# Problems in Two Variables

In this section, we will learn How to Approach LR the method of solving LR questions keeping in mind various do’s and do not.*Directions for questions 1 to 4: Read the following passage and solve the questions based on it.*

There are five boxes viz., T, U, W, X and Z, to be delivered on five consecutive days, Monday through Friday one box per day. The following conditions are to be kept in mind while formulating the delivery schedule of the boxes:

- Box X is not delivered on Monday
- If box T is delivered on Monday, then box X must be delivered on Friday.
- If box X is delivered on Tuesday, box U is delivered on Monday.
- Box W is delivered the day following the day of the delivery of box Z.

**Q 1. Box W could be delivered on any day except:**

- Monday
- Tuesday
- Wednesday
- Thursday

**Q 2. If box U is delivered the day after the box T is delivered, which of the following is a complete and accurate list of the days on which box X could be delivered?**

- Tuesday, Wednesday, Thursday
- Wednesday, Thursday, Friday
- Tuesday, Wednesday, Friday
- Wednesday, Friday

**Q 3. If box W is delivered on Wednesday, which of the following could be true?**

- Box U is delivered on Friday
- Box X is delivered on Tuesday
- Box X is delivered before box T
- Box X is delivered before box U

**Q 4. If the box U is delivered on Wednesday, which of the following must be true?**

- Box W is delivered before box T
- Box T is delivered before box X
- Box T is delivered before box U
- Box X is delivered before box W

**Solution**

**Scenario**

There are two sets of variables—1st set being five boxes and 2nd set being five days Monday through Friday.

As the boxes must be delivered in these five days, so one box is to be delivered everyday.

**Rules **

**There are two sets of rules given here:**

- Strict sequencing rule—These are the rules which give us the exact position of the variables.
- Loose sequencing rule—These are the rules which give us the relative positioning of one variable with respect to the other variable, without giving the exact position.
We will discuss about sequencing in Chapter 4 of this section.

**Now lets interpret the rules:**

- Box X is not delivered on Monday—This is a strict sequencing rule.
If box T is delivered on Monday, then box X must be delivered on Friday.This is a loose sequencing rule which introduces further conditions related to the delivery of box T and box X. But we should be very clear about certain facts related to this statement:

- If box T is not delivered on Monday, then box X can be delivered on any day. (except Monday)
- If box X is not delivered on Friday, then box T cannot be delivered on Monday.

- If box X is delivered on Tuesday, box U is delivered on Monday.
This is again a loose sequencing rule which introduces conditions related to the delivery of box X and box U. This statement also means that if U is not delivered on Monday, then X is not delivered on Tuesday.Box W is delivered the day following the day of the delivery of box Z.This is also a loose sequencing rule which introduces conditions related to the delivery of box W and box Z. It simply means that Z - W will be consecutive in the same order.For the sake of establishing the rules clearly in our mind, we should write it as follows:
- X - Not on Monday......(Rule 1)
- T (Monday) then - X (Friday)......(Rule 2)
- X (Tuesday) then - U (Monday)......(Rule 3)
- Z - W......(Rule 4)

**And then the five places:**

__Monday Tuesday Wednesday Thursday Friday__

**Ans. 1. (a)** Since box W is to be delivered on the day imme-diately after the day on which box Z is delivered, box W cannot be delivered on Monday.

**Ans. 2. (d)** Box U is to be delivered the day after box T. So, the order is: T - U. This is the second such pair after Z - W which is to be accommodated into the 5-days schedule.

Now start visualizing the days for X to be delivered:

__Monday __– Not possible

__Tuesday __– Not possible, otherwise one of the pairs from

T - U and Z - W will be required to split up.

__Wednesday __– If X is to be delivered then one of the pairs will be delivered on Monday – Tuesday and other pair on Thursday – Friday with either box T being delivered on Monday/Thursday or box Z being delivered on Monday/Thursday.

Now, T cannot be delivered on Monday otherwise X will be delivered on Friday (Rule 2).

If T is delivered on Thursday, then U will be delivered on Friday and compulsorily, Z will be delivered on Monday and W on Tuesday.

This delivery schedule is in accordance with the given rules. So, box X can be delivered on Wednesday.

__Thursday __– Not possible, otherwise one of the pairs from

T - U and Z - W will be required to split up.

__Friday__ – If X is to be delivered then one of the pairs will be delivered on Monday – Tuesday and other pair on Wednesday – Thursday with either box T being delivered on Monday/Wednesday OR box Z being delivered on Monday/Wednesday.

**The following delivery schedule scheme can be followed:**

Monday - T

Tuesday - U

Wednesday - Z

Thursday - W

Friday - X

Hence, box X can be delivered on either Wednesday or Friday.

**Ans. 3. (c)** Let us make the diagrams:

__Monday Tuesday Wednesday Thursday Friday__

W

If W is on Wednesday, then Z will be on Tuesday.

Monday Tuesday Wednesday Thursday Friday

Z W

**Checking the options:**

Option (a). Box U is delivered on Friday.

We are left with three boxes T, U and X and three days—Monday, Tuesday and Friday.

Now let us take the stock of the situation now:

For Monday—X cannot be delivered on Monday, so X has to be delivered on either Thursday or Friday.

If T is delivered on Monday, then X will be delivered on Friday and consequently, U on Thursday.

If U is delivered on Monday, then the rest two will be delivered on the remaining two days.

So, Box U cannot be delivered on Monday.

Option (b) We can eliminate this option very easily.

__Monday Tuesday Wednesday Thursday Friday__

Z W

Option (c)

__Monday Tuesday Wednesday Thursday Friday__

Z W

**Now moving ahead from here:**

If U is delivered on Monday, then X and T can be delivered on Thursday and Friday respectively.

Hence, option (c) could be true.

Work for students: Check the veracity of option (d).

**Ans. (a)** Box U is delivered on Wednesday.

__Monday Tuesday Wednesday Thursday Friday__

U

Now, Z will be delivered on either Monday or Thursday.

__Monday Tuesday Wednesday Thursday Friday...(1)__

Z W U

OR

__Monday Tuesday Wednesday Thursday Friday...(2)__

U Z W

We can see that (2) is not possible, because T has to be delivered on Monday (X cannot be delivered on Monday) and in that case X should be delivered on Friday, which is not possible.

So, (1) has to be true. T and X can be interchangeably delivered on Thursday or Friday without violating any condition.

So, option (a) is definitely true.

This question can be used to generate a set of new questions as well. One such question could have been made by replacing “must be” with “could be”. Of course in that case, more than one option would have become true out of the given four options.

However two variables LR set questions should be seen more as ‘Sequencing and Arrangement’ questions. We will discuss more about this in Chapter 4.