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Combined 8 bank_Senior Officer-2018

October 29, 2018

1094

[Set-A, Shapla-2018, 31.08.2018]

SECTION-C: GENERAL MATHEMATICS

Questions (41-60): Read the following questions carefully and choose the right answer:

41. There are 3 green, 4 orange and 5 white color bulbs in a bag. If a bulb is picked at random, what is the probability of having either a green or a white bulb?

(a) $\frac{3}{4}$ (b) $\frac{2}{3}$ (c) $\frac{4}{3}$ (d) $\frac{2}{5}$

Ans. (b) $\frac{2}{3}$

ব্যাখ্যা: Total bulbs = $3+4+5=12$

$\therefore$ The probability of having either a green or a white bulb= $\frac{{3 + 5}}{{12}} = \frac{8}{{12}} = \frac{2}{3}$

42. A motor-cycle covers 40 km with a speed of 20 km/hr. find the speed of the motor-cycle for the next 40 km journey so that the average speed of the whole journey will be 30 km/hr.

(a) $70$ km/hr (b) $52.5$ km/hr (c) $60$ km/hr (d) $60.5$ km/hr.

Ans. (c) $60$ km/hr

ব্যাখ্যা : Total distance cover in the whole journey = $(40+40)=80$ km.

Total time = $\frac{{80}}{{30}}$ hour

= $\left( {\frac{8}{3} \times 60} \right) = 160$ minutes [ $\because$ 1 hour= 60 minutes]

Time for first 40 km= $\frac{{40}}{{20}}$ = $2$ hour or 120 minutes.

Time for next 40 km= $\left( {160 - 120} \right) = 40$ minutes

So, speed for next 40 km= $\frac{{40}}{{40}} \times 60$ [ $\because$ 1 hour = 60 minutes]

= $60$ km/hr.

43. With a uniform speed, a car covers a distance in 8 hours. Had the speed been increased by 4 km/hr, the same distance could have been covered in 7 hours and 30 minutes. what is the distance covered ?

(a) 420 km (b) 480 km (c) 520 km (d) 640 km

Ans: (b) 480 km

ব্যাখ্যা: Let,distance =x km

7 hours 30 minutes = $7\frac{{30}}{{60}} = \frac{{15}}{2}$ hour

so, $\frac{{2x}}{{15}} - \frac{x}{8} = 4$

$\Rightarrow \frac{{16x - 15x}}{{120}} = 4$

$\Rightarrow x = 480$

44. The respective ratio between the speed of a car, a train and a bus is 5 : 9 : 4. The average speed of the car , bus and train is 72 km/hr together. What is the average speed of the car and the train together ?

(a) 82 km/h (b) 72 km/h (c) 67 km/h (d) 84 km/h

ব্যাখ্যা: Let, speed for the car, the train and the bus be 5x , 9x and 4x .

So, $\frac{{5x + 9x + 4x}}{3} = 72$

$\Rightarrow 18x = 72 \times 3$

$\Rightarrow x = \frac{{72 \times 3}}{{18}}$

$\therefore x = 12$

$\therefore$ Average speed of car and train

= $\frac{{5x + 9x}}{2} = \frac{{5 \times 12 + 9 \times 12}}{2}$

$= \frac{{60 + 108}}{2} = \frac{{168}}{2} = 84$

45.A, B and C are partners of a company. During aparticular year A received one-third of the profit, B received one-forth of the profit and C received the remaining TK. 5000. How much did A receive ?

(a) TK. 5000 (b) TK. 4000 (c) TK. 3000 (d) TK. 1000

Ans. (b) TK. 4000

ব্যাখ্যা : Let, Total potal portion 1

C’s portion = $1 - \left( {\frac{1}{3} + \frac{1}{4}} \right)$

$= 1 - \left( {\frac{{4 + 3}}{{12}}} \right)$

$= 1 - \frac{7}{{12}}$

$= \frac{{12 - 7}}{{12}}$

$= \frac{5}{{12}}$

So, $= \frac{5}{{12}}$ portion = TK. 5000

$\therefore$ 1 portion = TK. $\frac{{5000 \times 12}}{5}$

=TK. $4000$

The average ageof a group of 15 employees is 24 years. If 5 more employees join the group, the average age increases by 2 years. Find the average age of the new employees?

