IBA ফ্যাকাল্টি ভিত্তিক প্রস্তুতি

Question: In a factory, 20 workers can make 20 toys in 10 days working 10 hours per day. Then, in how many days can 25 workers make 30 toys working 20 hours per day?

Solution:
Given:

Number of workers initially = 20
Number of days initially = 10
Hours per day initially = 10
Number of toys initially = 20
Number of workers later = 25
Hours per day later = 20
Number of toys later = 30
Number of days later = ?

∴ (Number of workers initially × Number of days initially × Hours per day initially × Number of toys later)
= (Number of workers later × Number of days later × Hours per day later × Number of toys initially)
⇒ 20 × 10 × 10 × 30 = 25 × (Number of days later) × 20 × 20
⇒ 20 × 10 = 200 → 200 × 10 = 2,000 → 2,000 × 30 = 60,000
⇒ 25 × 20 × 20 = 10,000 × (Number of days later)
⇒ 60,000 = 10,000 × (Number of days later)

∴ Number of days later = 60,000 ÷ 10,000 = 6 days

Question: Working 4 hours a day, P can complete a work in 8 days and working 8 hours a day, Q can complete the same work in 4 days. Working 8 hours a day, they can jointly complete the work in-

Solution:
Working 4 hours a day, P can complete the work in 8 days
∴ P can complete the work in = 4 × 8 = 32 hours
∴ P can complete in 1 hour = 1/32 part

Working 8 hours a day, Q can complete the work in 4 days
∴ Q can complete the work in = 8 × 4 = 32 hours
∴ Q can complete in 1 hour = 1/32 part

(P + Q)'s 1 hour's work,
(1/32) + (1/32) = 2/32 = 1/16

∴ P and Q can complete the work in 16 hours
So, working 8 hours a day they require = 16/8
= 2 days

Question: Rahim's regular pay is Tk 40 per hour up to 40 hours. Overtime is paid at twice the regular rate. If he was paid Tk 2400 in total, how many hours of overtime did he work?

Solution:
Rahim’s regular wage for 40 hours = (40 × 40) = 1600 Taka.
Amount earned from overtime = (2400 - 1600) Taka = 800 Taka.
Since the overtime rate is twice the regular hourly wage,
Total overtime hours worked = 800 ÷ (40 × 2) hours
= 800 ÷ 80
= 10 hours

Question: In a particular country, one person is born every 6 seconds while one person dies every 10 seconds. Based on these birth and death rates, the net increase in population is one person after how many seconds?

Solution:
Let,
x be the number of seconds for the population to increase by one person.

We know,
Population growth = Birth rate - Death rate 

ATQ,
⇒ 1 person/6 seconds - 1 person/10 seconds = 1 person/x seconds
⇒ (5 - 3) person/30 seconds = 1 person/x seconds
⇒ 2 persons/30 seconds = 1 person/x seconds
⇒ 1 person/15 seconds = 1 person/x seconds

∴ x = 15 seconds

Question: To complete a work, P takes 25% more time than Q. If together they take 20 days to complete the work, how much time will Q take to do it?

Solution:
Let,
Q takes x days to complete the work

Then P will take 25% more time
i.e. 125% of x days
i.e. (5/4)x days

So, the one day’s work of P and Q together will be
(1/x) + {1/(5x/4)} = 1/20

⇒ (1/x) + (4/5x) = 1/20
⇒ (9/5x) = 1/20
⇒ x = 36

∴ Q takes 36 days to complete the work.

Question: Adnan can do 1/5 of a work in 8 days. In how many days will he complete the work?

Solution:
Adnan can do 1/5 of a work in 8 days.
∴ He will complete the work in = 8 × 5 = 40 days

∴ Adnan will complete the work in 40 days.

Question: 30 workers can manufacture 30 machines working 5 hours a day. How many workers need to be appointed extra to triple the production if they work 10 hours a day?

Solution:
5 hours to manufacture 30 machines by 30 workers
∴ 1 hour to manufacture 1 machine by = (30 × 5)/30 workers
∴ 10 hours to manufacture 90 machines by = (5 × 90)/10 workers
= 45 workers

∴ Extra workers required
= (45 - 30)
= 15 workers

Question: A truck can carry 24 motorcycles or 36 scooters at a time. If there are 10 motorcycles on the truck, how many scooters can be loaded onto it?

Solution:
Here,
24 motorcycles = 36 scooters
∴ 1 motorcycle = 36/24 scooters = 3/2 scooters
∴ 10 motorcycles = (36 × 10)/24 scooters = 15 scooters

∴ Maximum number of scooters that can still be loaded = 36 - 15 = 21 scooters

Question: Ali can type 60 pages in 20 minutes. Sara can type 12 pages in 12 minutes. Working together, how many pages can they type in 30 minutes?

