number of children must be a multiple of both 2 and 5. The least common multiple here is 10. So, there are at least 10 children. If there are ten children, then there are 2.5 x as many women (w/c = 5/2), so there would be 25 women. And the number of men is 11/5 * the number of children, so 11/5 * 10 = 22.
So, from (1) and the stem, the minimal ratio of w:c:m is: 25: 10: 22. This is the minimal ratio because if we were to simplify it further, we would have non-integers, and we would be cutting people into pieces. At this point (ie, before including (2)'s info), you could have that many number of women, children, and men, or 50 women, 20 children, 44 men, and so on.
In other words, there are at least 25 women, and the number of women must be a multiple of 25. And, from (2), we know that the number of women is less than 30. There is only one (positive) multiple of 25 less than 30. Therefore, we have 25 women, 10 children, and 22 men, and choose C.
_____
Suppose the ratio of women to men at a conference was 3:2. This means that the number of women will always be a multiple of 3, the number of men always a multiple of 2, and, because 3 + 2 = 5, the total number always a multiple of 5:
W M T
3 2 5
6 4 10
9 6 15
This is a key concept when moving from part to part to part to whole ratios. Children was the variable that was involved in both of our ratios. w:c = 5:2, therefore number of children a multiple of 2. Children to men was 5: 11, therefore the number of children also a multiple of 5. This is the key deduction that allowed qazpt to set up the ratio of w:c:m = 25:10:22.
Another important (and closely related) point is that when dealing with ratios of objects and people, you are always dealing with positive integers because in most of these problems it would be absurd to divide or cut up the object or the person.
