Time & Work, Chain Rule

PrepBank · পাতা ৩ / ১১ · ২০১–৩০০ / ১,০৭৬

২০১.

Pooja is twice as efficient as Aarti and takes 90 days less than Aarti to complete the job. Find the time in which they can finish the job together.

ক
ক) 30 days
খ
খ) 45 days
গ
গ) 60 days
ঘ
ঘ) 90 days

ব্যাখ্যা

Question: Pooja is twice as efficient as Aarti and takes 90 days less than Aarti to complete the job. Find the time in which they can finish the job together.

Solution:
পূজা কাজটি করতে পারে = 90 দিনে
আরতি কাজটি করতে পারে = 180 দিনে

পূজা ও আরতি 1 দিনে করতে পারে কাজটির = 1/90 + 1/180 অংশ
= (2 + 1)/180
= 3/180
= 1/60

পূজা ও আরতি 1/60 অংশ কাজ করতে পারে 1 দিনে
পূজা ও আরতি 1 অংশ কাজ করতে পারে (1× 60)/1 দিনে
= 60 দিনে

২০২.

If one pipe A can fill a tank in 20 minutes, then 5 pipes, each of 20% efficiency of A, can fill the tank in -

ক
ক) 20 minutes
খ
খ) 25 minutes
গ
গ) 28 minutes
ঘ
ঘ) 30 minutes

ব্যাখ্যা

Question: If one pipe A can fill a tank in 20 minutes, then 5 pipes, each of 20% efficiency of A, can fill the tank in -

Solution:
ধরি,
A পাইপের ইফিসিয়েন্সি = ১০০%
৫ টি পাইপের প্রতিটির ইফিসিয়েন্সি = ১০০%/৫ = ২০%

তাহলে, ৫ টি পাইপের মোট ইফিসিয়েন্সি = ২০% × ৫ = ১০০%

তাহলে, ৫ টি পাইপ এর সমষ্টিগত ইফিসিয়েন্সি A পাইপের সমান।
তাহলে ৫ টি পাইপ একত্রে ২০ মিনিটেই টাঙ্কটি পূর্ণ করতে পারবে।

২০৩.

Rahim and Karim can finish a work together in 6 hours. If Rahim takes 3 times as long as Karim to finish the job alone, how long will Karim take to finish the job alone?

ক
9 hours
খ
8 hours
গ
12 hours
ঘ
15 hours

ব্যাখ্যা

Question: Rahim and Karim can finish a work together in 6 hours. If Rahim takes 3 times as long as Karim to finish the job alone, how long will Karim take to finish the job alone?

Solution:
ধরি, করিম একা কাজটি শেষ করতে সময় নেয় = x ঘন্টা
রহিম একা কাজটি শেষ করতে সময় নেয় = 3x ঘন্টা

এখন,
করিমের 1 ঘন্টার কাজের পরিমাণ = 1/x অংশ
রহিমের 1 ঘন্টার কাজের পরিমাণ = 1/(3x) অংশ

তারা একসাথে 1 ঘন্টায় কাজ করতে পারে = (1/x) + 1/(3x)
= (3+1)/(3x)
= 4/(3x) অংশ

একসাথে কাজটি শেষ করতে সময় লাগে = 6 ঘন্টা

অর্থাৎ, 6 ঘন্টায় তারা পুরো 1টি কাজ শেষ করে

তাহলে,
(4/3x) × 6 = 1
⇒ 24/(3x) = 1
⇒ 8/x = 1
⇒ x = 8

∴ করিমের কাজটি একা শেষ করতে 8 ঘন্টা সময় লাগবে।

২০৪.

Karim and Bablu can build a wooden table in 3 days. Bablu can do it alone in 5 days. How many days would it take Karim to do this job alone?

ক
0.2
খ
7.5
গ
5.0
ঘ
6.4
ঙ
None of these

ব্যাখ্যা

Question: Karim and Bablu can build a wooden table in 3 days. Bablu can do it alone in 5 days. How many days would it take Karim to do this job alone?

Solution:
Let's say Karim's work rate per day = 1/x, where x is the days he takes alone
bablu's work rate per day = 1/5 (as he takes 5 days alone)
Together they take 3 days, so their combined rate = 1/3

ATQ,
(1/x) + (1/5) = 1/3
⇒ (5 + x)/5x = 1/3
⇒ 15 + 3x = 5x
⇒ 15 + 3x - 5x = 0
⇒ 15 - 2x = 0
⇒ 2x = 15
∴ x = 7.5

Therefore, Karim would take 7.5 days to build the table alone.

২০৫.

A tap can fill a tank in 6hours. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?

ক
ক) 3hrs 15min
খ
খ) 3hrs 45 min
গ
গ) 4hrs
ঘ
ঘ) 4hrs 15min

ব্যাখ্যা

Time taken by the tap to make the tank half full= 3 hrs.
Remaining part = 1/2
Part filled by 4 taps in 1 hour= (4×1/6) = 2/3
2/3 part is filled in 1 hour.
1/2 part is filled in (3/2×1/2) hr = 3/4 hr = 45 min.
Required time = 3hrs 45 min.

২০৬.

A mechanic repairs 4 cars in 5/3 hours. How many cars can he repair in 50 minutes?

ক
2
খ
21/3
গ
22/3
ঘ
25/6
ঙ
None of these

ব্যাখ্যা

Question: A mechanic repairs 4 cars in 5/3 hours. How many cars can he repair in 50 minutes?

Solution:
Here,
5/3 hours = (5/3) × 60 minutes
= 100 minutes

In 100 minutes he repairs = 4 cars
In 1 minutes he repairs = 4/100 cars
So, In 50 minutes he repairs = (4 × 50)/100 cars
= 2 cars

২০৭.

A garrison is provided with ration for 90 soldiers to last for 70 days. For how much more time would the whole ration last if 10 additional soldiers join them after 20 days?

ক
40 days
খ
36 days
গ
30 days
ঘ
56 days
ঙ
45 days

ব্যাখ্যা

Question: A garrison is provided with ration for 90 soldiers to last for 70 days. For how much more time would the whole ration last if 10 additional soldiers join them after 20 days?

Solution:
Let the whole ration now lasts for x days.
Equating the consumption on both sides, we get
(90 × 70) = (90 × 20) + (100 × x)
⇒ 630 = 180 + 100x
⇒ 100x = 450
⇒ x = 45 days.

২০৮.

A cistern can be filled by two pipes in 20 and 30 minutes respectively. Both pipes being opened, when the first pipe must be turned off so that the cistern may be filled in 10 minutes more.

ক
ক) after 8 minutes
খ
খ) after 9 minutes
গ
গ) after 12 minutes
ঘ
ঘ) after 15 minutes

ব্যাখ্যা

% Cistern is filled by 1st pipe in one minute
= 100/20
= 5%
% Cistern is filled by 2nd pipe in one minute
= 100/30
= 3.33%
% cistern filled by 1st and 2nd pipes in one minute = 8.33%

According to question,
Cistern is totally filled by 2nd pipe in last 10 minute.
That means 2nd pipe filled 33.3% of the cistern in last 10 minute and 66.66% of cistern is filled by 1st and 2nd pipe together in = 66.66/8.33 = 8 minutes
Thus, after 8 minute, 1st pipe must be turned off.

২০৯.

A garrison of 600 men has provisions to last 30 days. After 3 days, 300 additional men arrive. How many days will the remaining food now last?

ক
18 days
খ
20 days
গ
15 days
ঘ
12 days

ব্যাখ্যা

Question: A garrison of 600 men has provisions to last 30 days. After 3 days, 300 additional men arrive. How many days will the remaining food now last?

Solution:
After 3 days, food having for = (30 - 3) = 27 days
After arriving 300 men, total men = (600 + 300) = 900 men

600 men can eat the food for 27 days
∴ 1 man can eat the food for (27 × 600) days
∴ 900 men can eat the food for (27 × 600)/900 days
= 18 days

২১০.

A man can finish a work in 20 days, a woman can finish the same work in 30 days, and a boy can finish the same work in 60 days. 2 men, 8 women, and some boys working together can finish the work in 2 days. Find how many boys are working?

ক
10 boys
খ
8 boys
গ
9 boys
ঘ
15 boys

ব্যাখ্যা

Take the LCM of days = LCM of (20, 30, and 60) = 60
Let the total work = 60

1 Man's one day work = 60/20 = 3 unit
1 Woman's one day work = 60/30 = 2 unit
1 The boy's one day work = 60/60 = 1 unit........................ (i)

ATQ, (2Men + 8Women + x Boy) can finish a work in 2 days.

Or, (2Men + 8women)'s 1 day work = (2 × 3) + (8 × 2) = 22

Or, (2Men + 8women)'s 2 day work = 22 × 2 = 44
Remaining work = 60 - 44 = 16
That means the remaining 16 work is done by X boys in 2 days
Or, X boys' one day work = 16/2 = 8

By equation 1, 1 boy completes 1 unit of work per day
So, 8 boys are required to complete 8 units of work per day.

২১১.

30 workers can build 60 machines working 4 hours per day. How many extra workers are required to produce 90 machines working 6 hours per day?

ক
0 workers
খ
10 workers
গ
20 workers
ঘ
15 workers

ব্যাখ্যা

Question: 30 workers can build 60 machines working 4 hours per day. How many extra workers are required to produce 90 machines working 6 hours per day?

Solution:
4 hours to build 60 machines by 30 workers
1 hour to build 1 machine by = (30 × 4)/60 workers
6 hours to build 90 machines by = (2 × 90)/6 workers
= 30 workers

∴ extra workers = (30 - 30) = 0 workers

২১২.

A group of 8 men can paint a house in 20 days. If they want to complete the painting in 16 days, how many more men should they add to the group?

ক
6
খ
5
গ
3
ঘ
2

ব্যাখ্যা

Question: A group of 8 men can paint a house in 20 days. If they want to complete the painting in 16 days, how many more men should they add to the group?

Solution:
২০ দিনে কাজ শেষ করতে পারে ৮ জন
১ দিনে কাজ শেষ করতে পারে (২০ × ৮) জন
১৬ দিনে কাজ শেষ করতে পারে (২০ × ৮)/১৬ জন
= ১০ জন

∴ অতিরিক্ত (১০ - ৮) বা ২ জন যোগ করতে হবে।

২১৩.