Solution:
Ali can type in 1 min = 60/20 = 3 pages
Sara can type in 1 min = 12/12 = 1 page

∴ Working together they can type in 1 min = (3 + 1) pages = 4 pages

∴ They can type in 30 min = 4 × 30 pages = 120 pages

Question: If 5 identical machines, operating at a constant speed, can manufacture 200 pencils in one minute, how many pencils will 12 such machines produce in 10 minutes at the same rate?

Solution:
Given,
In 1 minute, 5 machines can produce 200 pencils
In 1 minute, 1 machine can produce 200/5 = 40 pencils

So, in 10 minutes, 12 machines can produce = (40 × 12 × 10) pencils
= 4800 pencils

Question: A team of workers can finish a project in 20 days. However, when 5 of them were absent, it took 25 days to complete the same work. How many workers were originally assigned to the project?

Solution:
Let,
the total number of people working originally = x
When 5 people were absent,
Total present workers = x - 5

x workers can complete the work in 20 days
∴ 1 worker can complete it in 20x days
∴ (x - 5) workers can complete it in 20x/(x - 5) days

ATQ,
20x/(x - 5) = 25
⇒ 4x/(x - 5) = 5
⇒ 5x - 25 = 4x
∴ x = 25

∴ The total number of people working originally = 25

Question: If a team of 5 workers can assemble a motorcycle in 6 hours, how many hours would it take a team of 10 workers to assemble the same motorcycle, working at the same constant rate?

Solution:
Given,
5 workers can assemble a motorcycle in 6 hours
∴ 1 worker can assemble a motorcycle in (6 × 5) = 30 hours
∴ 10 workers can assemble a motorcycle in 30/10 = 3 hours
= 3 × 60 minutes
= 180 minutes

Question: If 4 carpet weavers can complete 4 carpets in 4 days, how many carpets can 8 weavers produce in 8 days, assuming they work at the same constant rate?

Solution:
4 carpet-weavers in 4 days weave 4 carpets
∴ 1 carpet-weaver in 1 day weaves 4/(4 × 4) = 1/4 carpet

∴ 8 carpet-weavers in 8 days weave = (1/4 × 8 × 8) carpets
= 16 carpets

Question: A and B can finish a job together in x days. If A alone can complete the job in x + 3 days, and B alone can complete it in x + 12 days, what is the value of x?

Solution:
A's 1 day's work = 1/(x + 3) part
B's 1 day's work = 1/(x + 12) part
(A + B)'s 1 day's work = 1/x

ATQ,
1/(x + 3) + 1/(x + 12) = 1/x
⇒ (x + 12 + x + 3)/[(x + 3)(x + 12)] = 1/x
⇒ (2x + 15)/(x² + 15x + 36) = 1/x
⇒ 2x² + 15x = x² + 15x + 36
⇒ x² = 36
∴ x = 6

Question: Alif can fill 60 envelopes per minute, and Tonoy can fill 40 envelopes per minute. Working together, how long will they take to fill 500 envelopes?

Solution:
Given,
Alif can fill 60 envelopes in 1 minute
Tonoy can fill 40 envelopes in 1 minute

So together, they can fill in 1 minute = 60 + 40 = 100 envelopes

∴ 500 envelopes can be filled in 500 ÷ 100 = 5 minutes

Question: Rihan can write 120 pages in 30 hours. Zayan and Rihan together can write 240 pages in 40 hours. In what time can Zayan write 60 pages?

Solution:
Given,
In 30 hours Rihan can write 120 pages
∴ In 1 hour Rihan can write 120 ÷ 30 = 4 pages

Rihan and Zayan together can write 240 ÷ 40 = 6 pages per hour

∴ Zayan's 1 hour work = (Rihan + Zayan)'s 1 hour work - Rihan's 1 hour work
= 6 - 4 = 2 pages/hour

Zayan's time:
2 pages in 1 hour
∴ 1 page in 1/2 hour
∴ 60 pages in (1 × 60) ÷ 2
= 30 hours

Question: A can finish a task in 36 days, B in 54 days, and C in 72 days. All three start working together, but A leaves 8 days before the work is completed, and B leaves 12 days before completion. C works alone until the task is finished. How many days does it take to complete the work?

Solution: 
Let the work be completed in y days.
C works for y days

Therefore, A works for (y - 8) days
and, B works for (y - 12) days.

According to the question,
{(y - 8)/36} + {(y - 12)/54} + (y/72) = 1
⇒ {6(y - 8) + 4 (y - 12) + 3y}/216 = 1
⇒ 6(y - 8) + 4 (y - 12) + 3y = 216
⇒ 6y - 48 + 4y -  48 + 3y = 216
⇒ 13y = 216 + 96 = 312
⇒ y = 312/13
∴ y = 24