Three variables x, y and z take values 2, 3, 4 or 5 such that their values are always distinct. If M and N denote the largest possible value and the smallest possible value, respectively, for the expression $\{(x\times y)+z\}$; then $M-N$ is

The correct option is (c) - 13.

[as per provisional answerkey]

Solution

We are given three distinct variables x, y, and z from the set {2, 3, 4, 5}. We need to find the difference between the maximum value (M) and the minimum value (N) of the expression {$(x\times y)+z$}.

1. Finding the Maximum Value (M):
To maximize $(x\times y)+z$, we should assign the largest available numbers to the product (x * y) because multiplication scales faster than addition.
Let x = 5 and y = 4 (the two largest numbers).
Then z must be the next largest available number, which is 3.
$M=(5\times 4)+3=20+3=23$.
Note: If we chose z = 5, then x and y would be 4 and 3, giving $(4\times 3)+5=17$, which is smaller.

2. Finding the Minimum Value (N):
To minimize $(x\times y)+z$, we should assign the smallest available numbers to the product (x * y).
Let x = 2 and y = 3 (the two smallest numbers).
Then z must be the next smallest available number, which is 4.
$N=(2\times 3)+4=6+4=10$.
Note: If we chose z = 2, then x and y would be 3 and 4, giving $(3\times 4)+2=14$, which is larger.

3. Calculating $M-N$:
$M-N=23-10=13$.

Why the other options are incorrect

• Option (a) - 11: This result might be reached if one incorrectly calculates M as 22 or N as 12, failing to optimize the placement of the largest/smallest digits in the product vs. the sum.
• Option (b) - 12: This is a common error if the student assumes the difference between the highest possible product (20) and lowest possible product (6) is the only factor, or makes a minor subtraction error.
• Option (d) - 14: This value exceeds the actual range; it could occur if the distinctness constraint is ignored (e.g., using 5, 5, 4 for M or 2, 2, 3 for N), which violates the rule that values must be distinct.

Key Concept

To maximize an expression of the form $(x\times y)+z$, assign the largest values to the multiplicative terms; to minimize it, assign the smallest values to the multiplicative terms.