In a garden, there are 5 rose plants for every 7 marigold plants. Ther

Question

In a garden, there are 5 rose plants for every 7 marigold plants. There are 3 marigold plants for every 11 pegunias and 2 pegunias for every 9 cactus plants. What is the minimum number of plants in the garden?

A. 113
B. 793
C. 819
D. 919
E. 988

A. 113
B. 793
C. 819
D. 919
E. 988

CareerGeek · Answer

5R = 7M. 
3M = 11P
2P = 9C

So, R:M = 7:5
M:P = 11:3

Take LCM of values of M (5 & 11) = 55 and rewrite the ratios to get R:M:P

R:M = 77:55 and M:P = 55:15

—> R:M:P = 77:55:15
P:C = 9:2

Similarly, Take LCM of values of P (15 & 9) = 45 and rewrite the ratios to get R:M:P:C

R:M:P = 231:165:45 and P:C = 45:10

—> R:M:P:C = 231:165:45:10
Values of R,M,P,C are 231k, 165k, 45k, 10k respectively where k is any positive integer.

So, total should be a multiple of 231k + 165k + 45k + 10k = 451k

Minimum plants should be 451!!

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LeoN88 · Answer

D. 919 IMO Check Diagram: IMG_20190527_101634__01.jpg 5/7 :: 3/11 :: 2/9 ~ r/m :: m/p :: p::c form fractions and try to balance m:p

CareerGeek · Answer

Thanks!

jfranciscocuencag · Answer

Hello  nick1816

Could you please provide answer?

Kind regards!

Moinak94 · Answer

There is an easier method to calculate the ratios between entities.
Let's say you have A=N1/D1 ; B=N2/D2 ; C = N3/D3 ; D= N4/D4
So A:B:C:D can be expressed as (N1xN2xN3xN4):(D1xN2xN3xN4):(D1xD2xN3xN4):(D1xD2xD3xN4):(D1xD2xD3xD4) [Notice how the Numerators are eventually displaced by Denominators)

ScottTargetTestPrep · Answer

We can create the ratios:

R : M = 5 : 7

M : P = 3 : 11

P : C = 2 : 9

P, at a minimum, must be 22, so we have:

R : M = 5 : 7

M : P = 6 : 22

P : C = 22 : 99

M, at a minimum is 42, so we have:

R : M = 30 : 42

M : P = 42 : 154

P : C = 154 : 693

Therefore, there are a minimum of R + M + P + C = 30 + 42 + 154 + 693 = 919 plants in the garden.

Alternate Solution:

We can create the ratios:

R : M = 5 : 7

M : P = 3 : 11

P : C = 2 : 9

Let’s combine the first two ratios to form a triple ratio for R : M : P. Since M is the common flower in the two ratios, we need to re-express each ratio such that the number of M is the same. This is easily accomplished by noting that the LCM of the two M ratio numbers is 7 x 3 = 21. Thus, the first ratio can be re-expressed as R : M as 15 : 21 by multiplying the original ratio by 3. And, similarly, the second ratio can be re-expressed as M : P as 21 : 77 by multiplying the original ratio by 7. So now the triple ratio is R : M : P as 15 : 21 : 77.

Now, let’s combine the triple ratio with the third ratio to make a 4-way ratio. The common flower in these two ratios is P, where P is 77 in the triple ratio and P is 2 in the third ratio. Thus, their LCM is 77 x 2 = 154. So we will multiply the triple ratio by 2 and the third ratio by 77, getting R: M : P as 30 : 42 : 154 and the third ratio as P : C as 154 : 693.

Finally, we can write the 4-way ratio as R : M : P : C as 30 : 42 : 154 : 693. Since the LCM rule was followed for each step, we are assured that the sum 30 + 42 + 154 + 693 = 919 is a minimum value for the number of flowers in the garden.

Answer: D

menonrit · Answer

Hi Scott,

The working out is very clear. However, I just wanted to understand why "P" has to be a minimum of 22 and "M" has to be a minimum of 42.

ScottTargetTestPrep · Answer

Let’s first note that the number of any kind of flowers must be an integer; we can’t have fractional flowers.

If we just had the ratio M : P = 3 : 11; we wouldn’t be able to conclude that P must be at least 22. It could also be the case that there are 3 M flowers and 11 P flowers. However, we also have the ratio P : C = 2 : 9 and if P were 11, then there would have to be 49.5 C flowers, which is impossible. That’s why the smallest value of P is the least common multiple of the numbers 11 and 2, which is 22. Notice that when we take P to be 22, there are 6 M flowers and 99 C flowers; which works out fine.

A similar reasoning works for why the smallest value of M is 42. If we take P to be its smallest possible value, which is 22, it follows from the ratio M : P = 3 : 11 that M is a multiple of 6. Also, from the ratio R : M = 5 : 7, we understand that M must be a multiple of 7. Since M is a multiple of both 6 and 7, M is a multiple of 42, and the smallest positive multiple of 42 is 42.

bumpbot · Answer

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