## 26th SSC CGL tier II level Solution Set, 2nd on Time and work problems

This is the 26th solution set of 10 practice problem exercise for SSC CGL Tier II exam and the 2nd on Time and work problems. For maximum gains, the test should be taken first, that is obvious. But more importantly, to absorb the concepts, techniques and deductive reasoning elaborated through these solutions, one must solve many problems in a systematic manner using this conceptual analytical approach. One can learn well only by practicing learning by doing.

Before going through these solutions you should take the test by referring to * SSC CGL Tier II level Question Set 26 on Time and Work problems 2*.

### 26th solution set - 10 problems for SSC CGL Tier II exam: 2nd on Time Work problems - time 15 mins

**Problem 1.**

In 16 days A can do 50% of a job. B can do one-fourth of the job in 24 days. In how many days can they do three-fourths of the job while working together?

- 21
- 9
- 18
- 24

** Solution 1: Problem analysis and conceptual solution by work portion done in a day and working together concepts**

As number of days to complete a portion of a job is directly proportional to the portion of work done by a worker, by the first statement, A completes the whole job in,

$16\times{2}=32$ days, as $50\text{%}=\frac{1}{2}$

By the same concept, as B completes $\frac{1}{4}$th of the job in 24 days, the whole job is completed by B in,

$24\times{4}=96$ days.

This is the use of the first concept of direct proportionality of work done to number of days the worker worked.

**Solution 1: Problem solving second stage: Working together concept of summing up work portion done in a day**

When A and B work together, **total work portion done by them in a day** is given by **summing up the work portion done by each of them in a day**. **Inverting the total work portion done in a day**, you will get the **number of days required to complete the work by them** while working together.

To get the **work portion done in day** for a worker, just invert the number of days required by the worker to complete the work.

Using these concepts work done in a day by A and B working together is,

$\displaystyle\frac{1}{32}+\displaystyle\frac{1}{96}=\displaystyle\frac{4}{96}=\displaystyle\frac{1}{24}$.

This means, the whole work will be completed by the two working together in 24 days, and $\displaystyle\frac{3}{4}$th of the job will be completed in,

$24\times{\displaystyle\frac{3}{4}}=18$ days.

**Answer:** c: 18 days.

**Key concepts used:** * Work portion done to number of days of work direct proportionality -- Work portion done in a day as inverse of number of days to complete the work* --

*--*

**Working together concept to get portion of work done in a day by summing up portions of work done by each worker in a day**

**Number of days to complete the work as inverse of work portion done in a day.**If you are used to these common concepts of time and work, you can easily solve the problem in mind by being a little careful.

**Problem 2.**

If each of them had worked alone, B would have taken 10 hours more than what A would have taken to complete a job. Working together, they can complete the job in 12 hours. How many hours B would take to do 50% of the job?

- 30
- 20
- 10
- 15

** Solution 2: Problem analysis and execution: By Mathematical reasoning and Working together concept**

We have to introduce one variable for work completion time for either A or B, not two. By **general principle of mathematical problem solving**, that is supported by **common sense**. We would assume $b$ hours as the time taken by B to complete the work, because target duration involves B's completion time.

So completion time for A is,

$a=b-10$, which is in terms of $b$.

We would be dealing with a single variable.

As working together A and B complete the job in 12 hours, applying the * working together concept* of total work portion done by the two in a day as the sum of work portion done by each individually in a day,

$\displaystyle\frac{1}{b-10}+\displaystyle\frac{1}{b}=\displaystyle\frac{1}{12}$.

Cross-multiplying and rearranging terms we get the quadratic equation in $b$ as,

$b^2-34b+120=0$.

4 times 30 is 120 as well as 4 plus 30 is 34. So the factors of the quadratic equation are,

$(b-30)(b-4)=0$.

$b=4$ is not possible as $a=b-10$ will then be negative.

So, $b=30$ hours.

To complete 50% of the job then, B willl take half of 30, that is, 15 hours.