A pump can fill a tank with water in 6 hours. It took 13/2 hours to fill the tank for a leak. The leak can drain all the water of the tank in:

ক
39 hours
খ
52 hours
গ
69 hours
ঘ
78 hours

ব্যাখ্যা

short-cut:
The leak can drain all the water of the tank in:
= (13/2 × 6)/(13/2 - 6)
= 39/(1/2)
= 78 hours
--------------------------------------------------
৬ ঘণ্টায় পূর্ণ হয় ১ টি চৌবাচ্চা
১ ঘণ্টায় পূর্ণ হয় ১/৬ অংশ

আবার, ছিদ্র থাকায় ১৩/২ ঘণ্টায় পূর্ণ হয় ১ টি চৌবাচ্চা
অতএব, ১ ঘণ্টায় পূর্ণ হয় ২/১৩ অংশ

অতএব, ছিদ্র কর্তৃক ১ ঘণ্টায় খালি হয় (১/৬ - ২/১৩) অংশ = ১/৭৮ অংশ
১/৭৮ অংশ খালি হয় ১ ঘণ্টায়
সম্পূর্ণ অংশ খালি হয় ৭৮ ঘণ্টায়

২১৪.

A man and a boy together can do a certain amount of digging in 28 days. Their speeds in digging are in the ratio of 7 : 4. How many days will the boy take to complete the work if engaged alone?

ক
72 days
খ
77 days
গ
78 days
ঘ
79 days
ঙ
None

ব্যাখ্যা

Question: A man and a boy together can do a certain amount of digging in 28 days. Their speeds in digging are in the ratio of 7 : 4. How many days will the boy take to complete the work if engaged alone?

Solution:
Ratio of digging speeds of man and boy = 7 : 4
Ratio of times taken by man and boy = 4 : 7
Suppose the man takes 4x days while the boy takes 7x
∴ Man's 1 day's work = 1/4x part
∴ Boy's 1 day's work = 1/7x part

ATQ,
1/4x + 1/7x = 1/28
⇒ (7 + 4)/28x = 1/28
⇒ 11/28x = 1/28
⇒ 28x = 28 × 11
∴ x = 11

Hence, time taken by the boy to complete the work alone = (7 × 11) days
= 77 days

২১৫.

Rasel can fill advertising circulars into envelopes at the rate of 45 envelopes per minute and Ali requires a minute and a half to fill the same number of envelopes. Working together, how long will it take Rasel and Ali to fill 300 envelopes?

ক
4 minutes
খ
6 minutes
গ
8 minutes
ঘ
12 minutes
ঙ
15 minutes

ব্যাখ্যা

Question: Rasel can fill advertising circulars into envelopes at the rate of 45 envelopes per minute and Ali requires a minute and a half to fill the same number of envelopes. Working together, how long will it take Rasel and Ali to fill 300 envelopes?

Solution:
Given,
Rasel can fill 45 envelopes in 1 minutes
and,
Ali can fill 45 envelopes in (1 + 1/2) = 3/2 minutes
So Ali can fill in 1 minutes = (45 ÷ 3/2) envelopes
= (45 × 2/3) envelopes
= 30 envelopes

So Rasel and Ali together can fill in 1 minutes = (45 + 30) = 75 envelopes

∴ 75 envelopes can fill in 1 minutes
∴ 1 envelopes can fill in 1/75 minutes
∴ 300 envelopes can fill in (1 × 300)/75 minutes
= 4 minutes

২১৬.

A recipe requires 2 cups of flour for 8 cupcakes. How many cups of flour are needed for 12 cupcakes?

ক
5 cups
খ
2 cups
গ
4 cups
ঘ
3 cups

ব্যাখ্যা

Question: A recipe requires 2 cups of flour for 8 cupcakes. How many cups of flour are needed for 12 cupcakes?

Solution:
We know the amount of flour required for 8 cupcakes (2 cups).
Flour per cupcake = Total flour/Number of cupcakes = 2 cups/8 cupcakes = 0.25 cups per cupcake.

We need to find the flour for 12 cupcakes.
Flour Required = Unit value(flour per cupcake) × Number of cupcakes
Flour Required = 0.25 cups/cupcake × 12 cupcakes = 3 cups.

২১৭.

Worker A completes a task in 8 days, and worker B completes the same task in 10 days. If both A and B work together, in how many days they will complete the task?

ক
4 days
খ
40/9 days
গ
5 days
ঘ
40/7 days

ব্যাখ্যা

Question: Worker A completes a task in 8 days, and worker B completes the same task in 10 days. If both A and B work together, in how many days they will complete the task?

Solution:
Worker A completes the task in 8 days. So, in one day, he will complete 1/8 part of the task.

So, A's one day work = 1/8
Similarly, B's one day work = 1/10

∴ (A+B)'s one day work = 1/8 + 1/10 = (5 + 4)/40 = 9/40

9/40 of the task is completed in one day so both will complete the whole task in 40/9 days

২১৮.

It takes 6 days for 3 women and 2 men working together to complete a work. 3 men would do the same work 5 days sooner than 9 women. How many times does the output of a man exceed that of a woman?

ক
3 times
খ
4 times
গ
5 times
ঘ
6 times
ঙ
7 times

ব্যাখ্যা

Question: It takes 6 days for 3 women and 2 men working together to complete a work. 3 men would do the same work 5 days sooner than 9 women. How many times does the output of a man exceed that of a woman?

Solution:
It takes 6 days for 3 women and 2 men working together to complete a work:
As the rate of 1 woman is 1/w job/day,
then the rate of 3 women will be 3/w job/day.

As the rate of 1 man is 1/m job/day,
then the rate of 2 men will be 2/m job/day.

Combined rate of 3 women and 2 men in one day will be: 3/w + 2/m job/day.

As they do all the job in 6 days then in 1 day they do 1/6 of the job,
which is combined rate of 3 women and 2 men ⇒ 3/w + 2/m = 1/6 ..............(1)

3 men would do the same work 5 days sooner than 9 women:
As 1 man needs m days to do the job 3 men will need m/3 days to do the job.
As 1 woman needs w days to do the job 9 women will need w/9 days to do the job.
3 men would do the same work 5 days sooner means that 3 men will need 5 less days to do the job,
hence m/3 is 5 less than w/9 ⇒ m/3 + 5 = w/9 ...................(2)

Solving (1) and (2) we get:
m = 15 and w = 90

∴ w/m = 15/90 = 1/6
∴ m = 6w

২১৯.

In an army camp there was food for 150 soldiers for 30 days. But after 10 days 50 soldiers left the army. The rest of the food will last for -

ক
ক) 25 days
খ
খ) 28 days
গ
গ) 30 days
ঘ
ঘ) 32 days

ব্যাখ্যা

Question: In an army camp there was food for 150 soldiers for 30 days. But after 10 days 50 soldiers left the army. The rest of the food will last for -

Solution:
১০ দিন পর, ১৫০ জনের ২০ দিনের খাবার ছিল।
৫০ জন কমে যাওয়ায় এখন সৈনিক = ১৫০ -৫০ = ১০০ জন।

তাহলে,
১৫০ জনের খাবার আছে ২০ দিনের
১ জনের আছে (২০ × ১৫০) দিনের
১০০ জনের আছে (৩০০০/১০০) দিনের
= ৩০ দিনের

Shortcut:
ধরি, বাকি খাবারে ১০০ জনের x দিন চলবে।
150 : 100 = x : 20
100x = 3000
x = 30 days

২২০.

A car washer machine can wash 8 cars in 18 minutes. How many cars can be washed at the same rate in 3 hours?

ক
ক) 54
খ
খ) 72
গ
গ) 80
ঘ
ঘ) 87
ঙ
ঙ) 96

ব্যাখ্যা

In 18 minutes a car washer can wash 8 cars
In 1 minutes the car washer can wash 8/18 cars
So, in 3 hrs (180 minutes) car washer can wash = (8×180)/18 = 80 cars

২২১.

A barrack has enough food for 150 soldiers or 300 sailors. If 120 sailors have already taken the food, how many soldiers can be fed with the remaining food?

ক
115
খ
100
গ
120
ঘ
90

ব্যাখ্যা

Question: A barrack has enough food for 150 soldiers or 300 sailors. If 120 sailors have already taken the food, how many soldiers can be fed with the remaining food?

Solution:
এখানে,
300 জন নাবিকের খাবার = 150 জন সৈন্যের খাবার।

মোট নাবিক যাদের জন্য খাবার ছিল = 300 জন।
খাবার গ্রহণ করেছে = 120 জন।
অবশিষ্ট খাবার = (300 - 120) জন নাবিকের খাবার
= 180 জন নাবিকের খাবার।

এখন,
300 জন নাবিকের খাবার = 150 জন সৈন্যের খাবার।
∴ 1 জন নাবিকের খাবার = (150/300) জন সৈন্যের খাবার।
∴ 180 জন নাবিকের খাবার = (150 × 180)/300
= (1/2) × 180
= 90 জন সৈন্যের খাবার।

সুতরাং, অবশিষ্ট খাবার দিয়ে 90 জন সৈন্যকে দেওয়া যাবে।

২২২.

A man works twice as much as his assistant. The man can finish the job in 9 days. Find the number of days in which the man and his assistant can finish the job if they work together?

ক
ক) 6 days
খ
খ) 8 days
গ
গ) 12 days
ঘ
ঘ) 15 days

ব্যাখ্যা

Question: A man works twice as much as his assistant. The man can finish the job in 9 days. Find the number of days in which the man and his assistant can finish the job if they work together?

Solution:
Since, Man works twice as much as his assistant
Assistant can do the work in (2 × 9) days = 18 days
A man can do in 1 day = 1/9 part
Assistant can do in 1 day = 1/18 part
They together can do in 1 day = (1/9) + (1/18) = 3/18 = 1/6 part
They can do the job in = 1/(1/6) days = 6 days

২২৩.

To complete a work, A takes 50% more time than B. If together they take 18 days to complete the work, how much time shall B take to do it?

ক
20 days
খ
25 days
গ
30 days
ঘ
33 days
ঙ
35 days

ব্যাখ্যা

Question: To complete a work, A takes 50% more time than B. If together they take 18 days to complete the work, how much time shall B take to do it?

Solution:
Let
B takes x days to complete the work
Then A will take 50% more i.e 150% of x days i.e 3/2x days.

So the one day work of A and B together will be
(1/x) + {1/(3/2x)} = 1/18
⇒ (1/x) + (2/3x) = 1/18
⇒ 5/3x = 1/18
⇒ x = 30

∴ B takes 30 days to complete the work.

