## Solutions to SSC CGL Time and Work Problems Set 72

Learn to solve 10 Time and Work Problems in SSC CGL Solutions Set 72 in 15 mins. Time and work and Time work wages problems are selected to test you well.

Contents are,

- Solutions to Time and work problems
- Solutions to Time work wages problems
- Techniques to solve time and work problems quickly in mind.

For best results take the test first at,

* SSC CGL level Question Set 72 on Work time problems 7*.

### Solutions to SSC CGL Time and Work Problems Set 72—time to solve was 15 mins

#### Problem** 1.**

A, B and C undertake to do a job for Rs. 575. A and C are supposed to finish $\displaystyle\frac{19}{23}$rd of the work together. The amount that shall be paid to B is,

- Rs. 475
- Rs. 100
- Rs. 210
- Rs. 200

** Solution** 1:** Problem analysis and execution**

The principle of amount paid for work done which is a part of work and wage concept, is,

Amount is paid in proportion to work portion done.

As B is to do $\displaystyle\frac{4}{23}$ portion of the work for which total earning is supposed to be Rs. 575, B will be paid an amount,

$\displaystyle\frac{4}{23}\times{575}=4\times{25}=\text{Rs. }100$.

**Answer:** Option b: Rs. 100.

**Key concepts used:** Amount paid in proportion to work done concept -- work and wage concept.

#### Problem** 2.**

A skilled, a half-skilled and an unskilled laborer work for 7, 8 and 10 days respectively and they together get Rs. 369 for their work. If the ratio of their each day's work is, $\displaystyle\frac{1}{3}:\displaystyle\frac{1}{4}:\displaystyle\frac{1}{6}$, then how much does the skilled laborer get?

- Rs. 143.50
- Rs. 102.50
- Rs. 164
- Rs. 201.50

**Solution** 2:** Problem analysis and execution**

We repeat the principle of amount paid for work done which is a part of work and wage concept,

Amount is paid in proportion to portion of work done.

A, B and C does portion of work in a day, in the ratio of, $\displaystyle\frac{1}{3}:\displaystyle\frac{1}{4}:\displaystyle\frac{1}{6}$.

Introducing the cancelled out HCF $x$ as a multiplying factor to the ratio terms to get the actual values, we get work portion done in a day by A, B and C as,

$\displaystyle\frac{x}{3}$, $\displaystyle\frac{x}{4}$ and $\displaystyle\frac{x}{6}$.

As A, B and C work together for 7, 8 and 10 days respectively and complete the work for a total sum of Rs. 369, this payment amount is equivalent to the total of work portions done by them (as amount paid is proportional to work portion done).

So,

$\displaystyle\frac{7cx}{3}+\displaystyle\frac{8cx}{4}+\displaystyle\frac{10cx}{6}=369$, where $c$ is the work to payment amount proportionality constant

Or, $\displaystyle\frac{28+24+20}{12}cx=6cx=369$,

So,

$cx=61.5$, and payment to the skilled laborer is,

$\displaystyle\frac{7cx}{3}=7\times{\displaystyle\frac{61.5}{3}}=7\times{20.5}=\text{Rs. }143.50$.

**Answer:** Option a: Rs. 143.50.

**Key concepts used:** Payment for work done is in proportion to work portion done -- HCF reintroduction technique to get actual work portion done from work ratio -- Equivalence of total payment to total work also because of payment to work done proportionality.

#### Payment amount earned to work done proportionality

Payment is in RS., whereas work done is in work units, whatever it is. This is the ideal case for explaining the concept of proportionality.

In work wage problems, a basic rule is, payment amount will be proportional to the work done.

Assuming total work as $W$ in work units, and $E$ in money units as the payment earned for the work, by the principle of work wage proportionality we have,

$E=cW$, where $c$ is called the constant of proportionality.

The more is the work done, the more will be the earning amount. The *earning is not equal to the work done, it is proportional to it*.

It follows, for 1 unit of work done, earning will be $c$ units.

That's why, when A, B, and C, three workers do, $\displaystyle\frac{7x}{3}$, $\displaystyle\frac{8x}{4}$, and $\displaystyle\frac{10x}{6}$ portions of total work, assuming $c$ as the earning in Rs. for each portion of work done, we can state mathematically,

$\displaystyle\frac{7cx}{3}+\displaystyle\frac{8cx}{4}+\displaystyle\frac{10cx}{6}=369$, where Rs. 369 is the total payment earned.