**Answer:** d: 15.

**Key concepts used:** * Work portion done in a day as inverse of number of days to complete the work* -- Working together concept as work portion done in a day by two workers by summing up their individual work portion done in a day

**-- Formation and factorization of quadratic equation.**In this form of time and work problems, it is hard to avoid formation and factorization of a quadratic equation. But usually this is easy.

**Problem 3.**

Two workers P and Q are engaged to do a piece of work. Working alone P would take 8 hours more to complete the work than when working together. Working alone Q would take $4\frac{1}{2}$ hours more than when they work together. The time required to finish the work together is,

- 5 hours
- 6 hours
- 4 hours
- 8 hours

**Solution 3: Problem analysis and solution by work per unit time and working together concept**

Though the problem is quite interestingly framed, it is easy to set up the equation for per hour work portion done when P and Q work together as,

$\displaystyle\frac{1}{T}=\displaystyle\frac{1}{T+8}+\displaystyle\frac{1}{T+4.5}$, where $T$ is the working together work completion time you have to find.

The two denominators represent the two work completion times in terms of $T$ for P and Q.

Cross-multiply and simplify to form the desired quadratic equation as,

$(T+8)(T+4.5)=T(2T+12.5)$,

Or, $T^2=36$,

Or, $T=6$

An unexpected quick result if you have followed the right path.

Cancellation of $12.5T$ on both sides of the equation makes things simpler.

**Answer:** b: 6 hours.

**Key concepts used:** ** Work portion done per unit time** --

*.*

**Working together concept****Problem 4.**

A contractor employed 200 men to complete a certain work in 150 days. If only one-fourth of the work gets completed in 50 days, then how many more men the contractor must employ to complete the whole work in time?

- 100
- 300
- 200
- 600

**Solution 4 : Problem analysis and execution: Mandays concept**

200 men do one-fourth of the work in 50 days. Assuming that work rate (work portion done by a man in a day) remains same for all men, a total of $4\times{50}$, that is, 200 days would have required for 200 men to finish the job. Obviously the contractor misjudged the work rate capacity of the men. That's why to meet the target of 150 days he would need to employ more men.

The reason for the need of more men being clear, let's get on with our main task of calculating the number of extra men required.

In 50 days, $\displaystyle\frac{1}{4}$th of work is done by 200 men,

So the total work amount in terms of mandays is,

$50\times{200}\times{4}=40000$ mandays.

To complete three-fourth remaining part of this work, that is, 30000 mandays work, in remaining 100 days, number of men required will simply be,

$\displaystyle\frac{30000}{100}=300$.

The contractor has to employ then 100 more men to finish the job in 150 days.

**Answer:** a: 100.

**Key concepts used:** * Work amount in terms mandays concept* --

*--*

**Work rate assessment**

**Mandays technique to find the number of extra men required**

**.****Problem 5.**

A, B and C are engaged to do a work for Rs.5290. A and B together are supposed to do $\displaystyle\frac{19}{23}$rd of the work and B and C together $\displaystyle\frac{8}{23}$rd of the work. Then A should be paid,

- Rs.4250
- Rs.3450
- Rs.2290
- Rs.1950

**Solution 5: Problem analysis and execution: Earning share concept, Worker compensation proportional to work portion done**

As B and C together complete $\displaystyle\frac{8}{23}$rd of the work, the rest of the work must be completed by A alone.

So A completes,

$1-\displaystyle\frac{8}{23}=\displaystyle\frac{15}{23}$rd of the work.

Total amount of Rs.5290 is to be paid proportionate to the work amount done. So A will be paid,

$\displaystyle\frac{15}{23}\times{5290}=15\times{230}=\text{Rs.}3450$.

The first statement of work portion done by A and B is to create diversion and is not required for getting the answer. But we can satisfy our curiosity by calculating that with A doing $\displaystyle\frac{15}{23}$rd portion of work, B would have done,

$\displaystyle\frac{19}{23}-\displaystyle\frac{15}{23}=\displaystyle\frac{4}{23}$rd of work and so C's work portion will be,

$\displaystyle\frac{8}{23}-\displaystyle\frac{4}{23}=\displaystyle\frac{4}{23}$rd portion of whole work.