২২৪.

Rahim finishes 40% of a task in 10 days. He then asks Karim to help and together they complete the remaining work in 10 days. How many days would Karim take to complete the whole task alone?

ক
24 days
খ
30 days
গ
42 days
ঘ
50 days

ব্যাখ্যা

Question: Rahim finishes 40% of a task in 10 days. He then asks Karim to help and together they complete the remaining work in 10 days. How many days would Karim take to complete the whole task alone?

Solution:
রহিম 40% বা 2/5 অংশ কাজ করে 10 দিনে।
∴ রহিম 1 দিনে কাজ করে = (2/5) ÷ 10 = 1/25 অংশ
∴ রহিম শেষ 10 দিনেও কাজ করেছে = 10 × (1/25) = 2/5 অংশ

মোট কাজ বাকি ছিল = 1 - 2/5 = 3/5 অংশ
এই 3/5 অংশ কাজের মধ্যে রহিম করেছে 2/5 অংশ
বাকি কাজ টুকু করিম করেছে = 3/5 - 2/5 = 1/5 অংশ

করিম 1/5 অংশ কাজ করে = 10 দিনে
∴ করিম সম্পূর্ণ (1 অংশ) কাজ একা করতে সময় নেবে = 10 × 5 দিন
= 50 দিন

২২৫.

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?

ক
11 carpets
খ
20 carpets
গ
12 carpets
ঘ
10 carpets
ঙ
The answer is not available

ব্যাখ্যা

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

২২৬.

The five individuals - A, B, C, D, and E - have an average age of 25 years. A and B together average 28 years, while C and D have an average of 32. Can you deduce the age of E?

ক
10 years
খ
8 years
গ
5 years
ঘ
None of the above

ব্যাখ্যা

Question: The five individuals - A, B, C, D, and E - have an average age of 25 years. A and B together average 28 years, while C and D have an average of 32. Can you deduce the age of E?

Solution:
The average age of A, B, C, D and E is = 25 years.
The total age of A, B, C, D and E is = (25 × 5) = 125 years.

The average age of A and B is = 28 years
The total age of A and B is = (28 × 2) = 56 years

The average of C and D is = 32 years
The total age of C and D is = (32 × 2) = 64 years

∴ Age of E is = (125 - 56 - 64) = 5 years

২২৭.

48 men can complete a piece of work in 18 days. In how many days can 36 men complete the same piece of work?

ক
20 days
খ
24 days
গ
26 days
ঘ
28 days

ব্যাখ্যা

Question: 48 men can complete a piece of work in 18 days. In how many days can 36 men complete the same piece of work?

Solution:
Let the required number of days be x
Less men, More days (Indirect proportion)
∴ 36 : 48 : : 18 : x
⇔ 36/48 = 18/x
⇔ 36x = 18 × 48
⇔ x = (18 × 48)/36
∴ x = 24

২২৮.

A worker was hired for 7 days. Each day, he was paid Tk. 10 more than what he was paid for the previous day of work. The total amount he was paid in the first 4 days of work equaled the total amount he was paid in the last 3 days. What was his starting pay?

ক
Tk. 90
খ
Tk. 138
গ
Tk. 150
ঘ
Tk. 160

ব্যাখ্যা

Question: A worker was hired for 7 days. Each day, he was paid Tk. 10 more than what he was paid for the previous day of work. The total amount he was paid in the first 4 days of work equaled the total amount he was paid in the last 3 days. What was his starting pay?

Solution:
ধরি,
১ম দিন ছিলো x টাকা

২য় দিন = x + 10
৩য় দিন = (x + 20)
৪র্থ দিন = (x + 30)
৫ম দিন = (x + 40)
৬ষ্ঠ দিন = (x + 50)
৭ম দিন = (x + 60) টাকা

প্রশ্নমতে,
x + (x + 10) + (x + 20) + (x + 30) = (x + 40) + (x + 50) + (x + 60)
⇒ 4x + 60 = 3x + 150
⇒ 4x - 3x = 150 - 60
∴ x = 90

২২৯.

G is twice as fast as S in doing work. If G can do a work in 30 days less than S, how many days will they take to complete the work together?

ক
25
খ
20
গ
22
ঘ
15

ব্যাখ্যা

Question: G is twice as fast as S in doing work. If G can do a work in 30 days less than S, how many days will they take to complete the work together?

Solution:
G, S এর চেয়ে দ্বিগুণ গতিতে কাজ করে।
ধরি,
S একটি কাজ করে x দিনে
G সেই কাজ করে x/2 দিনে

প্রশ্নমতে,
x - x/2 = 30
⇒ 2x - x = 60
∴ x = 60

S কাজটি করে 60 দিনে
∴ S কাজটি 1 দিনে করে 1/60 অংশ

G কাজটি করে 60/2 দিনে = 30 দিনে
∴ G কাজটি 1 দিনে করে 1/30 অংশ

S ও G একত্রে 1 দিনে করে 1/60 + 1/30 অংশ
= (1 + 2)/60 অংশ
= 3/60 অংশ
= 1/20 অংশ

∴ S ও G একত্রে কাজটি 20 দিনে করতে পারবে।

২৩০.

A 10% monthly salary increase resulted in a Tk. 9,000 per year increase in salary for an employee. What was has monthly salary before the increase?

ক
ক) 5000
খ
খ) 7500
গ
গ) 9000
ঘ
ঘ) 12000

ব্যাখ্যা

12 মাসে বেতন বাড়ে = 9000 টাকা
1 মাসে বেতন বাড়ে = 9000/12
= 750 টাকা

প্রশ্নমতে
10% = 750 টাকা
1% = 750/10
100% = (750 × 100)/10
= 7500 টাকা

২৩১.

If P and Q together can complete a piece of work in 16 days and P alone in 24 days, in how many days can Q alone complete the work?

ক
72 days
খ
48 days
গ
42 days
ঘ
36 days

ব্যাখ্যা

Question: If P and Q together can complete a piece of work in 16 days and P alone in 24 days, in how many days can Q alone complete the work?

Solution:
P and Q complete a work in = 16 days
One day's work of (P + Q) = 1/16

P complete the work in = 24 days;
One day's work of P = 1/24

Then, Q's one day's work = (1/16) - (1/24)
= (3 - 2)/48
= 1/48

So, Q alone can complete the work in = 48 days.

২৩২.

Mr. Rafin can complete a job in 15 days. while Mr. Shafin takes twice as long to do the same job. If they work together, how many days will they require to complete the job?

ক
ক) 12
খ
খ) 15
গ
গ) 10
ঘ
ঘ) 6

ব্যাখ্যা

Question: Mr. Rafin can complete a job in 15 days. while Mr. Shafin takes twice as long to do the same job. If they work together, how many days will they require to complete the job?

Solution:
রাফিন সাহেব সময় নেয় ১৫ দিন
∴ শাফিন সাহেব সময় নেয় ৩০ দিন

তারা ১ দিনে মোট করতে পারে = (১/১৫ + ১/৩০)
= ৩/৩০
= ১/১০

∴ সম্পূর্ণ অংশ শেষ করতে তাদের সময় লাগবে = ১০ দিন

২৩৩.

A can finish a work in 15 days, and B can do it in 25 days. They work together for 5 days. What fraction of the work remains unfinished?

ক
3/5
খ
7/15
গ
2/3
ঘ
None of the above

ব্যাখ্যা

Question: A can finish a work in 15 days, and B can do it in 25 days. They work together for 5 days. What fraction of the work remains unfinished?

Solution:
Work done by A in 1 day = 1/15
Work done by B in 1 day = 1/25

Combined work in 1 day = 1/15 + 1/25
= (5 + 3) / 75
= 8/75

Work done in 5 days = 5 × (8/75) = 40/75 = 8/15

Fraction of work left = 1 - 8/15 = 7/15

২৩৪.

In a dairy farm, 40 cows eat 40 bags of husk in 40 days. In how many days one cow will eat one bag of husk?

ক
ক) 1
খ
খ) 1/40
গ
গ) 40
ঘ
ঘ) 80

ব্যাখ্যা

Let,
The required number of days be x.
Less Cows, More days[Indirect proportion]
Less Bags, Less days [Direct proportion]
Cows(1 : 40), bags(40 : 1)}::40 : x
∴ 1 × 40 × x = 40 × 1 × 40
= 40.

২৩৫.

X can do a piece of work in 20 days and Y can do the 1/7th of the same work in 5 days. In how many days together can they complete the 11/14th of the total work?

ক
ক) 15 days
খ
খ) 20 days
গ
গ) 10 days
ঘ
ঘ) 12 days

ব্যাখ্যা

X can do in 1 day = 1/20 part
Y can do in 1 day = 1/(7×5) = 1/35 part
X & Y together can do in 1 day = 1/20 + 1/35 = 11/140
11/140 part of the work is done in 1 day
So, 11/14 part of the work is done in = (11/14) × (140/11) = 10 days

২৩৬.

If 30 oranges cost Tk. 210, how much does 10 oranges cost?

ক
Tk. 7
খ
Tk. 0.7
গ
Tk. 30
ঘ
Tk. 70

ব্যাখ্যা

Question: If 30 oranges cost Tk. 210, how much does 10 oranges cost?

Solution:
The cost of 30 oranges 210
Unit value (cost of 1 orange) = 210/30 = 7

The cost of 10 oranges = (7 × 10) = 70

২৩৭.

Joy and Bijoy can do a piece of work in 20 days and 12 days respectively. Joy started the work alone and then after 4 days Bijoy joined him till the completion of the work. How long did the work last?

ক
ক) 14 days
খ
খ) 10 days
গ
গ) 8 days
ঘ
ঘ) 6 days

ব্যাখ্যা

Work done by Joy in 4 days = 4 ×(1/20) = 1/5
Remaining work = (1 - 1/5) = (5- 1)/5 = 4/5

(Joy + Bijoy)'s 1 day's work = (1/20) + (1/12)
= (3 + 5)/60
= 8/60
= 2/15

Now, 2/15 is work done by Joy and Bijoy in 1 day.
So, 4/5 work will be done by Joy and Bijoy in = (15/2) × (4/5) = 6 days
Hence, total time taken =(6 + 4) days = 10 days.

২৩৮.

A tap can fill a tank in 6 hours. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?

