In a pet shop, there are $3$ hamsters for every $2$ guinea pigs, and there are $2$ giant cloud rats for every $3$ guinea pigs. If $N$ is the total number of hamsters, guinea pigs, and giant cloud rats, and $N>0$, then what is the smallest possible value of $N$?
From your problem, we can represent the number of hamsters, guinea pigs, and giant cloud rats with the letters H, G, and R respectively. Therefore, the following equations are true:
2H = 3G
3R = 2G
N = R + G + H
Now we can solve for R and H in the upper equations with respect to G and substitute them into the lower equation:
H = 3/2G
R = 2/3G
N = 2/3G + G + 3/2G
Simplifying gives:
N = 19/6G
Since N must be a whole number, G has to be a multiple of 6 for this equation to make sense. Therefore, we will use 6 for G to find that N=19.
Checking your work will indicate that there are 6 Guinea Pigs, 9 Hamsters, and 4 Giant Cloud Rats, and this is indeed the smallest combination of the three with all else being true.
+118658
I have set up the 2 ratios here. You must keep the same ratios but make the number of guinea pigs the same.
The lowest common multiple of 2 and 3 is 6
| H | G | R | G | ||
| 3*3 | 2*3 | 2*2 | 3*2 | ||
| 9 | 6 | 4 | 6 |
So this is the ratio of the animals
| H | R | G | Total |
| 9 | 4 | 6 | 18 |
| N=18k (where k is an integer) |
The number of animals in total is a multiple or 18. The smallest number is 18 :)
From your problem, we can represent the number of hamsters, guinea pigs, and giant cloud rats with the letters H, G, and R respectively. Therefore, the following equations are true:
2H = 3G
3R = 2G
N = R + G + H
Now we can solve for R and H in the upper equations with respect to G and substitute them into the lower equation:
H = 3/2G
R = 2/3G
N = 2/3G + G + 3/2G
Simplifying gives:
N = 19/6G
Since N must be a whole number, G has to be a multiple of 6 for this equation to make sense. Therefore, we will use 6 for G to find that N=19.
Checking your work will indicate that there are 6 Guinea Pigs, 9 Hamsters, and 4 Giant Cloud Rats, and this is indeed the smallest combination of the three with all else being true.
+129839 R = cloud rats 2
(3/2)R = guinea pigs 3
(3/2) [(3/2)R] = hamsters 9/2
So...since we can't have 9/2 hamsters.....we need to multiply by 2
R = cloud rats 2 * 2 = 4
(3/2)R = guinea pigs 3 * 2 = 6
(3/2)(3/2)R = hamsters 9/2 *2 = 9
So...the smallest that N can be = 4 + 6 + 9 = 19
