[email protected] wrote:At a certain school, the ratio of the number of second graders to the number of fourth graders is 8 to 5, and the ratio of the number of first graders to the number of second graders is 3 to 4. If the ratio of the number of third graders to the number of fourth graders is 3 to 2, what is the number of first graders to the number of third graders?
a) 16 to 15
b) 9 to 5
c) 5 to 16
d) 5 to 4
e) 4 to 5
Is there a better explanation than the one provided in the OG?
Solution:
Background: Ratios show the relationship between members of two or more categories. As an example, if the ratio of horses to ponies is 7:4 and the ratio of ponies to goats is 4:2, then we can directly express the ratio of horses to goats as 7:2 because the original ratios had matching values for ponies. A more complicated pair of ratios would be horses to ponies is 7:4 and ponies to goats is 2:1. We must make the ponies numbers equal in both ratios before being able to directly express the ratio of horses to goats. We would thus multiply the second ratio (ponies to goats) by 2 to get the equivalent ratio of 4:2. Now that the ponies numbers match, we can directly express the ratio of horses to goats as 7:2.
We are given the following:
1) The ratio of the number of second graders to the number of fourth graders is 8 to 5.
2) The ratio of the number of first graders to the number of second graders is 3 to 4.
3) The ratio of the number of third graders to the number of fourth graders is 3 to 2.
That is:
1) 2nd : 4th = 8 : 5
2) 1st : 2nd = 3 : 4
3) 3rd : 4th = 3 : 2
From the first and second ratios, we see that they both contain 2nd graders, so let's make 2nd graders the same number. We see that the number for 2nd graders in the first ratio is 8 and that for the 2nd graders in the second ratio is 4. We multiply the second ratio by 2, obtaining 6 : 8, and now both ratios have a matching 8 for 2nd graders.
1)
2nd : 4th =
8 : 5
2) 1st :
2nd = 6 :
8
Now we see that the ratio of 4th graders to 1st graders must be 5 to 6. That is:
4th : 1st = 5 : 6
We will now compare this with the third ratio we set up originally:
3rd : 4th = 3 : 2
We see that both ratios contain 4th graders, so let's make 4th graders the same number. We must change both ratios. We multiply the ratio 5 : 6 by 2, to get an equivalent ratio of 10 : 12. Then we multiply the ratio 3 : 2 by 5, to get an equivalent ratio of 15 : 10. That is:
4th : 1st =
10 : 12
3rd :
4th = 15 :
10
Now that 4th graders have a matching 10 in each ratio, we can express the ratio of 1st graders to 3rd graders as 12 : 15. That is:
1st : 3rd = 12 : 15
We can reduce this ratio by dividing both 12 and 15 by 3:
1st : 3rd = 4 : 5
Answer:
E