ক
ক) 3 hrs 15 min
খ
খ) 3 hrs 45 min
গ
গ) 4 hrs 15 min
ঘ
ঘ) None of these

ব্যাখ্যা

Time taken by one tap to fill half the tank = 3 hrs.
Part filled by one tap in 1 hour = 1/6
Part filled by four taps in 1 hour = (4×1/6) = 2/3
Remaining part = (1−1/2) = 1/2
2/3 of the tank is filled by four taps in 1 hour.
So, 1/2 of the tank is filled in = 3/2×1/2=3/4 hours
3/4 hours = 3/4 × 60 = 45 min
So, the total time taken = 3 hrs + 45 min = 3 hrs 45 min or 225 min
-------------------------------------------------
Alternative way:
A tap can fill a tank in 6 hours.
A tap can fill half of a tank in 6/2 or 3 hours.
4 tap can fill half of a tank in 3/4 hours = 45 min
Total time taken = 3 hours 45 min

২৩৯.

A, B and C completed a work costing Tk. 1800. A worked for 6 days, B for 4 days and C for 9 days. If their daily wages are in the ratio of 5 : 6 : 4, how much amount will be received by C?

ক
ক) Tk. 720
খ
খ) Tk. 480
গ
গ) Tk. 600
ঘ
ঘ) Tk. 820

ব্যাখ্যা

Let the daily wages of A, B and C be Tk. 5x, Tk. 6x and Tk. 4x respectively.
Then, ratio of their amounts
= (5 × 6) : (6 × 4) : (4 × 9)
= 30 : 24 : 36
= 5 : 4 : 6
∴ C's amount
= Tk. (1800 × 6/15)
= Tk. 720

২৪০.

A painter needs 5 liters of paint to cover 20 square meters of wall. How much paint is needed to cover 10 square meters?

ক
3.5 liters
খ
3 liters
গ
2.5 liters
ঘ
2 liters

ব্যাখ্যা

Question: A painter needs 5 liters of paint to cover 20 square meters of wall. How much paint is needed to cover 10 square meters?

Solution:
We know the paint needed for 20 square meters (5 liters).
Paint per square meter = Total paint / Area covered = 5 liters / 20 square meters = 0.25 liters per square meter.
Paint needed for 10 square meters = Unit value (paint per square meter) × Area to be covered
= 0.25 liters/square meter × 10 square meters
= 2.5 liters.

২৪১.

Arif bought 17 pens of three colors- black, green and red. They cost Tk. 5, Tk. 10 and Tk. 25 each respectively. The total amount that Arif paid was Tk. 205. If Arif bought twice as many green pens as red pens, how many black pens did he buy?

ক
4
খ
5
গ
7
ঘ
None

ব্যাখ্যা

Question: Arif bought 17 pens of three colors- black, green and red. They cost Tk. 5, Tk. 10 and Tk. 25 each respectively. The total amount that Arif paid was Tk. 205. If Arif bought twice as many green pens as red pens, how many black pens did he buy?

Solution:
Let,
Arif buys R red pens
Arif buys G green pens
Arif buys B black Pens
If Arif bought twice as many green pens as red pens
∴ G = 2R

Now,
B + R + G = 17
⇒ B + R + 2R = 17
⇒ 3R = 17 - B

ATQ,
5B + 10G + 25R = 205
⇒ 5B + 20R + 25R = 205
⇒ 5B + 45R = 205
⇒ 5B + 45{(17 - B)/3} = 205
⇒ 5B + 15(17 - B) = 205
⇒ 5B + 255 - 15B = 205
⇒ 10B = 50
∴ B = 5

২৪২.

A and B can together finish a work 30 days. They worked together for 20 days and then B left. After another 20 days, A finished the remaining work. In how many days A alone can finish the work?

ক
ক) 40
খ
খ) 80
গ
গ) 54
ঘ
ঘ) 60

ব্যাখ্যা

(A +B)'s 20 day's work = 20 × 1/30 = 2/3
Remaining work = 1 − 2/3 = 1/3
Now,1/ 3 work is done by A in 20 days
∴ The whole work will be done by A
in 20×3 = 60days

২৪৩.

The price of 357 mangoes is Tk. 1517.25. Find the approximate price of 49 dozens of such mangoes?

ক
Tk. 3000
খ
Tk. 3500
গ
Tk. 4000
ঘ
Tk. 2500

ব্যাখ্যা

Question: The price of 357 mangoes is Tk. 1517.25. Find the approximate price of 49 dozens of such mangoes?

Solution:
We know that 1 dozen = 12 piece
49 dozens = 49 × 12 = 588 mangoes
Let,
x is the price for 588 mangoes.
Now,
Put the same unit on the same side
Price and mangoes are directly proportional to each other, so
⇒ 357 mangoes : 588 mangoes = 1517.25 : x
∴ x = (1517.25 × 588)/ 357 = 2499
Hence, The approximate value, x = Tk. 2500

২৪৪.

Time taken by A to finish a piece of work is twice the time taken B and thrice the time taken by C. If all three of them work together, it takes them 2 days to complete the entire work. How much work was done by C alone?

ক
8 days
খ
6 days
গ
4 days
ঘ
9 days

ব্যাখ্যা

Time taken by A = x days
Time taken by B = x/2 days
Time Taken by C = x/3 days

⇒ {(1/x) + (2/x) + (3/x) = 1/2
⇒ 6/x = 1/2
⇒ x = 12

Time taken by C = x/3 = 12/3 = 4 days

২৪৫.

6 men and 8 women can complete a work in 10 days. 26 men and 48 women can finish the same work in 2 days. 15 men and 20 women can do the same work in _____ days.

ক
6 days
খ
4 days
গ
8 days
ঘ
2 days

ব্যাখ্যা

Let work done by 1 man in 1 day = m and work done by 1 woman in 1 day = b
Work is done by 6 men and 8 women in 1 day = 1/10
⇒ 6m + 8b = 1/10
⇒ 60m + 80b = 1 --- (1)
Work done by 26 men and 48 women in 1 day = 1/2
⇒ 26m + 48b = 1/2
⇒ 52m + 96b = 1--- (2)
Solving equation 1 and equation 2.
We get m = 1/100 and b = 1/200
Work is done by 15 men and 20 women in 1 day
= 15/100 + 20/200 =1/4
⇒ Time is taken by 15 men and 20 women in doing the work = 4 days.

২৪৬.

A, B, and C can do a piece of work in 20, 30 and 60 days respectively. In how many days can A do the work if he is assisted by B and C on every third day?

ক
12 days
খ
15 days
গ
16 days
ঘ
18 days

ব্যাখ্যা

A's 2 day's work = (1/20) × 2
= 1/10
(A + B + C)'s 1 day's work = (1/20) + (1/30) + (1/60)
= 6/60
= 1/10
(A + B + C) work in 3 days = (1/10) + (1/10)
= 1/5.
Now, 1/5 work is done in 3 days.
∴ Whole work is done in (3 × 5)
= 15 days.

২৪৭.

A takes twice as much time as B or thrice as much time as C to finish a piece of work. Working together, they can finish the work in 2 days. B can do the work alone in:

ক
4 days
খ
6 days
গ
8 days
ঘ
12 days

ব্যাখ্যা

Question: A takes twice as much time as B or thrice as much time as C to finish a piece of work. Working together, they can finish the work in 2 days. B can do the work alone in:

Solution:
Let
A, B and C take x, x/2 and x/3 days respectively to finish the work.

Now
(1/x) + (2/x) + (3/x) = 1/2
(1 + 2 + 3)/x = 1/2
⇒ 6/x = 1/2
⇒ x = 12.

So, B takes to finish the work = 12/2 days

২৪৮.

A salesman receives daily wage of Tk. 450 and earns a commission of 10% on all sales he makes. How much taka worth of sales does he need to make in order to bring his total daily income of Tk. 1200?

ক
Tk. 9000
খ
Tk. 8500
গ
Tk. 7500
ঘ
Tk. 6500

ব্যাখ্যা

Question: A salesman receives daily wage of Tk. 450 and earns a commission of 10% on all sales he makes. How much taka worth of sales does he need to make in order to bring his total daily income of Tk. 1200?

Solution:
He has to earn only with commission (1200 - 450) Taka = 750 Taka

He earns 10 taka when sells 100 taka
∴ He earns 1 taka when sells 100/10 taka = 10 taka
∴ He earns 750 taka when sells 10 × 750 taka = 7500 taka

২৪৯.

If 10 persons can do a job in 20 days, then 20 persons with twice the efficiency can do the same job in -

ক
ক) 10 days
খ
খ) 8 days
গ
গ) 5 days
ঘ
ঘ) 4 days

ব্যাখ্যা

Question: If 10 persons can do a job in 20 days, then 20 persons with twice the efficiency can do the same job in -

Solution:
10 persons in one day can do = 1/20
20 persons in one day can do = 2 × 1/20 = 1/10

with double efficiency, in one day they can do = 2 × 1/10 = 1/5

so, the work will be finished in just 5 days

২৫০.

Four gardeners with four grass mowers mow 400 sq. m of ground in 4 hours. How long would it take for eight gardeners with eight grassmowers to mow 800 sq.m of the ground?

ক
2 hours
খ
4 hours
গ
6 hours
ঘ
8 hours

ব্যাখ্যা

Question: Four gardeners with four grassmowers mow 400 sq. m of ground in 4 hours. How long would it take for eight gardeners with eight grass mowers to mow 800 sq. m of the ground?

Solution:
Since each gardener would work with one grassmower,
So, we shall consider 1 gardener and 1 grass mower as one unit.
Let the required time be x hours

ATQ,
8 × 400 × x = 4 × 800 × 4
⇒ x = (4 × 800 × 4)/(8 × 400)
∴ x = 4 hours

২৫১.

Three men, four women, and six children can complete a work in seven days. A woman does double the work a man does and a child does half the work a man does. How many women alone can complete this work in 7 days?

ক
4 days
খ
5 days
গ
6 days
ঘ
7 days

ব্যাখ্যা

Question: Three men, four women, and six children can complete a work in seven days. A woman does double the work a man does and a child does half the work a man does. How many women alone can complete this work in 7 days?

Solution:
Let, women's 1 day's work = x
1 man's 1 day's work = x/2
1 child's 1 day's work = x/4

ATQ,
3x/2 + 4x + 6x/4 = 1/7
⇒ (6x + 16x + 6x)/4 = 1/7
⇒ 28x/4 = 1/7
⇒ 7x = 1/7
∴ x = 1/49

1 woman alone can complete 1/49 of the work in 1 day.
So, 1 woman alone can complete the work in 49 days.
To complete the work in 7 days, the number of women required = 49/7 = 7 days.

