A farmer had a rectangular land containing 205 trees. He distributed that land among his four daughters – Abha, Bina, Chitra and Dipti by dividing the land into twelve plots along three rows (X,Y,Z) and four Columns (1,2,3,4) as shown in the figure below:
The plots in rows X, Y, Z contained mango, teak and pine trees respectively. Each plot had trees in non-zero multiples of 3 or 4 and none of the plots had the same number of trees. Each daughter got an even number of plots. In the figure, the number mentioned in top left corner of a plot is the number of trees in that plot, while the letter in the bottom right corner is the first letter of the name of the daughter who got that plot (For example, Abha got the plot in row Y and column 1 containing 21 trees). Some information in the figure got erased, but the following is known:
Which of the following is the correct sequence of trees received by Abha, Bina, Chitra and Dipti in that order?
Who got the plot with the smallest number of trees and how many trees did that plot have?
Let each plot in the grid be represented by its row label and column label. For example, (X, 2) represents the plot in row X and column 2.
From (8), Chitra and Dipti did not get plots which were adjacent to each other.
From the figure, we can see that Chitra has the plot (X, 1). Hence, Dipti cannot have the plots (X, 2) and (Y, 2).
Also, Chitra ha s the plot (Z, 2). Hence, Dipti cannot have the plots (Z, 3) and (Y, 3). From (6), Dipti has two adjoining plots in the same row. Hence, the only possibility for Dipti to have such plots is if she has the plots (X, 3) and (X, 4).
It is given that each daughter got an even number of plots. Also, from (4), Abha and Bina had a higher number of plots than Dipti.
Since Dipti already has 2 plots, Abha and Bina must have at least 4 plots each. Chitra already has 2 plots. Hence, Abha and Bina cannot have a higher n umber of plots. Hence, the number of plots that Abha, Bina, Chitra and Dipti must be 4, 4, 2 and 2, respectively.
We already know the positions of all the plots of Chitra and Dipti. Hence, the remaining plots must belong to Abha or Bina.
From (5), the corner plot, (Z, 4) must belong to Bina.
From (7), Bina got a plot in each row. In the first row, Chitra got (X, 1) and Dipti got (X, 3) and (X, 4). Hence, Bina must have gotten (X, 2). Bina has a total of 4 plots and we know the positions of three plots.
For Bina to have a plot in each row and each column, she must still have plot(s) in row Y and column 3. Since she can have only one more plot, she must have a plot at the intersection of this row and column.
Hence, Bina must have gotten the plot (Y, 3). A should have the remaining two plots, i.e., (Y, 2) and (Z, 3). Let the number of trees in (Y, 2) be a.
From (3), the number of trees in (Y, 3) must be 2a and the number of trees in (Y, 4) must be 4a.
From (2), 4a cannot be more than 32 and since (Y, 4) is owned by Abha, it cannot be 32. Hence, a can be at most 7.
Also, a should be a multiple of 3 or 4. Hence, the possible values for a are 3, 4 and 6. However, a cannot be 3, since 4a will be 12 and (X, 1) has 12 trees (each plot has distinct number of trees).
Also, a cannot be 6, since 2a will be 12. Hence, a must be 4.
The number of trees in (Y, 2), (Y, 3) and (Y, 4) must be 4, 8 and 16. The total number of trees in row Y is 21 + 4 + 8 + 16 = 49.
From (9), the total number of trees in row X = \(49\times 2 = 98\).
The number of trees in row Z = 205 - 49 - 98 = 58.
The total number of trees in the plots that Abha got is 21 + 4 + 16 + 9 = 50 (adding the trees in (Y, 1), (Y, 2), (Y, 4) and (Z, 3).
From (1), Chitra must have 30 trees and Dipti must have 56 trees. Since Chitra has 30 trees, and Chitra has 12 trees in (X, 1), there must be 18 trees in (Z, 2) (the only other plot that Chitra got).
The number of trees in (Z, 2), (Z, 3) and (Z, 4) are 18, 9 and 28 respectively. Since there must be 58 trees in row Z, the number of trees in (Z, 1) must be 3.
The number of trees with Bina must be 205 - 50 - 56 - 30 = 69. Bina has 3 trees in (Z, 1), 8 trees in (Y, 3) and 28 trees in (Z, 4).
In the last plot that Bina owns, i.e., in (X, 2), there must be 69 - 3 - 8 - 28 = 30 trees. In row X, in the plots that Dipti owns, (X, 3) and (X, 4), there must be a total of 56 trees.
Since the maximum possible number of trees in only 32, the maximum possible number of trees in these two plots can be if they have 32 trees in one plot and 24 trees in the other plot (since 30 and 28 trees are already present in other plots).
Hence, the plots (X, 3) and (X, 4) must have 32 and 24 trees in any order.
The following table provides the distribution of plots and trees
The total number of mango trees are 98
Option: 2Question 2:
The correct sequence of trees received by Abha, Bina, Chitra and Dipti are 50, 69, 30, 56.
Option: 1Question 3:
The number of pine trees received by Chitra = 18
Option: 3Question 4:
Bina got the plot with the smallest number of trees, which had 3 trees
Option: 4Question 5:
Bina did not 32 pine trees. She got 31 pine trees
Option: 2Question 6:
Column 4 has the highest number of trees