# Simplification

### Important Formulas

1. ‘BODMAS’ Rule:This rule depicts the correct sequence in which the operations are to be executed, so as to find out the value of given expression.

Here B – Bracket,

O – of,

D – Division,

M – Multiplication,

S – Subtraction

Thus, in simplifying an expression, first of all the brackets must be removed, strictly in the order (), {} and ||.

After removing the brackets, we must use the following operations strictly in the order:

(i) of (ii) Division (iii) Multiplication (iv) Addition (v) Subtraction.

2. Modulus of a Real Number:Modulus of a real number a is defined as

 |a| = a, if a > 0 –a, if a < 0

Thus, |5| = 5 and |-5| = -(-5) = 5.

3. Virnaculum (or Bar):When an expression contains Virnaculum, before applying the ‘BODMAS’ rule, we simplify the expression under the Virnaculum.

1

A man has Rs. 480 in the denominations of one-rupee notes, five-rupee notes and ten-rupee notes. The number of notes of each denomination is equal. What is the total number of notes that he has ?
 A. 45 B. 60 C. 75 D. 90

Explanation:

Let number of notes of each denomination be x.

Then x + 5x + 10x = 480 16x = 480 x = 30.

Hence, total number of notes = 3x = 90.

2.

There are two examinations rooms A and B. If 10 students are sent from A to B, then the number of students in each room is the same. If 20 candidates are sent from B to A, then the number of students in A is double the number of students in B. The number of students in room A is:
 A. 20 B. 80 C. 100 D. 200

Explanation:

Let the number of students in rooms A and B be x and y respectively.

Then, x – 10 = y + 10 xy = 20 …. (i)

and x + 20 = 2(y – 20) x – 2y = -60 …. (ii)

Solving (i) and (ii) we get: x = 100 , y = 80. The required answer A = 100.

3.

The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:
 A. Rs. 3500 B. Rs. 3750 C. Rs. 3840 D. Rs. 3900

Explanation:

Let the cost of a chair and that of a table be Rs. x and Rs. y respectively.

 Then, 10x = 4y   or   y = 5 x. 2 15x + 2y = 4000 15x + 2 x 5 x = 4000 2 20x = 4000 x = 200.

 So, y = 5 x 200 = 500. 2

Hence, the cost of 12 chairs and 3 tables = 12x + 3y

= Rs. (2400 + 1500)

= Rs. 3900.

4.

If ab = 3 and a2 + b2 = 29, find the value of ab.
 A. 10 B. 12 C. 15 D. 18

Explanation:

2ab = (a2 + b2) – (ab)2

= 29 – 9 = 20 ab = 10.

5.

The price of 2 sarees and 4 shirts is Rs. 1600. With the same money one can buy 1 saree and 6 shirts. If one wants to buy 12 shirts, how much shall he have to pay ?
 A. Rs. 1200 B. Rs. 2400 C. Rs. 4800 D. Cannot be determined E. None of these

Explanation:

Let the price of a saree and a shirt be Rs. x and Rs. y respectively.

Then, 2x + 4y = 1600 …. (i)

and x + 6y = 1600 …. (ii)

```Divide equation (i) by 2, we get the below equation.

=> x +  2y =  800. --- (iii)

Now subtract (iii) from (ii)

x +  6y = 1600  (-)
x +  2y =  800
----------------
4y =  800
----------------

Therefore, y = 200.

Now apply value of y in (iii)

=>  x + 2 x 200 = 800

=>  x + 400 = 800

Therefore x = 400

```

Solving (i) and (ii) we get x = 400, y = 200. Cost of 12 shirts = Rs. (12 x 200) = Rs. 2400.

6.

A sum of Rs. 1360 has been divided among A, B and C such that A gets of what B gets and B gets of what C gets. B’s share is:
 A. Rs. 120 B. Rs. 160 C. Rs. 240 D. Rs. 300

Explanation:

Let C’s share = Rs. x

 Then, B’s share = Rs. x ,   A’s share = Rs. 2 x x = Rs. x 4 3 4 6 x + x + x = 1360 6 4 17x = 1360 12 x = 1360 x 12 = Rs. 960 17

 Hence, B’s share = Rs. 960 = Rs. 240. 4

7.