২৫২.

A worker’s regular pay is Tk. 15 per hour up to 40 hours. Overtime is paid at twice the regular rate. If he was paid Tk. 1,050, how many hours of overtime did he work?

ক
9 hours
খ
12 hours
গ
10 hours
ঘ
15 hours

ব্যাখ্যা

Question: A worker’s regular pay is Tk. 15 per hour up to 40 hours. Overtime is paid at twice the regular rate. If he was paid Tk. 1,050, how many hours of overtime did he work?

Solution:

ধরি,
তিনি x ঘণ্টা overtime কাজ করেছেন।

সাধারণ বেতন (40 ঘণ্টার জন্য) = 15 × 40 = 600 টাকা
Overtime বেতন = 15 × 2 × x = 30x টাকা

মোট বেতন = সাধারণ বেতন + overtime বেতন

অর্থাৎ,
600 + 30x = 1,050
⇒ 30x = 1,050 - 600
⇒ 30x = 450
⇒ x = 450/30
∴ x = 15

∴ তিনি 15 ঘন্টা overtime করেছেন।

২৫৩.

A tap can fill a tank in 4 hours. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?

ক
ক) 1 hour 30 minutes
খ
খ) 2 hour 30 minutes
গ
গ) 3 hour 30 minutes
ঘ
ঘ) 4 hour 30 minutes

ব্যাখ্যা

A tap can fill a tank in 4 hours.
Therefore the tap can fill half the tank in 2 hours.

Remaining = 1/2

After half the tank is filled, three more similar taps are opened.
Hence, the total number of taps becomes 4.

Part filled by one tap in 1 hour = 1/4
Part filled by four taps in 1 hour = 4 × (1/4) = 1
i.e., 4 taps can fill the remaining half in 30 minutes.

Total time taken
= 2 hour + 30 minute = 2 hour 30 minutes.

২৫৪.

A man, a woman, and a child together receive 84 Taka for 6 days' work. A woman's daily wage is twice a child's wage, and a man earns twice as much as a woman's. How much does a woman earn per day?

ক
3 Taka
খ
4 Taka
গ
5 Taka
ঘ
6 Taka

ব্যাখ্যা

Question: A man, a woman, and a child together receive 84 Taka for 6 days' work. A woman's daily wage is twice a child's wage, and a man earns twice as much as a woman's. How much does a woman earn per day?

Solution:
Let,
C = Daily wage of the child (in Taka).
W = Daily wage of the woman (in Taka).
M = Daily wage of the man (in Taka).

A.T.Q,
W = 2C
M = 2(2C) = 4C

Total payment of 6 days = 6 (C + W + M)
84 = 6(C + 2C + 4C)
42C = 84
∴ C = 2

So, the woman earns = 2 × 2 = 4 taka

২৫৫.

Since A is three times as efficient as B, he completes a task 40 days sooner. Find the time required for both to complete it working together.

ক
15 days
খ
24 days
গ
14 days
ঘ
30 days

ব্যাখ্যা

Question: Since A is three times as efficient as B, he completes a task 40 days sooner. Find the time required for both to complete it working together.

Solution:
Let A alone takes x days
and B alone takes 3x days to complete the job.

ATQ,
3x - x = 40
⇒ 2x = 40
⇒ x = 20

So, A alone takes 20 days and B alone takes 3 × 20 = 60 days to complete the job.

(A + B)'s 1 days work = (1/20 + 1/60) = 1/15 part

∴ A and B together can do the work in 15 days.

২৫৬.

If 5 workers can collect 60 kg wheat in 3 days, how many kilograms of wheat will 8 workers collect in 5 days ?

ক
160 kg
খ
140 kg
গ
120 kg
ঘ
200 kg

ব্যাখ্যা

Question: If 5 workers can collect 60 kg wheat in 3 days, how many kilograms of wheat will 8 workers collect in 5 days ?

Solution:
5 workers 3 days collection = 60 kg
1 worker 1 day collection = (60/15) kg
8 workers 5 days collection = ( 60 × 40 ) / 15 kg
= 160 kg

২৫৭.

A is 30% more efficient than B. How much time will they, working together, take to complete a job which A alone could have done in 23 days?

ক
ক) 13 days
খ
খ) 15 days
গ
গ) 10 days
ঘ
ঘ) None

ব্যাখ্যা

Ratio of times taken by A and B = 100 : 130 = 10 : 13.
Let, B takes x days to do the work.
ATQ, x : 23 = 13 : 10
Or, x = (23×13 / 10)
Or, x = 299/10
A’s 1 day’s work = 1/23
B’s 1 day’s work = 10/299
(A + B)’s 1 day’s work = ( 1/23 + 10/299 ) = 23/299 = 1/13 part
So, A and B together can complete the work in 13 days.

২৫৮.

Five machines working together can process 1200 items in 4 days. If one machine breaks down, how many items can the remaining machines process in 6 days?

ক
1080 items
খ
1200 items
গ
1440 items
ঘ
1620 items

ব্যাখ্যা

Question: Five machines working together can process 1200 items in 4 days. If one machine breaks down, how many items can the remaining machines process in 6 days?

Solution:
Total items processed by 5 machines in 4 days = 1200
Items processed by 1 machine in 4 days = (1200/5) items
Items processed by 1 machine in 1 day = {1200/(5 × 4)} items
= 60 items

Therefore,
Items processed by 4 machines in 6 days = 60 × 6 × 4
= 1440 items

২৫৯.

A fort had provision of food for 150 men for 45 days. After 10 days, 25 men left the fort. The number of days for which the remaining food will last, is:

ক
ক) 29
খ
খ) 37
গ
গ) 42
ঘ
ঘ) 54
ঙ
ঙ) 60

ব্যাখ্যা

After 10 days : 150 men had food for 35 days.
Suppose 125 men had food for x days.
Now, Less men, More days (Indirect Proportion)
∴ 125 : 150 :: 35 : x
⇒ 125 × x = 150 × 35
⇒ x = (150 × 35)/125
= 42

২৬০.

Parimal got two and a half time as many marks in English as in History. If his total marks in the two subjects are 140, the marks obtained by him in English are:

ক
40
খ
75
গ
90
ঘ
100

ব্যাখ্যা

Question: Parimal got two and a half time as many marks in English as in History. If his total marks in the two subjects are 140, the marks obtained by him in English are:

Solution:
ধরি,
পরিমল ইতিহাসে পায় x নম্বর
∴ পরিমল ইংরেজিতে পায় 2.5x নম্বর

প্রশ্নমতে,
x + 2.5x = 140
⇒ 3.5x = 140
⇒ x = 140/3.5
∴ x = 40

∴ পরিমল ইংরেজিতে পায় 2.5x = 2.5 × 40 = 100 নম্বর

২৬১.

A and B together can paint a wall in 8 days. A can do it alone in 12 days. How many days would it take B to do this job alone?

ক
24 days
খ
20 days
গ
15 days
ঘ
14 days
ঙ
None

ব্যাখ্যা

Question: A and B together can paint a wall in 8 days. A can do it alone in 12 days. How many days would it take B to do this job alone?

Solution:
A's 1 day's work 1/12
A and B's 1 day's work 1/8
∴ B's 1 day's work = 1/8 - 1/12
= (6 - 4)/48
= 2/48
= 1/24

1/24 part of job done by B in 1 day
∴ Full job done by B in (24/1) days
= 24 days

২৬২.

A worker earns Tk. 250 on the first day and spends Tk. 200 on the second day, earns Tk. 250 on the third day and again spends Tk. 200 on the fourth day and so on. On which day would he have had Tk. 1000.

ক
20^th day
খ
30^th day
গ
31^th day
ঘ
40^th day

ব্যাখ্যা

Question: A worker earns Tk. 250 on the first day and spends Tk. 200 on the second day, earns Tk. 250 on the third day and again spends Tk. 200 on the fourth day and so on. On which day would he have had Tk. 1000?

Solution:
১ম দিনে আয় করে = ২৫০ টাকা
২য় দিনে ব্যয় করে = ২০০ টাকা
∴ প্রতি ২ দিনে জমা হয় = ২৫০ - ২০০ = ৫০ টাকা

শুধু ১ম দিনে আয় করে তার হাতে থাকে ২৫০ টাকা
তাহলে, ১০০০ - ২৫০ = ৭৫০ টাকা আরও জমা করতে হবে।

৫০ টাকা জমা হয় ২ দিনে
∴ ৭৫০ টাকা জমা হয় = (২ × ৭৫০)/৫০ = ৩০ দিনে

অর্থাৎ, ৩০ দিনের শেষে তার হাতে জমা থাকে = ৭৫০ টাকা
৩১তম দিনে সে আবার আয় করে = ২৫০ টাকা
⇒ মোট = ৭৫০ + ২৫০ = ১০০০ টাকা

∴ ৩১তম দিনে তার কাছে ১০০০ টাকা ছিল।

২৬৩.

12 men can do a piece of work in 24 days. How many days are needed to complete the work, if 8 men do this work?

ক
ক) 28
খ
খ) 36
গ
গ) 48
ঘ
ঘ) 52

ব্যাখ্যা

12 men can do a piece of work in 24 days
⇒ M₁ = 12 and D₂ = 24
8 men can do this work in D₂ days
⇒ M₂ = 8
M₁D₁ = M₂D₂
⇒ 12 × 24 = 8 × D₂
⇒ D₂ = (12 × 24)/8
= 36 days.

২৬৪.

Three workers can do a job in 12 days. Two of the workers work twice as fast as the third. How long would it take one of the faster workers to do the job himself?

ক
ক) 24
খ
খ) 30
গ
গ) 32
ঘ
ঘ) None

ব্যাখ্যা

Question: Three workers can do a job in 12 days. Two of the workers work twice as fast as the third. How long would it take one of the faster workers to do the job himself?

Solution:
তিনজন শ্রমিক একটি কাজ ১২ দিনে করতে পারে।
তাদের মধ্যে দুজন তৃতীয়জন থেকে দ্বিগুণ গতিতে কাজ করতে পারে।

ধরি, তৃতীয়জন 2x দিনে কাজটি সম্পন্ন করে।
বাকি দুজন প্রত্যেকে কাজটি সম্পন্ন করে x দিনে

তিনজন একসাথে একদিনে কাজ করে = (1/2x) + (1/x) + (1/x)
= (1 + 2 + 2)/2x
= 5/2x অংশ

তিনজনের সম্পূর্ণ কাজটি করতে সময় লাগে = 2x/5 অংশ কাজ

2x/5 = 12
⇒ 2x = 60
⇒ x = 30

দ্বিগুণ গতিতে কাজ করা ব্যক্তিরা প্রত্যেকে ৩০ দিনে কাজটি করতে পারে।

২৬৫.

