Question

A and B can complete a job in 4 and 5 days respectively when work separately. How many days can
they take to complete the same job, if they work together?​

Answer 1

Step-by-step explanation:

Let A's one day work be

x

1

and B's one day work

y

1

.

It is given that A and B can complete work together in 5 days, therefore, we have:

A+B=

x

1

+

y

1

=

5

1

⇒

xy

y

+

xy

x

=

5

1

⇒

xy

y+x

=

5

1

⇒5(x+y)=xy.....(1)

A worked at twice his speed means

x

2

B worked at half of his speed means

2y

1

Together they complete work in 4 days, therefore, we have:

x

2

+

2y

1

=

4

1

⇒

2xy

4y

+

2xy

x

=

4

1

⇒

2xy

4y+x

=

4

1

⇒4(x+4y)=2xy

⇒2(x+4y)=xy

⇒2x+8y=xy.....(2)

Substitute equation 2 in 1:

5x+5y=2x+8y

⇒5x−2x=8y−5y

⇒3x=3y

⇒x=y

x=y means that number of days worked by A alone is equal to B alone.

Now, substitute the value in equation 1:

x

1

+

x

1

=

5

1

⇒

x

1+1

=

5

1

⇒

x

2

=

5

1

⇒x=5×2

⇒x=10

Hence, it would take 10 days for A alone to complete the job.

Answer 2

Given:

No. of days A take to complete work = 4

No. of days B take to complete work = 5

To find:

No. of days A and B take to complete work together.

Solution:

Amount of work A takes to complete in 1 day = $\frac{1}{4}$

Amount of work B takes to complete in 1 day = $\frac{1}{5}$

Let $x$ be the no. of days A and B take to complete the work together.

Then, amount of work both of them take to complete it together = $\frac{1}{x}$

The total amount of work they do to complete the work together is obtained by adding the work done by A and B separately in 1 day, i.e.,

$\frac{1}{x}=\frac{1}{4}+\frac{1}{5}$

Here, the denominators are not same and hence, we cannot perform simple addition on these fractions. We take the LCM of both the denominators,i.e., $4$ and $5$ to obtain a common denominator.

A suitable number is multiplied with these denominators to obtain LCM. This suitable number is also multiplied with the numerator too.

$LCM(4,5)=20$

$\frac{1}{x}=\frac{1}{4}* \frac{5}{5}+\frac{1}{5} *\frac{4}{4}$

$\frac{1}{x}=\frac{5}{20}+\frac{4}{20}$

Now, the denominators are same and thus, we can simply add them.

$\frac{1}{x}=\frac{9}{20}$

∴ $x=\frac{20}{9}=2.2$ ≈ $2$ days

Hence, A and B take 2 days to complete the job if they work together.

A and B take 2 days to complete the job if they work together.