One-third of Rahul’s savings in National Savings Certificate is equal to one-half of his savings in Public Provident Fund. If he has Rs. 1,50,000 as total savings, how much has he saved in Public Provident Fund ?
 A. Rs. 30,000 B. Rs. 50,000 C. Rs. 60,000 D. Rs. 90,000

Explanation:

Let savings in N.S.C and P.P.F. be Rs. x and Rs. (150000 – x) respectively. Then,

 1 x = 1 (150000 – x) 3 2 x + x = 75000 3 2 5x = 75000 6 x = 75000 x 6 = 90000 5 Savings in Public Provident Fund = Rs. (150000 – 90000) = Rs. 60000

8.

A fires 5 shots to B’s 3 but A kills only once in 3 shots while B kills once in 2 shots. When B has missed 27 times, A has killed:
 A. 30 birds B. 60 birds C. 72 birds D. 90 birds

Explanation:

Let the total number of shots be x. Then,

 Shots fired by A = 5 x 8

 Shots fired by B = 3 x 8

 Killing shots by A = 1 of 5 x = 5 x 3 8 24

 Shots missed by B = 1 of 3 x = 3 x 2 8 16 3x = 27 or x = 27 x 16 = 144. 16 3

 Birds killed by A = 5x = 5 x 144 = 30. 24 24

9.

Eight people are planning to share equally the cost of a rental car. If one person withdraws from the arrangement and the others share equally the entire cost of the car, then the share of each of the remaining persons increased by:
A.
 1 7
B.
 1 8
C.
 1 9
D.
 7 8

Explanation:

 Original share of 1 person = 1 8

 New share of 1 person = 1 7

 Increase = 1 – 1 = 1 7 8 56 Required fraction = (1/56) = 1 x 8 = 1 (1/8) 56 1 7

10.

To fill a tank, 25 buckets of water is required. How many buckets of water will be required to fill the same tank if the capacity of the bucket is reduced to two-fifth of its present ?
 A. 10 B. 35 C. 62.5 D. Cannot be determined E. None of these

Explanation:

Let the capacity of 1 bucket = x.

Then, the capacity of tank = 25x.

 New capacity of bucket = 2 x 5 Required number of buckets = 25x (2x/5)

 = 25x x 5 2x

 = 125 2

= 62.5

11

In a regular week, there are 5 working days and for each day, the working hours are 8. A man gets Rs. 2.40 per hour for regular work and Rs. 3.20 per hours for overtime. If he earns Rs. 432 in 4 weeks, then how many hours does he work for ?
 A. 160 B. 175 C. 180 D. 195

Explanation:

Suppose the man works overtime for x hours.

Now, working hours in 4 weeks = (5 x 8 x 4) = 160. 160 x 2.40 + x x 3.20 = 432 3.20x = 432 – 384 = 48 x = 15.

Hence, total hours of work = (160 + 15) = 175.

12.

Free notebooks were distributed equally among children of a class. The number of notebooks each child got was one-eighth of the number of children. Had the number of children been half, each child would have got 16 notebooks. Total how many notebooks were distributed ?
 A. 256 B. 432 C. 512 D. 640 E. None of these

nswer: Option C

Explanation:

Let total number of children be x.

 Then, x x 1 x = x x 16 x = 64. 8 2 Number of notebooks = 1 x2 = 1 x 64 x 64 = 512. 8 8

13.

A man has some hens and cows. If the number of heads be 48 and the number of feet equals 140, then the number of hens will be:
 A. 22 B. 23 C. 24 D. 26

Explanation:

Let the number of hens be x and the number of cows be y.

Then, x + y = 48 …. (i)

and 2x + 4y = 140 x + 2y = 70 …. (ii)

Solving (i) and (ii) we get: x = 26, y = 22. The required answer = 26.

14.

 (469 + 174)2 – (469 – 174)2 = ? (469 x 174)
 A. 2 B. 4 C. 295 D. 643

Explanation:

 Given exp. = (a + b)2 – (a – b)2 ab

 = 4ab ab

= 4 (where a = 469, b = 174.)

15.

David gets on the elevator at the 11th floor of a building and rides up at the rate of 57 floors per minute. At the same time, Albert gets on an elevator at the 51st floor of the same building and rides down at the rate of 63 floors per minute. If they continue travelling at these rates, then at which floor will their paths cross ?
 A. 19 B. 28 C. 30 D. 37

Explanation:

Suppose their paths cross after x minutes.

Then, 11 + 57x = 51 – 63x 120x = 40

 x = 1 3

 Number of floors covered by David in (1/3) min. = 1 x 57 = 19. 3

So, their paths cross at (11 +19) i.e., 30th floor.