If 5 workers can collect 60 kg wheat in 3 days, how many kilograms of wheat will 8 workers collect in 5 days?

ক
ক) 100 kg
খ
খ) 120 kg
গ
গ) 160 kg
ঘ
ঘ) 200 kg

ব্যাখ্যা

Question: If 5 workers can collect 60 kg wheat in 3 days, how many kilograms of wheat will 8 workers collect in 5 days?

Solution:
৫ জন লোক ৩ দিনে গম সংগ্রহ করতে পারে ৬০ কেজি
১ জন লোক ১ দিনে গম সংগ্রহ করতে পারে ৬০/(৫ × ৩) কেজি
৮ জন লোক ৫ দিনে গম সংগ্রহ করতে পারে (৬০ × ৮ × ৫)/(৫ × ৩) কেজি
= ১৬০ কেজি

২৬৬.

Two workers A and B are engaged to do a work. A working alone takes 9 hours more to complete the job than if both worked together. If B worked alone, he would need 4 hours more to complete the job than they both working together. What time would they take to do the work together?

ক
ক) 5 hours
খ
খ) 6 hours
গ
গ) 7 hours
ঘ
ঘ) 8 hours

ব্যাখ্যা

Question: Two workers A and B are engaged to do a work. A working alone takes 9 hours more to complete the job than if both worked together. If B worked alone, he would need 4 hours more to complete the job than they both working together. What time would they take to do the work together?

Solution:
Let A and B together take x hours to complete the work
Then, A alone takjes (x + 9) hrs and B alone takes (x + 4) hrs to complete the work.

ATQ,
{1/(x + 9)} + {1/(x + 4)} = 1/x [ A ও B এর ১ ঘণ্টার কাজের অংশের যোগফল = তাদের একত্রে ১ ঘণ্টায় করা অংশ ]
⇒ (x + 4 + x + 9)/(x + 9)(x + 4) = 1/x
⇒ (x + 4 + x + 9)/(x² + 9x + 4x + 36) = 1/x
⇒ (2x + 13)/(x² + 13x + 36) = 1/x
⇒ 2x² + 13x = x² + 13x + 36
⇒ x² = 36
∴ x = 6

∴ They take 6 hours to do the work together

২৬৭.

If A and B together can complete a piece of work in 50 days and B alone in 100 days, in how many days can A alone complete the work?

ক
50
খ
80
গ
100
ঘ
120
ঙ
150

ব্যাখ্যা

(A + B)'s one day's % work = 100/50 = 2%
B's one day's % work = 100/100 = 1%
A's one day's % work = 2 - 1 = 1%
Thus, A need 100/1 or 100 days to complete the work.
------------------------------------------------------------
Alternative way:
A and B complete a work in = 50 days
One day's work of (A + B) = 1/50 portion of work
B complete the work in = 100 days;
One day's work of B = 1/100 portion of work
Then, A's one day's work = 1/50 - 1/100 = 1/100
Thus, A can complete the work in 100 days.

২৬৮.

A and B together can complete a work in 6 days. They start together, but after 4 days, B left work. If the work is completed after 4 more days, B alone could do the work in -

ক
6 days
খ
12 days
গ
14 days
ঘ
15 days

ব্যাখ্যা

Question: A and B together can complete a work in 6 days. They start together, but after 4 days, B left work. If the work is completed after 4 more days, B alone could do the work in

Solution:
(A + B)'s one day's work = 1/6 part
(A + B) works 4 days together = 4/6 part
= 2/3 part

Remaining work = 1 - (2/3)
= 1/3 part

1/3 part of the work is completed by A in 4 days.
Hence, one day's work of A = 1/(3 × 4) part
= 1/12 part

Then, one day's work of B = (1/6) - (1/12) part
= (2 - 1)/12 part
= 1/12 part

So, B alone can complete the whole work in 12 days.

২৬৯.

Rakib can do a job in 15 minutes and his friend takes twice as long to do the same job. If they work together, how long it take to complete the job?

ক
10 minutes
খ
12 minutes
গ
9 minutes
ঘ
5 minutes

ব্যাখ্যা

Question: Rakib can do a job in 15 minutes and his friend takes twice as long to do the same job. If they work together, how long it take to complete the job?

Solution:
রাকিবের সময় লাগে 15 মিনিট
তার বন্ধুর সময় লাগে 30 মিনিট

1 মিনিটে তারা কাজ সম্পন্ন করে = (1/15) + (1/30)
= (2 + 1)/30
= 3/30
= 1/10 অংশ

1/10 অংশ কাজ করতে সময় লাগে = 1 মিনিট
∴ সম্পূর্ণ কাজ করতে সময় লাগে = 1/(1/10) = 10 মিনিট

অর্থাৎ, একসাথে কাজ করলে তারা 10 মিনিটে কাজ শেষ করবে।

২৭০.

If Michael can shovel all the snow of a standard driveway in 12 minutes, and Emon can shovel all the snow of a standard driveway in 36 minutes, then working together, how many minutes would it take for them both to shovel all the snow of a standard driveway?

ক
ক) 5
খ
খ) 9
গ
গ) 15
ঘ
ঘ) 18

ব্যাখ্যা

মাইকেল 12 মিনিটে করে = 1 টি কাজ
মাইকেল। মিনিটে করে = 1/12অংশ কাজ

ইমন 36 মিনিটে করে = 1 টি কাজ
ইমন। মিনিটে করে = 1/36অংশ কাজ

মাইকেল ও ইমন একত্রে। মিনিটে করে
=(1/12) + (1/36) অংশ
= (3 + 1)/36
= 4/36
= 1/9 অংশ

তারা 1/9 অংশ কাজ করে। 1 মিনিটে
1 বা সম্পূর্ণ অংশ কাজ করে (9 × 1)মিনিটে
= 9 মিনিটে

২৭১.

A, B and C can do a piece of work in 20, 30 and 60 days respectively. In how many days can A do the work if he is assisted by B and C on every third day?

ক
12 days
খ
15 days
গ
16 days
ঘ
18 days

ব্যাখ্যা

Question: A, B and C can do a piece of work in 20, 30 and 60 days respectively. In how many days can A do the work if he is assisted by B and C on every third day?

Solution:

A's 2 day's work = (1/20) × 2 = 1/10
(A + B + C)'s 1 day's work = 1/20 + 1/30 + 1/60 = 6/60 = 1/10

Work done in 3 days = 1/10 + 1/10 = 2/10 = 1/5

Now,
1/5 work is done in 3 days.
∴ Whole work will be done in (3 × 5) = 15 days.

২৭২.

A man can do 1 work in 8 hours. How many work he can do at 40 hours?

ক
ক) 2
খ
খ) 4
গ
গ) 5
ঘ
ঘ) 7

ব্যাখ্যা

Question: A man can do 1 work in 8 hours. How many work he can do at 40 hours?

Solution:
In 8 hours 1 work is done
In 1 hour 1/8 work is done
∴ In 40 hours 40/8 work = 5 work can be done

২৭৩.

5 mat-weavers can weave 5 mats in 5 days. At the same rate, how many mats would be woven by 10 mat-wevers in 10 days?

ক
18 mats
খ
20 mats
গ
22 mats
ঘ
24 mats

ব্যাখ্যা

Question: 5 mat-weavers can weave 5 mats in 5 days. At the same rate, how many mats would be woven by 10 mat-wevers in 10 days?

Solution:
5 mat-weavers in 5 days weave = 5 mats
∴ 1 mat-weavers in 1 days weave = 5/(5 × 5) mats
∴ 10 mat-weavers in 10 days weave = (5 × 10 × 10)/(5 × 5) mats
= 20 mats

২৭৪.

If two men or four women or six boys can finish a job in 72 days, then find out the number of days in which that work can be done by one man, two women, and three boys?

ক
48 days
খ
38 days
গ
58 days
ঘ
28 days

ব্যাখ্যা

Question: If two men or four women or six boys can finish a job in 72 days, then find out the number of days in which that work can be done by one man, two women, and three boys?

Solution:
Given,
2 men = 4 women = 6 boys
1 man = 2 women = 3 boys
∴ 1 man + 2 women + 3 boys = (1 + 1 + 1) men
= 3 men

Here,
2 men can do in = 72 days
∴ 1 men can do in = (72 × 2) days
= 144 days
∴ 3 men can do in = (144 ÷ 3) days
= 48 days

২৭৫.

A group of workers promise to complete a piece of work in 8 days, but 6 of them do not report for work. If it took the remaining workers 10 days to complete the work, then what was the number of workers originally hired?

ক
ক) 25
খ
খ) 27
গ
গ) 30
ঘ
ঘ) 33

ব্যাখ্যা

Question: A group of workers promise to complete a piece of work in 8 days, but 6 of them do not report for work. If it took the remaining workers 10 days to complete the work, then what was the number of workers originally hired?

Solution:
Let, the workers promised were = x
The workers worked were = x - 6

[x জনের 8 দিনের করা কাজ = (x - 6) জনের 10 দিনে করা কাজের পরিমাণ একই]
ATQ,
8x = 10x - 60
⇒ 10x - 8x = 60
⇒ 2x = 60
⇒ x = 60/2
∴ x = 30

∴ The number of workers originally hired was 30

২৭৬.

If A can do 1/4 of a work in 3 days and B can do 1/6 of the same work in 4 days, how much will A get if both work together and paid Tk 3000 in all?

ক
Tk 1800
খ
Tk 1600
গ
Tk 2200
ঘ
Tk 2000
ঙ
None of these

ব্যাখ্যা

Question: If A can do 1/4 of a work in 3 days and B can do 1/6 of the same work in 4 days, how much will A get if both work together and paid Tk 3000 in all?

Solution:
Whole work is done by A in (3 × 4) = 12 days
Whole work is done by B in (4 × 6) = 24 days

A's 1 day's work : B's 1 day's work
= A's wages : B's wages
= 1/12 : 1/24
= 2 : 1

A's share = 3000 × (2/3) = Tk 2000

২৭৭.