This is the reason of total work given by two statements becoming $\displaystyle\frac{27}{23}$, the extra $\displaystyle\frac{4}{23}$ coming from C contributing twice.

**Answer:** b: Rs.3450.

**Key concepts used:** * Earning share concept* --

**Worker compensation proportional to work portion done**

**.**#### Problem 6.

Ruchi does $\displaystyle\frac{1}{4}$th of a job in 6 days and Bivas completes rest of the same job in 12 days. Then they together complete the job in,

- $9\frac{3}{5}$ days
- $9$ days
- $7\frac{1}{3}$ days
- $8\frac{1}{8}$ days.

**Solution 6 : Problem analysis and solution: Working together concept**

The first step is to accurately evaluate portion of job completed by each separately in 1 day.

As Ruchi does $\displaystyle\frac{1}{4}$th of the job in 6 days, her work rate in terms of work portion done in a day is,

$\displaystyle\frac{1}{4}\times{\displaystyle\frac{1}{6}}=\displaystyle\frac{1}{24}$.

Bivas completes the rest of the job, that is, $\displaystyle\frac{3}{4}$th of the job in 12 days. So the portion of job he completes in a day is,

$\displaystyle\frac{3}{4}\times{\displaystyle\frac{1}{12}}=\displaystyle\frac{1}{16}$.

Together they complete the portion of job in 1 day is then,

$\displaystyle\frac{1}{24}+\displaystyle\frac{1}{16}=\displaystyle\frac{5}{48}$.

And number of days they take to complete the job is inverse of this portion of total work done in a day by the two, which is,

$\displaystyle\frac{48}{5}=9\frac{3}{5}$ days.

**Answer:** a: $9\frac{3}{5}$ days.

**Key concepts used:** Work portion done directly proportional to number of days of work -- * Work rate in terms of work portion done in a day is work portion done divided by number of days of work* --

*.*

**Working together per unit time concept -- Number of days of completion of work is inverse of work portion done in a day**Easy to solve in mind with a little care.

** Problem 7.**

P and Q together can do a job in 6 days and Q and R finishes the same job in $\displaystyle\frac{60}{7}$ days. Starting the work alone P worked for 3 days. Then Q and R continued for 6 days to complete the work. What is the difference in days in which R and P can complete the job, each working alone?

- 15
- 8
- 12
- 10

**Solution 7: Problem analysis and solution: Work rate technique and Working together concept**

Assume, $p$, $q$ and $r$ to be the portion of work done by P, Q and R respectively in 1 day, each working alone.

By the first statement then,

$6(p+q)=W$, where $W$ is the total work amount.

So, $(p+q)=\displaystyle\frac{1}{6}W$

By the second statement similarly,

$\displaystyle\frac{60}{7}(q+r)=W$,

Or, $(q+r)=\displaystyle\frac{7}{60}W$.

And by the third statement,

$3p+6(q+r)=W$,

Or, $3p=W\left(1-\displaystyle\frac{7}{10}\right)=\displaystyle\frac{3}{10}W$

So, $10p=W$.

It means P completes the work in 10 days working alone.

Subtracting $(q+r)$ from $(p+q)$, you get,

$p-r=W\left(\displaystyle\frac{1}{6}-\displaystyle\frac{7}{60}\right)=\displaystyle\frac{1}{20}W$,

Or, $20p-20r=W$,

Or, $20r=2W-W=W$.

This means R will complete the work in 20 days working alone, and the desired difference in days is,

$20-10=10$.

**Answer:** d: 10.

**Key concepts used:** * Work rate technique* --

*--*

**Working together concept**

**Sequencing of events -- Algebraic simplification techniques.**The solution is speeded up because of bypassing the need of evaluating $q$.

** Problem 8.**

A man is twice as fast as a woman who is twice as fast as a boy in doing a piece of work. If one each of them work together and finish the work in 7 days, in how many days would a boy finish the work when working alone?