(a) 35 (b) 30 (c) 24 (d) 32

Ans. (d) 32

ব্যাখ্যা : Total ages of $15$ employees = $\left( {24 \times 15} \right)$ years = $360$ years

Total ages of $15 +5$ = $20$ employees

= $\left( {26 \times 20} \right)$ years

= $520$ years

$\therefore$ Average ages of 5 new employees = $\frac{{520 - 360}}{5} = \frac{{160}}{5} = 32$ years

47.By selling 32 guavas for TK. 30 at the rate of TK. 1.066 per guava a man loss 25%. How many guavas should be sold for TK. 18 to gain 20% of profit in the transaction?

(a) 24 (b) 12 (c) 18 (d) 36

Ans. (b) 12

ব্যাখ্যা: Selling price at 25% loss = TK. (100-25)= TK. 75

Selling price at 20% profit = TK. (100 + 20) = TK. 120

Now, 75% = TK. 30

$\therefore$ 1% = TK. $\frac{{30}}{{75}}$

$\therefore$ 120% = TK. $\frac{{30 \times 120}}{{75}}$

=TK. $48$

So, By Tk. 48 to sell 32

$\therefore$ By TK 1 to sell $\frac{{32}}{{48}}$

$\therefore$ By TK 18 to sell $\frac{{32 \times 18}}{{48}} = 12$

A sold a watch to B at a gain of 20% and B sold it to C at a loss of 10%. If C bought the watch for Tk. 216, at what price did A purchase it?

(a) TK. 200 (b) TK. 216 (c) TK. 250 (d) TK. 176

Ans. (a) TK. 200

Let, Cost price of A =TK. 100

Selling price of A and cost price of B = TK. (100 + 20) = TK. 120

Selling price of B and cost price of C = TK. $\left( {120 \times \frac{{90}}{{100}}} \right)$

=TK. $108$

Now,

108%= TK. 216

$\therefore$ 1%= $\frac{{216}}{{108}}$

$\therefore$ 100% =TK. $\frac{{216 \times 100}}{{108}}$

=TK. 200

A square is inscribed in a circle of diameter 2a and another square is circumscribing circle. The difference between the areas of outer and inner squares is-

(a) ${a^2}$ (b) $2{a^2}$ (c) $3{a^2}$ (d) $4{a^2}$

Ans. (b) $2{a^2}$

ব্যাখ্যা: Diameter of the circle and a side of the outer square =2a

$\therefore$ Area of outer square = ${\left( {2a} \right)^2} = 4{a^2}$

Diagonal of inner square =2a

So, $\sqrt 2 \times a$ side =2a

a $\Rightarrow side = \frac{{2a}}{{\sqrt 2 }}$

$\therefore$ Area of inner square = ${\left( {\frac{{2a}}{{\sqrt 2 }}} \right)^2} = 2{a^2}$

$\therefore$ Required difference = $4{a^2} - 2{a^2} = 2{a^2}$

The average of the three numbers x, y and z is 45, x is greater than the average of y and z by 9. The average of y and z is greater than y by 2. Then the difference of x and z is-

