Question

(i) Sunil takes 5 days more than Anil to complete a certain work. 4 days after starting
the work, Anil left the work. The remaining work was done by Sunil in 5 days.
Find how many days each will take to complete the work.​

Answer 1

$\huge{\text{\underline{Question:-}}}$

Sunil takes 5 days more than Anil to complete a certain work. 4 days after startingvthe work, Anil left the work. The remaining work was done by Sunil in 5 days. Find how many days each will take to complete the work.

$\huge{\text{\underline{Solution:-}}}$

Let Anil takes x number of days to complete the work.

$\implies$1 day = 1 / x of the work

Sunil takes 5 more days than Anil to complete the work.

$\implies$x + 5 days

$\implies$1 day = 1 / (x + 5) of the work

★ Together:-

1 day = 1 / x + 1 / (x + 5)

1 day = (x + 5 + x) / x (x + 5)

1 day = (2x + 5) / (x² + 5x) of work done

★ To find 4 days of work done:-

1 day = (2x + 5) / (x² + 5x)

4 days = 4 (2x + 5) / (x² + 5x)

4 days = (8x + 20) / (x² + 5x) of work done.

★ To find the amount of work left to be done:-

Work left = 1 - (8x + 20) / (x² + 5x)

Work left = (x² + 5x - 8x - 20) / (x² + 5x)

Work left = (x² - 3x - 20) / (x² + 5x)

★ Point to remember:-

Given that Anil needs 5 days to complete the rest of the work.

(x² - 3x - 20) / (x² + 5x) ÷ 1 / (x + 5) = 5

(x² - 3x - 20) / (x² + 5x) × (x + 5) = 5

(x² - 3x - 20) / x = 5

x² - 3x - 20 = 5x

x² - 8x - 20 = 0

(x - 10) (x + 2) = 0

x = 10 or x = - 2 (Number of days cannot be negative)

$\large{\boxed{\text{x = 10}}}$

Therefore, x = 10.

★ To find the number of days each take to complete the work:-

Anil = $\large{\boxed{\text{ x = 10 days}}}$

Sunil = x + 5 = 10 + 5 = 15

$\large{\boxed{\text{= 15 days}}}$

Hence, Anil takes 10 days and Sunil takes 15 days to complete the work.

__________________________________

Answer 2

$\sf{\underline{\underline{\huge{\rm{AnsWer:}}}}}$

Number of days taken by Anil to complete the work = 10 days

Number of days taken by Sunil to complete the work = 15 days

$\sf{\underline{\underline{\large{\mathfrak{StEp\:by\:stEp\:explanation:}}}}}$

GiVeN :

• Sunil takes 5 days more than Anil to complete a certain work.
• 4 days after startingthe work, Anil left the work.
• The remaining work was done by Sunil in 5 days

To FiNd :

• Number of days taken by Sunil and Anil to complete the work individually.

SoLuTiOn:

Let the number of days taken by Anil to complete the work be x.

$\sf{\therefore{Anil's\:one\:day\:work\:=\:{\dfrac{1}{x}}}}$ ---> (1)

$\sf{\underline{\underline{As\:PeR\:tHe\:FiRsT\:cOnDiTiOn:}}}$

• Sunil takes 5 days more than Anil to complete a certain work.

$\sf{\therefore{Number\:of\:days\:taken\:by\:Sunil\:to\:complete\:the\:work\:=\:x\:+\:5\:days}}$

$\sf{\therefore{Sunil's\:one\:day\:work\:=\:{\dfrac{1}{x\:+\:5}}}}$ ---> (2)

$\sf{\underline{\underline{As\:PeR\:tHe\:SeCoNd\:cOnDiTiOn:}}}$

• 4 days after starting
• 4 days after startingthe work, Anil left the work.

Anil's + Sunil's work :

$\longrightarrow$ $\sf{Anil's\:one\:day\:work\:+\:Sunil's\:one\:day\:work}$

$\longrightarrow$ $\sf{\dfrac{1}{x}}$ + $\sf{\dfrac{1}{x\:+\:5}}$

$\longrightarrow$ $\sf{\dfrac{x\:+\:5\:+\:x}{x\:(x\:+\:5)}}$

$\longrightarrow$ $\sf{\dfrac{2x\:+\:5}{x^2\:+\:5x}}$

$\longrightarrow$$\sf{\therefore{Anil's\:and\:Sunil's\:one\:day\:work\:=\:{\dfrac{2x\:+\:5}{x^2\:+\:5x}}}}$

4 days work :

To find the work completed in 4 days we can simply multiply one day's work by 4,

$\longrightarrow$$\sf{4\:({\dfrac{2x\:+\:5}{x^2\:+\:5x})}}$

$\longrightarrow$$\sf{\dfrac{8x\:+\:20}{x^2\:+\:5x}}$

$\longrightarrow$$\sf{\therefore{Work\:completed\:in\:4\:days\:=\:{\dfrac{8x\:+\:20}{x^2\:+\:5x}}}}$

Now given given that,

• Anil left the work

So amount of work remaining will be difference of 4 day's work and one

$\longrightarrow$$\sf{\:1\:-\:{\dfrac{8x\:+\:20}{x^2\:+\:5x}}}$

$\longrightarrow$ $\sf{\dfrac{x^2\:+\:5x\:-\:8x\:-\:20}{x^2\:+\:5}}$

$\longrightarrow$ $\sf{\dfrac{x^2\:-\:3x\:-\:20}{x^2\:+\:5x}}$

Remaining work was done :

Sunil completed the remaining work in 5 days.

Constituting it mathematically,

$\longrightarrow$$\sf{\dfrac{x^2\:-\:3x\:-\:8}{x^2\:+\:5x}}$ ÷ $\sf{\dfrac{1}{x\:+\:5}}$ = 5

$\longrightarrow$$\sf{\dfrac{x^2\:-\:3x\:-\:20}{x^2\:+\:5x}}$ $\sf{\times{x\:+\:5}}$ = 5

$\longrightarrow$$\sf{\dfrac{x^2\:-\:3x\:-\:20}{x}}$ = 5

$\longrightarrow$$\sf{x^2\:-\:3x\:-\:20\:=\:5x}$

$\longrightarrow$$\sf{x^2\:-\:3x\:-\:5x\:-\:20\:=0\:}$

$\longrightarrow$$\sf{x^2\:-\:8x\:-20\:=\:0}$

$\longrightarrow$$\sf{x\:(x^2\:=\:8x\:+20}$

$\longrightarrow$$\sf{(x-10)\:\:(x+2)\:=\:0\:}$

$\longrightarrow$$\sf{(x-4)^2\:-\:36}$

$\longrightarrow$$\sf{x\:y\:-\:2\:\:OR\:\:x\:=\:10}$

• Calculation from the website :

• Wolframalpha

x = - 2 is not acceptable since the number of days cannot be negative.

•°• Number of days taken by Anil to complete the work individually = 10 days.

Number of days taken by Sunil,

$\longrightarrow$ $\sf{x+5}$

$\longrightarrow$ $\sf{10\:+\:5}$

$\longrightarrow$ $\sf{15}$

•°• Sunil took 15 days to complete the work individually.