This is a linear equation and by solving it we will get the value of $cx$ which will be sufficient for us to get the payment earned by each of the workers.

For mathematically minded we have given the mathematical explanation here, which is good to know. This is the base concept and this how the proportionality concept works.

For making rough calculations for MCQ tests though, constant $c$ can easily be omitted in this problem.

#### Logic for solving in mind quickly

In actual test, we don't do the deductions as we did. Instead, applying the concepts of earning to work done proportionality, first we conclude that Rs. 369 will be divided in the ratio of $\displaystyle\frac{7}{3}:\displaystyle\frac{8}{4}:\displaystyle\frac{10}{6}$, as the three types of workers worked for 7, 8 and 10 days to earn the total amount of Rs. 369.

Now applying ratio and proportion concepts, we find out the value earned for 1 unit of proportional share as,

$\displaystyle\frac{369}{\displaystyle\frac{7}{3}+\displaystyle\frac{8}{4}+\displaystyle\frac{10}{6}}$.

You may need to write this step and calculate accurately and quickly.

The denominator can be mentally added to the value of 6, the common factor of 3 eliminated with 369, and final value of each share obtained as Rs. 61.5. Multiply this with $\displaystyle\frac{7}{3}$, the share portion of the skilled laborer, to get Rs. 143.50 as the answer.

Try and see if you can solve the problem this way. If you are more comfortable with mental calculations, you may not write the single step above, and evaluate the sum of fractions as 6 and divide 369 to get the value of each share, all in mind.

You can do it if you are **clear on the concepts** applicable. If you **practice solving in mind**, your conceptual strength as well as problem solving speed will increase.

#### Problem 3.

If a man earns Rs. 2000 for his first 50 hours of work in a week and is then paid one and half times his regular hourly rate, then hours he must work to make Rs. 2300 in a week is,

- 6 hours
- 4 hours
- 5 hours
- 7 hours

#### Solution 3: Problem analysis and solving

After doing the first 50 hours of weekly work at the rate of $\displaystyle\frac{2000}{50}=\text{Rs. }40$ per hour, if he does overtime, his earning rate increases by one and half times to Rs. 60 per hour.

To earn Rs. 300 more then, he has to work for 5 more hours.

**Answer:** Option c: 5 hours.

**Key concepts used:** Work and wage concept.

#### Problem** 4.**

A man and a woman working together can do a certain job in 18 days. Their skills in doing the work is in the ratio of $3:2$. How many days will the woman take to finish the job alone?

- 27 days
- 45 days
- 30 days
- 36 days

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

The meaning of, "Work skills of a man and a woman are in the ratio of $3:2$" is,

In the time duration a man does 3 units of work the woman does 2 units of work.

Then if $M$, $W$ are the portion of total work $K$ done by the man and the woman respectively in 1 day,

$\displaystyle\frac{M}{W}=\frac{3}{2}$,

Or, $M=\displaystyle\frac{3}{2}W$.

As a man and a woman together complete the work $K$ in 18 days,

$18(M+W)=K$,

Or, $18\times{\displaystyle\frac{3}{2}}W+18W=K$,

Or, $45W=K$,

A woman completes the work alone in 45 days.

**Answer:** Option b: 45 days.

**Key concepts used:** Work rate technique -- Worker equivalence concept -- Working together concept.

#### Problem** 5.**

P and Q can do a work in 6 days. Q and R can finish the same job in $\displaystyle\frac{60}{7}$ days. P started the work and worked for 3 days. Q and R continued for 6 days to finish the job. Then the difference of days that R and P take to complete the work alone is,

- 15
- 10
- 12
- 8

**Solution** 5:** Problem analysis and execution**

Let's assume $p$, $q$ and $r$ to be the portion of total work $k$ done in a day by P, Q and R respectively. We have to find, difference of $\displaystyle\frac{1}{r}$ and $\displaystyle\frac{1}{p}$.

Assuming work variables as work portion done in a day by the workers is what we call **Work rate technique** that simplifies fraction calculations which is transferred to the single variable of work, $k$.

By the first, second and third statements then,

$p+q=\displaystyle\frac{1}{6}k$,

$q+r=\displaystyle\frac{7}{60}k$, and

$3p+6(q+r)=k$.