(a) 3 (b) 5 (c) 7 (d) 11

Ans. (c) 7

ব্যাখ্যা: According to question ,

$\frac{{x + y + z}}{3} = 45$

$\Rightarrow x + y + z = 45 \times 3$

$\therefore x + y + z = 135............(1)$

And, $x = \frac{{y + z}}{2} + 9$

$\Rightarrow x = \frac{{y + z + 18}}{2}$

$\Rightarrow 2x = y + z + 18.............(2)$

Again, $\frac{{y + z}}{2} = y + 2$

$\Rightarrow y + z = 2y + 4$

$\Rightarrow 2y - y = z - 4$

From (2),

$2x = y + z + 18$

$\Rightarrow 2x = z - 4 + z + 18\left[ {\therefore y = z - 4} \right]$

$\Rightarrow 2x - 2z = 14$

$\Rightarrow 2\left( {x - z} \right) = 14$

$\Rightarrow x - z = 7$

51.A boat travel with a speed of 10 km/hr in still water. If the speed of the stream is 3 km/hr then find time taken by boat to travel 52 km downstream.

(a) 2 hrs (b) 4 hrs (c) 6 hrs (d) 9 hrs

Ans. (b) 4 hrs

ব্যাখ্যা: Speed of the downstream= $(10+3)$ km/hr

=13 km/hr

$\therefore$ Required time for =52 km downstream

= $\frac{{52}}{{13}} = 4$ hours

Rahima can row 16 km/hr in still water. It takes her thriceas long to row up as to row down the river. Find the difference between her speed in still water and that of the stream.

(a) 8 km/hr (b) 16 km/hr (c) 24 km/hr (d) 12 km/hr

Ans: (a) 8 km/hr

ব্যাখ্যা: Let, speed of the stream x km/h

Row up speed = $(16-x)$ km/h

Row down speed = $(16+x)$ km/h

So, $\left( {16 - x} \right)3 = 16 + x$

$\Rightarrow 48 - 3x = 16 + x$

$\Rightarrow - 3x - x = 16 - 48$

$\Rightarrow - 4x = - 32$

$\therefore x = 4$

A train leaves a station A at 7 am and reaches another station B ar 11 am. Another train leaves B at 8 am and reaches A at 11:30 am. The two trains cross one another at-

(a) 8:36 am (b) 8:56 am (c) 9:00 am (d) 9:24 am

Ans: (d) 9:24 am

ব্যাখ্যা: Let, Distance A to B is x km

Speed from A to B = $\frac{2}{x}$

Speed from B to A = $\frac{{\frac{x}{7}}}{2}$ km/h

= $\frac{{2x}}{7}$ km/h

Suppose,they meet t hours after $7$ am

So, $\frac{{tx}}{4} + \frac{{2x(t - 1)}}{7} = x$

$\Rightarrow \frac{{7tx + 8tx - 8x}}{{28}} = x$

$\Rightarrow 15tx = 28x + 8x$

$\Rightarrow t = \frac{{36x}}{{15x}}$

$\therefore t = 2.4$

Or,2 hours 24 minutes

$\therefore$ Two trains cross at 7 am + 2 hours 24 minutes

Or, 9:24 am.

A train 150 m long crosses a milestone in 15 seconds and crosses another train of the same length travelling in the oppsite direction in 12 seconds. The speed of the second train in km/hr is-

(a) 52km/hr (b) 56 km/hr (c) 54 km/hr (d) 58 km/hr

Ans. (c) 54 km/hr

ব্যাখ্যা: Speed of the train = $\frac{{150}}{{15}} = 10$ m/sec

= $\left( {10 \times \frac{{18}}{5}} \right) = 36$ km/h

In $12$ second distance cover = $\left( {150 + 150} \right)$ m

=300 meter

$\therefore$ Speed of the two train = $\frac{{300}}{{12}} = 25$ m/sec

= $\left( {25 \times \frac{{18}}{5}} \right) = 90$ km/h

$\therefore$ Speed of the 2^nd train = $(90-36)$ km/h

= $54$ km/h

There are 2 numbers in the ratio of 4 : 5. If 4 is substracted from both numbers the ratio becomes 3 : 4. What will be the ratio if 4 is added in the both numbers ?