A certain hotel has 1,400 single rooms and 420 double rooms. Each room is cleaned by one person. If one person can clean a single room every 15 minutes and a double room every 20 minutes, how many cleaning persons are needed to clean all the rooms if each person works for exactly 7 hours?

ক
65
খ
90
গ
80
ঘ
70
ঙ
265

ব্যাখ্যা

Question: A certain hotel has 1,400 single rooms and 420 double rooms. Each room is cleaned by one person. If one person can clean a single room every 15 minutes and a double room every 20 minutes, how many cleaning persons are needed to clean all the rooms if each person works for exactly 7 hours?

Solution:
Number of single rooms= 1400
Number of double rooms = 420
Time taken by a person to clean a single room = 15 mins = 15/60 hour = 1/4 hour
Time taken by a person to clean a double room = 20 mins = 20/60 hour = 1/3 hour

Total time to clean 1400 single and 420 double rooms
= 1400 × (1/4) + 420 × (1/3)
= 350 + 140 hours
= 490 hours

Each person works for = 7 hours

∴ Number of persons required = 490/7 = 70

২৭৮.

A wall of 100 meters can be built by 7 men or 10 women in 10 days. How many days will 14 men and 20 women take to build a wall of 600 metres?

ক
ক) 15
খ
খ) 16
গ
গ) 17
ঘ
ঘ) 20

ব্যাখ্যা

Let the required number of days be x
7 men = 10 women
(14 men and 20 women) = (20 + 20) women = 40 women
More length, More days (Direct proportion)
More women, Less days (Indirect proportion)
Length 100:600 & Women 40:10} :: 10:x
∴ 100×40×x = 600×10×10
⇔ x = (600×10×10)(100×40)
⇔ x = 15

২৭৯.

A can finish a work in 10 days and B can finish the same work in 15 days. If they work alternatively, find the time taken to finish the job.

ক
14 days
খ
15 days
গ
12 days
ঘ
14.5 days

ব্যাখ্যা

Question: A can finish a work in 10 days and B can finish the same work in 15 days. If they work alternatively, find the time taken to finish the job.

Solution:
Note: Assume the total work = LCM of the given days

Take the LCM of days = LCM of (10 and 15) = 30
Let the total work = 30

Note: One day work = (total work/ days)

Now,

A's one day work = 30/10 = 3 unit
B's one day work = 30/15 = 2 unit

That means A and B works 5 unit works in 2 days
i.e., 2 days = 5 unit work

To complete the 30 work, multiply both sides with 6.
i.e., 2 days × 6 = 5 unit work × 6

Or, 12 days = 30 works
So, A and B require 12 days to finish the work.

২৮০.

Taposh and Sumon can finish a work together in 10 hours. If Taposh takes 4 times as long as Sumon to finish the job alone, how long will Sumon take to finish the job alone?

ক
10 hours
খ
12. 5 hours
গ
14.5 hours
ঘ
20 hours

ব্যাখ্যা

Question: Taposh and Sumon can finish a work together in 10 hours. If Taposh takes 4 times as long as Sumon to finish the job alone, how long will Sumon take to finish the job alone?

Solution:
ধরি, সুমন একা কাজটি শেষ করতে সময় নেয় = x ঘণ্টা
তাপস একা কাজটি শেষ করতে সময় নেয় = 4x ঘণ্টা

সুমনের 1 ঘণ্টার কাজের পরিমাণ = 1/x অংশ
তাপসের 1 ঘণ্টার কাজের পরিমাণ = 1/(4x) অংশ

তারা একসাথে 1 ঘণ্টায় কাজ করতে পারে = (1/x) + 1/(4x)
= (4 + 1) / (4x)
= 5 / (4x) অংশ

একসাথে কাজটি শেষ করতে সময় লাগে = 10 ঘণ্টা
অর্থাৎ, 10 ঘণ্টায় তারা পুরো 1টি কাজ শেষ করে

শর্তমতে,
(5/4x) × 10 = 1
⇒ 50/4x = 1
⇒ 4x = 50
⇒ x = 50 / 4
⇒ x = 12.5

∴ সুমনের কাজটি একা শেষ করতে 12.5 ঘণ্টা সময় লাগবে।

২৮১.

Three workers can do a job in 12 days. Two of the workers work thrice as fast as the third. How long would it take one of the faster workers to do the job himself?

ক
18 days
খ
28 days
গ
20 days
ঘ
22 days

ব্যাখ্যা

Question: Three workers can do a job in 12 days. Two of the workers work thrice as fast as the third. How long would it take one of the faster workers to do the job himself?

Solution:
Let,
The first and second worker need P days.
Third worker needs 3P days.

ATQ,
(1/P) + (1/P) + (1/3P) = 1/12
⇒ (3 + 3 + 1)/3P = 1/12
⇒ 7/3P = 1/12
⇒ 3P = 84
⇒ P = 84/3
∴ P = 28

So the one faster worker alone will take 28 days to complete the job.

২৮২.

A takes 50% longer than B to do the work. If they take 18 days collectively, how much time would B require to finish it alone?

ক
30 days
খ
18 days
গ
22 days
ঘ
26 days

ব্যাখ্যা

Question: A takes 50% longer than B to do the work. If they take 18 days collectively, how much time would B require to finish it alone?

Solution:
Let B takes x days to complete the work
Then A will take 50% more i.e 150% of x days i.e 3/2x days.

So the one day work of A and B together will be
(1/x) +{1/(3/2x)} = 1/18
⇒ (1/x) + (2/3x) = 1/18
⇒ 5/3x = 1/18
⇒ x = 30

∴ B takes 30 days to complete the work.

২৮৩.

A is thrice efficient as B and C is twice as efficient as B. What is the ratio of number of days taken by A,B and C, when they work individually?

ক
ক) 2:6:3
খ
খ) 2:3:6
গ
গ) 1:2:3
ঘ
ঘ) 3:1:2
ঙ
ঙ) None of these

ব্যাখ্যা

A : B : C
Ratio of efficiency - 3 : 1 : 2
Ratio of No. of days - 1/3 : 1/1 : 1/2
or 2 : 6 : 3

২৮৪.

Six identical machines can produce 540 articles in 12 hours. How many articles would 8 such machines produce in 15 hours?

ক
720
খ
800
গ
900
ঘ
1080

ব্যাখ্যা

Question: Six identical machines can produce 540 articles in 12 hours. How many articles would 8 such machines produce in 15 hours?

Solution:
Total articles produced by 6 machines in 12 hours = 540.
Articles produced by 1 machine in 12 hours = 540/6
Articles produced by 1 machine in 1 hour = 540/(6×12) = 7.5 articles

So, Articles produced by 8 machines in 15 hours = 7.5 × 8 × 15
= 900 articles

২৮৫.

A certain number of people were supposed to complete a work in 24 days. The work, however, took 32 days since 9 people were absent throughout. How many people were supposed to be working originally?

ক
28
খ
32
গ
36
ঘ
38

ব্যাখ্যা

Question: A certain number of people were supposed to complete a work in 24 days. The work, however, took 32 days since 9 people were absent throughout. How many people were supposed to be working originally?

Solution:
Let,
the total number of people were working originally = x
When 9 people were absent,
Total present workers were = x - 9

x workers can complete it = 24 days.
∴ 1 workers can complete it = 24x days.
∴ (x - 9) workers can complete it = 24x/(x - 9) days.

ATQ,
24x/(x - 9) = 32
⇒ 3x/(x - 9) = 4
⇒ 4x - 36 = 3x
∴ x = 36

The total number of people were working originally 36.

২৮৬.

If 36 men can do a piece of work in 25 hours, in how many hours will15 men do it?

ক
40
খ
50
গ
60
ঘ
70
ঙ
None of the above

ব্যাখ্যা

Let the required no of hours be x. Then
Less men , More hours (Indirect Proportion)
Therefore, 15:36 :: 25:x
=> (15 × x) = (36 × 25)
=> x = 60
Hence, 15 men can do it in 60 hours.

২৮৭.

To complete a work, X takes 25% more time than Y. If together they take 20 days to complete the work, how much time shall Y take to do it?

ক
38 days
খ
40 days
গ
30 days
ঘ
36 days

ব্যাখ্যা

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

Solution:
Let Y takes x days to complete the work
Then X will take 25% more i.e 125% of x days i.e 5/4x days.
So the one day work of X and Y together will be
(1/x) +{1/(5/4x)} = 1/20
⇒ (1/x) + (4/5x) = 1/20
⇒ 9/5x = 1/20
⇒ x = 36
∴ Y takes 36 days to complete the work.

২৮৮.

P can build a wall in 24 days and Q can do it in 20 days. With help of R, they completed the work in 6 days. Find in how many days R alone can do the work.

ক
29/4 days
খ
40/3 days
গ
44/5 days
ঘ
56/3 days

ব্যাখ্যা

Question: P can build a wall in 24 days and Q can do it in 20 days. With help of R, they completed the work in 6 days. Find in how many days R alone can do the work.

Solution:
P can do 1/24 part per day
Q can do 1/20 part per day
R can do per day = (1/6) - {(1/24) + (1/20)}
= (1/6) - (11/120)
= (20/120) - (11/120)
= 3/40
Time taken by R alone = 40/3 days

২৮৯.

A contractor undertook to complete a road project in 40 days. However, on the first day, 20 workers were absent. The remaining workers finished the project in 60 days. How many workers were originally supposed to work on the project?

ক
36
খ
50
গ
60
ঘ
80

ব্যাখ্যা

Question: A contractor undertook to complete a road project in 40 days. However, on the first day, 20 workers were absent. The remaining workers finished the project in 60 days. How many workers were originally supposed to work on the project?

Solution:
ধরি, শুরুতে শ্রমিকের সংখ্যা ছিল = x জন
মোট কাজ = x × 40

যেহেতু 20 জন শ্রমিক অনুপস্থিত ছিল, তাই বর্তমান শ্রমিক সংখ্যা = (x - 20) জন।
তারা কাজটি শেষ করে 60 দিনে।
∴ বর্তমান মোট কাজ = (x - 20) × 60

শর্তমতে,
40x = 60(x - 20)
⇒ 40x = 60x - 1200
⇒ 60x - 40x = 1200
⇒ 20x = 1200
⇒ x = 1200/20
⇒ x = 60

∴ শুরুতে 60 জন শ্রমিক নিয়োগ করা হয়েছিল।

২৯০.

A map has a scale of 1 cm to 3 km, what length on actual ground does a 3 cm length on the map represent?