Subtracting second equation from the first we get,

$p-r=\displaystyle\frac{1}{20}k$.

Now substituting value of $(q+r)$ from second equation to the third we get,

$3p+\displaystyle\frac{7}{10}k=k$,

Or, $p=\displaystyle\frac{1}{10}k$,

Or, P does $\displaystyle\frac{1}{10}$th of total work in 1 day (by definition of $p$), and so does the total work in 10 days, the inverse of work portion done in a day.

Substituting the value of $p$ in the fourth equation, $p-r=\displaystyle\frac{1}{20}k$,

$\displaystyle\frac{1}{10}k-r=\displaystyle\frac{1}{20}k$,

Or, $r=\displaystyle\frac{1}{20}k$,

Or, Q does the work $k$ in 20 days.

The difference in days of completing the work alone by R and P is 10 days.

**Answer:** Option b: 10.

**Key concepts used:** Work rate technique -- Working together concept -- linear equation solving.

Try to solve the problem by assuming the work variables as total days to complete the work alone for each worker.

#### Problem 6.

Three persons undertake to complete a piece of work for Rs. 1200. The first person can complete the work in 8 days, second person in 12 days and the third person in 16 days. They complete the work with the help of a fourth person in 3 days. How much does the fourth person get?

- Rs. 200
- Rs. 180
- Rs. 225
- Rs. 250

**Solution** 6:** Problem analysis and execution**

The total work portion completed by the first, second and the third person in 1 day is,

$\displaystyle\frac{1}{8}+\displaystyle\frac{1}{12}+\displaystyle\frac{1}{16}=\displaystyle\frac{6+4+3}{48}=\frac{13}{48}$.

In three days of working together then they will complete 39 out of 48 portions of work, leaving 9 out of 48 portion, that is, $\displaystyle\frac{3}{16}$th of the work for the fourth person to complete.

Thus by the payment earned to work done proportionality concept, the fourth person will earn $\displaystyle\frac{3}{16}$th of the total earning of Rs. 1200, which is, Rs. 225.

**Answer:** Option c: Rs. 225.

**Key concepts used:** **Work rate concept** of work portion done in a day as inverse of total number of days to complete the work; there is a subtle difference between this classic work rate concept and **Work rate technique** that we have used in the last problem; here we have used the classic concept because total number of days to complete the work are given -- **Working together concept** as the sum of work portion done in a day by each, resulting in work portion done by the workers working together in a day -- **Work and wage concept** as payment earned by a worker is proportional to the work portion done by him or her.

#### Problem** 7.**

If 10 men or 20 women or 40 boys can do a job in 7 months, then 5 men, 5 women and 5 boys can do half the work in,

- 6 months
- 5 months
- 4 months
- 8 months

**Solution** 7:** Problem analysis and execution**

As the time period of completion of the job by each type of worker team is same, that is, 7 months, we can assume 4 boys do the work of 1 man, and 2 boys do the work of 1 woman. This is application of **Worker equivalence concept**.

In the second case then, 5 men are equivalent to 20 boys and 5 women equivalent to 10 boys. With 5 more boys then, effectively a team of 35 boys will complete the work.

The work amount being, $40\times{7}=280$ boy-months, half of this work will be completed by,

$\displaystyle\frac{140}{35}=4$ months.

**Answer:** Option c: 4 months.

** Key concepts used:** **Worker equivalence concept**, we convert numbers of faster workers to the equivalent numbers of slowest worker so that fractional work is avoided -- **Mandays** concept as a measure of work.

#### Problem** 8.**

Three men A, B, and C can complete a work together in 6 hours. After working together for 2 hours, C left and A and B completed the work in 7 hours more. C alone does the work in,

- 16 hours
- 15 hours
- 14 hours
- 17 hours

**Solution** 8:** Problem analysis and execution**

Applying work rate technique, let us assume work rates in terms of work portion done in an hour by A, B and C respectively as, $a$, $b$ and $c$ and the total work amount as $k$.

By the first statement then,

$a+b+c=\displaystyle\frac{1}{6}k$.

By the second statement, as A, B, and C complete $\displaystyle\frac{1}{3}$rd of the work in 2 hours when C leaves, we get,

$7(a+b)=\displaystyle\frac{2}{3}k$,

Or, $(a+b)=\displaystyle\frac{2}{21}k$.