(a) 1 : 2 (b) 2 : 3 (c) 5 : 6 (d) 1 : 4

Ans. (c) 5 : 6

ব্যাখ্যা: Let, 4x and 5x be the numbers

So, $\frac{{4x - 4}}{{5x - 4}} = \frac{3}{4}$

$\Rightarrow 16x - 16 = 15x - 12$

$\Rightarrow 16x - 15x = - 12 + 16$

$\therefore x = 4$

$\therefore$ Required ratio = $4 \times 4 + 4:5 \times 4 + 4$

$= 20:24 = 5:6$

The income of ‘A’ is 20% higher than that of ‘B’. The income of ‘B’ is 25% less than of ‘C’. What percent less is a’s income from C’s income ?

(a) 7% (b) 8% (c) 9% (d) 10%

Ans. (d) 10%

ব্যাখ্যা: Let, Income of C =TK. 100

Income of B =TK. $(100-25)$ = TK. 75

Income of A= TK. $\left( {75 \times \frac{{120}}{{100}}} \right)$ =TK. $90$

$\therefore$ Difference = TK $\left( {100 - 90} \right)$

=TK. $10$ or $10\%$

The ratio of two numbers is 7 : 4. If 8 is added both the numbers ratio becomes 13 : 8. What is the smaller number?

(a) 40 (b) 56 (c) 38 (d) 52

Ans. (a) 40

ব্যাখ্যা: Let, Larger no. Be $7x$

Smaller no. Be $4x$

So, $\frac{{7x + 8}}{{4x + 8}} = \frac{{13}}{8}$

$\Rightarrow 56x + 64 = 52x + 104$

$\Rightarrow 56x - 52x = 104 - 64$

$\Rightarrow 4x = 40$

$\therefore x = 10$

$\therefore$ Smaller no. Be = $\left( {4 \times 10} \right) = 40$

a and B can do a work in 12 days . B can do the same work in 18 days. In how many days A can complete the $\frac{2}{3}$ of the same work?

(a) 36 days (b) 24 days (c) 16 days (d) 27 days

Ans. (b) 24 days

ব্যাখ্যা: (A+B) can do in $1$ day = $\frac{1}{{12}}$ part

B can do in $1$ day = $\frac{1}{{18}}$ part

$\therefore$ A can do in $1$ day = $\left( {\frac{1}{{12}} - \frac{1}{{18}}} \right)$ part

A can $\frac{1}{{36}}$ part in $1$ day

A can $1$ part in $36$ day

A can $\frac{2}{3}$ part in $\frac{{36 \times 2}}{3}$ day

= 24 days

A is twice as good as B and together they finish a piece of work in $16$ days. The number of days taken by A alone to finish the work is-

(a) 20 days (b) 21 days (c) 22 days (d) 24 days

Ans. (d) 24 days

ব্যাখ্যা: Let, A can do in $2x$ days

B can do in x days

So, $\frac{1}{{2x}} + \frac{1}{x} = \frac{1}{{16}}$

$\Rightarrow \frac{{1 + 2}}{{2x}} = \frac{1}{{16}}$

$\Rightarrow \frac{3}{{2x}} = \frac{1}{{16}}$

$\Rightarrow 2x = 48$

$\therefore x = 24$

A box contains $5$ pink, 3 green and 2 yellow balls. Three balls are picked up randomly. What is the probability that none of the ball drawn is green?

(a) $\frac{3}{{16}}$ (b) $\frac{7}{{24}}$ (c) $\frac{5}{{13}}$ (d) $\frac{4}{{23}}$

Ans. (b) $\frac{7}{{24}}$

ব্যাখ্যা: The number of ways of selecting 3 balls from 10 = ${10_C}_{_3} = \frac{{10 \times 9 \times 8}}{{3 \times 2}} = 120$

The number of selecting 3 non-green balls from $(5 + 2)$ or, $7$ non-green = ${7_C}_{_3} = \frac{{7 \times 6 \times 5}}{{3 \times 2}} = 35$

$\therefore$ Required Probability = $\frac{{35}}{{120}} = \frac{7}{{24}}$

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