ক
ক) 9 km
খ
খ) 1 km
গ
গ) 6 km
ঘ
ঘ) 6 cm
ঙ
ঙ) 1 cm

ব্যাখ্যা

Question: A map has a scale of 1 cm to 3 km, what length on actual ground does a 3 cm length on the map represent?

Solution:
Given that,
1 cm represents 3 km
∴ 3 cm represent (3 × 3) km
= 9 km

২৯১.

If 10 men or 18 women can do a work in 50 days then how many days would it take 25 men and 15 women to do twice the work?

ক
25 days
খ
30 days
গ
32 days
ঘ
40 days

ব্যাখ্যা

Question: If 10 men or 18 women can do a work in 50 days then how many days would it take 25 men and 15 women to do twice the work?

Solution:
Here,
10 men = 18 women
∴ 1 men = 18/10 women
∴ 25 men = (18 × 25)/10 women
= 45 women

∴ 25 men and 15 women = (45 + 15) = 60 women

ATQ,
18 women can do the work in 50 days
∴ 1 women can do the work in = (50 × 18) days
∴ 60 women can do the work in (50 × 18)/60 days
= 15 days

Hence to do twice the work time need = (15 × 2) = 30 days

২৯২.

Peter is twice as good as workman as Tom. When they work together they can finish a task in 16 days. If Tom works alone, in how many days he will complete the task?

ক
46 days
খ
48 days
গ
50 days
ঘ
52 days

ব্যাখ্যা

Question: Peter is twice as good as workman as Tom. When they work together they can finish a task in 16 days. If Tom works alone, in how many days he will complete the task?

Solution:
(Peter + Tom)'s one-day work = 1/16
As per the question, Peter can finish twice as much work as finished by Tom in a given duration of time.

Therefore, 2/3 of their one day's work will be completed by Peter and 1/3 of their one day work will be completed by Tom.

So, Tom's one day work will be = (1/16) × (1/3) = 1/48
So, Tom will take 48 days to complete the task.

২৯৩.

If 4 workers can assemble a car in 6 hours, how long would it take 15 workers to assemble the same car?

ক
102 minutes
খ
80 minutes
গ
96 minutes
ঘ
72 minutes

ব্যাখ্যা

Question: If 4 workers can assemble a car in 6 hours, how long would it take 15 workers to assemble the same car?

Solution:
Given,
4 workers can assemble a car in 6 hours
∴ 1 workers can assemble a car in (6 × 4) hours
∴ 15 workers can assemble a car in (24/15) hours
= (24 × 60)/15 minutes
= 96 minutes

২৯৪.

A can finish a work in 18 days and B can do the same work in 15 days. B worked for 10 days and left the job. In how many days, A alone can finish the remaining work?

ক
ক) 4
খ
খ) 5.5
গ
গ) 6
ঘ
ঘ) 8

ব্যাখ্যা

B's 10day's work = 1/15 × 10 = 2/3
Remaining work =
(1−2)/3 = 1/3
Now,
1/18 work is done by A in 1 day
∴ 1/3 work is done by A in 18 × 1/3 = 6 days

২৯৫.

Working alone, R can complete a certain kind of job in 9 hours. R and S, working together at their respective rates, can complete one of these jobs in 6 hours. In how many hours can S, working alone, complete one of these jobs?

ক
18
খ
12
গ
9
ঘ
6

ব্যাখ্যা

Question: Working alone, R can complete a certain kind of job in 9 hours. R and S, working together at their respective rates, can complete one of these jobs in 6 hours. In how many hours can S, working alone, complete one of these jobs?

Solution:
working together R and S can complete the job in 6 hours
which means
1/R + 1/S =1/6
R can complete the job in 9 hours
∴ 1/9 + 1/S = 1/6
⇒ 1/S = 1/6 - 1/9
⇒ 1/S = 1/18
∴ S working alone can complete the job in 18 hours

২৯৬.

One woman and one man can build a wall together in two hours, but the woman would need the help of two girls in order to complete the same job in the same amount of time. If one man and one girl worked together, it would take them four hours to build the wall. Assuming that rates for men, women and girls remain constant, how many hours would it take one woman, one man, and one girl, working together, to build the wall?

ক
10/7
খ
1
গ
5/7
ঘ
12/7
ঙ
None

ব্যাখ্যা

Question: One woman and one man can build a wall together in two hours, but the woman would need the help of two girls in order to complete the same job in the same amount of time. If one man and one girl worked together, it would take them four hours to build the wall. Assuming that rates for men, women and girls remain constant, how many hours would it take one woman, one man, and one girl, working together, to build the wall?

Solution:
One woman and one man can build a wall together in 2 hrs.
so, 1w + 1w = 1/2 [walls per hour]

The woman would need the help of 2 girls to complete the same job in the same amount of time.
so, 1w + 2g = 1/2 [walls per hour]
From (I) and (II), we deduce that: 1m = 2g

"If one man and one girl worked together, it would take them 4 hours to build the wall."
This gives the equation: 1m + 1g = 1/4 [walls per hour]
⇒ 2g + g = 1/4
⇒ 3g = 1/4
⇒ g = 1/12

we now know:
1m = 2g = 1/6
1w = 4g = 1/3

Now, 1 woman, 1 man, and 1 girl working together to build the wall

২৯৭.

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-

ক
5 days
খ
3 days
গ
2 days
ঘ
4 days
ঙ
none

ব্যাখ্যা

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

২৯৮.

If 4/9th of a bucket is filled in 1 minute, the rest of it will be filled in?

ক
ক) 0.75 minute
খ
খ) 1.5 minute
গ
গ) 1.25 minute
ঘ
ঘ) 1.95 minute

ব্যাখ্যা

Remaining part = 1- (4/9) = 5/9

Let the required time be x minutes
More volume to be filled, More time taken (Direct proportion)

∴ 4/9:5/9::1:x
⇒ (4/9)x = 5/9
⇒ x = (5/9)×(9/4)
⇒ x = 5/4

⇒ x = 1.25

২৯৯.

A, B and C working individually can complete a task in 30 days, 15 days and 10 days respectively. If A starts working alone and B and C helps A on every 2nd and 3rd day respectively, how long will it takes to be completed?

ক
ক) 15 days
খ
খ) 14 days
গ
গ) 12 days
ঘ
ঘ) None

ব্যাখ্যা

প্রশ্ন: A, B and C working individually can complete a task in 30 days, 15 days and 10 days respectively. If A starts working alone and B and C helps A on every 2nd and 3rd day respectively, how long will it takes to be completed?

সমাধান:
A একা ১ দিনে করে ১/৩০ অংশ
B একা ১ দিনে করে ১/১৫ অংশ = ২/৩০ অংশ
C একা ১ দিনে করে ১/১০ অংশ = ৩/৩০ অংশ

∴ B একা ১ দিনে করে A এর ২ দিনের কাজ
∴ C একা ১ দিনে করে A এর ৩ দিনের কাজ

A ও B একত্রে ১ দিনে করে A এর (১ + ২) = ৩ দিনের কাজ
A ও C একত্রে ১ দিনে করে A এর (১ + ৩) = ৪ দিনের কাজ
A, B ও C একত্রে ১ দিনে করে A এর (১ + ২ + ৩) = ৬ দিনের কাজ

এখানে কাজ আছে A এর ৩০ দিনের
এখন,
১ম দিন কাজ করে A একা সম্পন্ন করে ১ দিনের কাজ
২য় দিন কাজ করে AB সম্পন্ন করে মোট (১ + ৩) = ৪ দিনের কাজ
৩য় দিন কাজ করে AC সম্পন্ন করে মোট (৪ + ৪) = ৮ দিনের কাজ
৪র্থ দিন কাজ করে AB সম্পন্ন করে মোট (৮ + ৩) = ১১ দিনের কাজ
৫ম দিন কাজ করে A সম্পন্ন করে মোট (১১ + ১) = ১২ দিনের কাজ
৬ষ্ঠ দিন কাজ করে ABC সম্পন্ন করে মোট (১২ + ৬) = ১৮ দিনের কাজ
৭ম দিন কাজ করে A সম্পন্ন করে মোট (১৮ + ১) = ১৯ দিনের কাজ
৮ম দিন কাজ করে AB সম্পন্ন করে মোট (১৯ + ৩) = ২২ দিনের কাজ
৯ম দিন কাজ করে AC সম্পন্ন করে মোট (২২ + ৪) = ২৬ দিনের কাজ
১০ম দিন কাজ করে AB সম্পন্ন করে মোট (২৬ + ৩) = ২৯ দিনের কাজ
১১তম দিন কাজ করে A সম্পন্ন করে মোট (২৯ + ১) = ৩০ দিনের কাজ

∴ সম্পূর্ণ কাজ করতে ১১ দিন লাগবে।

বিকল্প পদ্ধতি,

বিকল্প পদ্ধতি,
A একা ১ দিনে করে ১/৩০ অংশ
B একা ১ দিনে করে ১/১৫ অংশ
C একা ১ দিনে করে ১/১০ অংশ

প্রথম ১০ দিনের জন্য হিসেব করে পাই,
প্রথম ১০ দিনে A কাজ করে ১০ দিন তথা = ১০/৩০ অংশ কাজ
প্রথম ১০ দিনে B কাজ করে ৫ দিন তথা = ৫/১৫ অংশ কাজ
প্রথম ১০ দিনে C কাজ করে ৩ দিন তথা = ৩/১০ অংশ কাজ

প্রথম ১০ দিনে A,B,C কাজ করে ১০/৩০ + ৫/১৫ + ৩/১০ = (১০ + ১০ + ৯)/৩০ = ২৯/৩০ অংশ

বাকি (১ - ২৯/৩০) = ১/৩০ অংশ কাজ করতে ১ দিন সময় লাগবে।
∴ মোট দিন = ১০ + ১ = ১১ দিন

৩০০.

39 persons can repair a road in 12 days, working 5 hours a day. In how many days will 30 persons, working 6 hours a day, complete the work?

ক
10
খ
13
গ
14
ঘ
15
ঙ
16

ব্যাখ্যা

Question: 39 persons can repair a road in 12 days, working 5 hours a day. In how many days will 30 persons, working 6 hours a day, complete the work?

Solution:
Let the required number of days be x.
Less persons, More days (Indirect Proportion)
More working hours per day, Less days (Indirect Proportion)

∴ 30 × 6 × x = 39 × 5 × 12
⇒ x = (39 × 5 × 12)/(30 × 6)
∴ x = 13