Ratio

[MBA.Aspirant]

Pea, tomato, and spinach plants are planted in a field. For every 2 pea plants, there are 3 tomato plants, and for every 5 pea plants, there are 6 spinach plants. Spinach is sown 18 plants to a row and tomatoes 12 plants to a row, with no partial or mixed-crop rows. What is the minimum number of spinach plants in the field?

(A) 36

(B) 72

(C) 120

(D) 144

(E) 180

the common ratio is T:P:S 15:10:12

and we know that the no. of spinach has to be a multiple of 18 and tomato 12, how to proceed from here?

Thanks

[diebeatsthegmat]

i just plug in and i found the answer is D 144 and i did this in 5 mins to understand the question
and find the answer. there must be another way to solve this with less time
pea/tomato=2/3=x/12*a
pea/spinach=5/6 =x/18*b

( there are only 18 and 12 plants for spinach and tomato)
plug in and see than only with spinach =144 so there are 120 peas in the field and 180 tomatos in the fielf
and 180 can absolutely be divided by 12

i guessif anybody has another shorter and less time consuming, please explain.... thanks

[GMATGuruNY]

Pea, tomato, and spinach plants are planted in a field. For every 2 pea plants, there are 3 tomato plants, and for every 5 pea plants, there are 6 spinach plants. Spinach is sown 18 plants to a row and tomatoes 12 plants to a row, with no partial or mixed-crop rows. What is the minimum number of spinach plants in the field?

(A) 36

(B) 72

(C) 120

(D) 144

(E) 180

the common ratio is T:P:S 15:10:12

and we know that the no. of spinach has to be a multiple of 18 and tomato 12, how to proceed from here?

Thanks

To combine the two given ratios, the element common to both ratios -- the pea plants -- must be represented by the same value.
P:T = 2:3 = 10:15.
P:S = 5:6 = 10:12.
Now that P=10 in each ratio, the ratios can be combined:
P:T:S = 10:15:12.

We need to determine the minimum factor by which all the values in the ratio above must be multiplied.

Since T=15, and the tomato plants are planted 12 to a row, the minimum number of tomato plants must be a multiple both of 15 and of 12.
The LCM of 15 and 12 is 60.
Since 60/T = 60/15 = 4, all the values in the ratio above must be multiplied at least by a factor of 4.

Since S=12, and the spinach plants are planted 18 to a row, the minimum number of spinach plants must be a multiple both of 12 and of 18.
The LCM of 12 and 18 is 36.
Since 36/S = 36/12 = 3, all the values in the ratio above must be multiplied at least by a factor of 3.

Thus, all the values in the ratio above must be multiplied by 4*3 = 12.

12 * (10:15:12) = 120:180:144.

Thus, S=144.

The correct answer is D.

Another approach would be to plug in the answers, which represent the minimum number of spinach plants.
The correct answer will yield a value for T that is a multiple of 18 (since the tomato plants are 18 to a row).

Answer choice A: S=36.
10:15:12 = 30:45:36.
If S=36, then T=45.
45 is not a multiple of 18.
Eliminate A.

Answer choice B: S=72.
10:15:12 = 60:90:72.
If S=72, then T=90.
90 is not a multiple of 18.
Eliminate B.

Answer choice C: S=120.
10:15:12 = 100:150:120.
If S=120, then T=150.
150 is not a multiple of 18.
Eliminate C.

Answer choice D: S=144.
10:15:12 = 120:180:144.
If S=144, then T=180.
180 is a multiple of 18.

The correct answer is D.