- 7
- 6
- 49
- 42

**Solution 8: Problem analysis and solution: Work rate technique and Worker equivalence concept**

Assume, $m$, $w$ and $b$ to be the portion of work done in a day by a man, a woman and a boy respectively when working alone. This is use of work rate technique. This approach reduces fraction calculation and thus speeds up solution.

So by the given efficiency statements, as in a day, a man does twice the work portion done by a woman, and a woman does twice the work portion done by a boy,

$m=2w=4b$.

Basically this means 1 man is equivalent to 4 boys and 1 woman is equivalent to 2 boys. This is **Worker equivalence concept**. Worker efficiency leads to worker equivalence.

So by the working together statement,

$7(m+w+b)=W$ where $W$ is the work amount.

Or, $7(4b+2b+b)=49b=W$,

This means, a boy working alone would complete the work in 49 days.

**Answer:** c: 49.

**Key concepts used:** ** Work rate technique **--

**Worker equivalence concept -- Working together concept -- Worker efficiency concept**

**.**** Problem 9.**

While A can do a job working alone in 27 hours, B can do it in 54 hours also working alone. Find the share of C (in Rs.) if A, B and C get paid Rs.4320 for completing the job in 12 hours working together.

- 1440
- 960
- 1280
- 1920

**Solution 9: Problem analysis and solution: Earning share proportional to work portion done and Working together concept**

In 12 hours, work portion done by A and B is,

$12\left(\displaystyle\frac{1}{27}+\displaystyle\frac{1}{54}\right)=\displaystyle\frac{2}{3}$.

So rest $\displaystyle\frac{1}{3}$rd portion of the work is completed by C.

As share of earning is proportional to work portion done, and total work is worth Rs.4320, the earning by C is one-third of Rs.4320,

$\displaystyle\frac{1}{3}\times{4320}=1440$.

**Answer:** a: 1440.

**Key concepts used:** * Earning share concept* --

**Earning to work done proportionality -- Working together concept -- Work portion left concept**

**.**** Problem 10.**

While A and B together finish a work in 15 days, A and C take 2 more days than B and C working together to finish the same work. If A, B and C complete the work in 8 days, in how many days would C complete it working alone?

- $20$ days
- $40$ days
- $24$ days
- $17\frac{1}{7}$ days

**Solution 10: Problem analysis and solution: Strategic problem definition, Work rate technique and Working together concept**

The **strategy of problem definition** is to **form first** the algebraic relations that contain **maximum amount of certain information**. Out of four given statements, the fourth statement carrying maximum amount of certain information, we'll first form corresponding equation as,

$8(a+b+c)=W$.

By work rate technique we have assumed variables $a$, $b$ and $c$ to be the work portion done per day by A, B and C respectively and $W$ as the total work amount.

Next we'll form the equation corresponding to the first statement as it involves no uncertainty,

$15(a+b)=W$.

It is easy to see that $c$ can be evaluated from these two equations by eliminating $(a+b)$.

From first equation,

$(a+b+c)=\displaystyle\frac{W}{8}$, and from second equation,

$(a+b)=\displaystyle\frac{W}{15}$.

Subtracting the second result from the first,

$c=W\left(\displaystyle\frac{1}{8}-\displaystyle\frac{1}{15}\right)=W\left(\displaystyle\frac{7}{120}\right)$.

Inverse of this work rate of C is the number of days to complete the work by C working alone. It is then,

$\displaystyle\frac{120}{7}=17\frac{1}{7}$.

**Answer:** d: $17\frac{1}{7}$ days.

**Key concepts used:** **Strategy of problem definition -- Work rate technique -- Working together concept -- Solving in mind****.**

This is a good example of diversionary tactics in a question. The second and the third statements are not required at all in finding the answer.

**Task for you:** What would be the number of days to complete the work for B working alone?

**Note:** Observe that most, if not all, of the problems can be solved quickly in mind if you use the right concepts and techniques. Problem analysis and clear problem definition play an important role in such quick solutions.

### Useful resources to refer to

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