Substituting this value of $(a+b)$ in the first equation,

$\displaystyle\frac{2}{21}k+c=\displaystyle\frac{1}{6}k$,

Or, $c=\left(\displaystyle\frac{1}{6}-\displaystyle\frac{2}{21}\right)k$

Or, $c=\displaystyle\frac{1}{14}k$.

So, C will complete the job working alone in 14 hours.

**Answer:** Option c: 14 hours.

**Key concepts used:** Work rate technique -- Working together concept -- Leftover work concept.

#### Problem** 9.**

A can do a work alone in 4 days and B can do the same work alone in 6 days. If they work in alternate days with A starting the work, in how many days will the work be completed?

- $4\displaystyle\frac{1}{2}$ days
- $3\displaystyle\frac{1}{2}$ days
- $3\displaystyle\frac{1}{4}$ days
- $4\displaystyle\frac{2}{3}$ days

**Solution** 9:** Problem solving execution**

By the given statement, in first 2 consecutive days, the work portion done by the two will be,

$\displaystyle\frac{1}{4}+\displaystyle\frac{1}{6}=\displaystyle\frac{5}{12}$.

Taking two more days, that is in 4 days then they will complete $\displaystyle\frac{10}{12}$ portion of work, leaving $\displaystyle\frac{2}{12}$ portion of work.

On the fifth day it is A's turn to work and his work rate in a day is, $\displaystyle\frac{1}{4}=\frac{3}{12}$ portions of work.

So A will complete the rest of the work in $\displaystyle\frac{2}{3}$rd of a day more, and the total work will be completed in, $4\displaystyle\frac{2}{3}$ days.

**Answer:** Option d: $4\displaystyle\frac{2}{3}$ days.

**Key concepts used:** Work rate concept -- Working together concept -- Boundary condition concept; you need to be aware of the ending situation in such problems and treat it differently from the other pairs of days.

** Problem** 10.

A can do a work alone in 4 days while B can destroy the whole work alone in 16 days. If the two start to work simultaneously, then the work will be completed in,

- $5\displaystyle\frac{1}{3}$ days
- $5\displaystyle\frac{1}{2}$ days
- $10$ days
- $6$ days

#### Solution 10: Problem analysis and solving execution

Work destruction means we have to subtract the work portion destroyed in a day from the work done in a day by other workers to get the effective positive work done in a day. This in this case is,

$\displaystyle\frac{1}{4}-\displaystyle\frac{1}{16}=\displaystyle\frac{3}{16}$th of the total work.

The number of days to complete the work will be the inverse of this work portion done in a day,

$\displaystyle\frac{16}{3}=5\displaystyle\frac{1}{3}$ days.

**Answer:** Option a: $5\displaystyle\frac{1}{3}$ days.

**Key concepts used:** **Work destruction concept** as work portion destroyed per day and is to be subtracted from work portion completed per day by other workers to get the effective positive work done by the workers in a day; we can call this **Negative work concept** -- Work rate concept -- Working together concept.

### Solving in mind quickly

For problem 2, we have put down the logic of solving the problem in mind. For the other problems, we have stated the solution steps in such a way that number of steps are reduced as far as possible, as well as you can choose which steps you need to write down and which step you can do in mind without making any mistake.

Being safe and correct is as important as solving the problem quickly.

The truth is,

To solve most of the MCQ problems in mind, you must be

clear on the concepts, fairly quick inmental math calculations, especiallyfraction calculations, andpractice solving in mindextensively.

There is no shortcut or tricks. It is all intelligent and practiced problem solving.

### Useful resources to refer to

#### Guidelines, Tutorials and Quick methods to solve Work Time problems

**7 steps for sure success in SSC CGL Tier 1 and Tier 2 competitive tests**

**How to solve Arithmetic problems on Work-time, Work-wages and Pipes-cisterns**

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**How to solve a hard CAT level Time and Work problem in a few confident steps 3**

**How to solve a hard CAT level Time and Work problem in a few confident steps 2**

**How to solve a hard CAT level Time and Work problem in few confident steps 1**

**How to solve Work-time problems in simpler steps type 1**

**How to solve Work-time problem in simpler steps type 2 **

**How to solve a GATE level long Work Time problem analytically in a few steps 1**

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