A broken clover has 2 leaves, a normal clover has 3 leaves and a lucky clover has 4 leaves. In a bunch of these clovers , there are twice as many normal clovers as broken clovers and 5 times as many normal clovers as lucky clovers. The bunch has a total of N leaves. N is greater than 200. What is the lowest value of N?
Actually n=5l, so n=42 and l=9 does not work.
For a base unit of clovers, there would be 10 normals, 5 brokens, and 2 luckies. so 10 normals = 30 leaves, 5 brokens = 10 leaves, and 2 luckies = 8 leaves. So 48 leaves in a base unit. So basically what is the first multiple of 48 that is greater than 200. 240 is the answer. Guo out. Brian says hi
Guys its
E Strategy: Determine the possible numbers of normal leaves.
The number of normal clovers is a multiple of 5, since there are 5 times as many normal
clovers as lucky clovers. The number of normal clovers is also even, since there are twice
as many of them as there are of broken clovers. The number of normal clovers is therefore
a multiple of 10.
For each 10 normal clovers, there are 2 lucky clovers and 5 broken clovers.
Split the collection of N clovers into groups so that each group has 10 normals, 2 luckies,
and 5 brokens. The number of leaves in each group is 10×3 + 2×4 +5×2 = 48.
The total number of leaves is a multiple of 48, and the least multiple of 48 that is greater than
200 is 5×48 = 240. The least value of N is 240.
To find the lowest value of N, we need to determine the minimum number of each type of clover in the bunch. Let's start by assigning variables to the number of each type of clover:
Let's say the number of broken clovers is B.
The number of normal clovers is N.
The number of lucky clovers is L.
According to the given information:
1) A broken clover has 2 leaves, so the total number of leaves contributed by broken clovers is 2B.
2) A normal clover has 3 leaves, so the total number of leaves contributed by normal clovers is 3N.
3) A lucky clover has 4 leaves, so the total number of leaves contributed by lucky clovers is 4L.
From the given information:
1) There are twice as many normal clovers as broken clovers: N = 2B.
2) There are 5 times as many normal clovers as lucky clovers: N = 5L.
Now, let's find the minimum values for B, N, and L.
We know that N is greater than 200, so let's start by assuming N is 201:
N = 201
Since N = 2B, we can substitute the value of N:
201 = 2B
B = 201/2
B = 100.5
However, B represents the number of broken clovers, which must be a whole number. Since we can't have half a clover, we need to round up to the nearest whole number:
B = 101
Now that we have the value of B, we can substitute it back into N = 2B to find the value of N:
N = 2 * 101
N = 202
We can also find the value of L by substituting N = 201 into the equation N = 5L:
201 = 5L
L = 201/5
L = 40.2
Similar to B, L represents the number of lucky clovers, which must also be a whole number. Rounding up:
L = 41
So, the minimum values are:
B = 101 (broken clovers)
N = 202 (normal clovers)
L = 41 (lucky clovers)
To find the minimum value of N, we calculate the total number of leaves:
Total leaves = 2B + 3N + 4L
Total leaves = 2(101) + 3(202) + 4(41)
Total leaves = 202 + 606 + 164
Total leaves = 972
Therefore, the lowest value of N is 972, which is the total number of leaves in the bunch of clovers.
n = 2b
n = 5l
2b+3n+4l > 200
2(n/2) + 3n + 4(n/5) > 200
n + 3n + 4n/5 > 200
24n/5 > 200
24n > 1000
n > 41.666
n >= 42
b >= 21
l >= 9
2*21 + 3*42 + 4*9 = 42 + 126 + 36 = 204
So, N >= 204 leaves in total