Question

Tarun can print one – fourth of painting in 8 days. How many days will he take to complete the painting?

Answer 1

Hint: To solve the above question, we will first assume that Tarun takes ‘b’ days to complete \[{{a}^{th}}\] part of the painting and Tarun takes ‘x’ days to complete the full painting (y parts). Then, we will develop a relation between (b and x) and (a and y). From this relation, we will find out the value of x.

Complete step-by-step answer:
To start with, we will assume that Tarun takes ‘b’ days to complete \[{{a}^{th}}\] part of the painting. Thus, we can say that in one day, Tarun will print \[\dfrac{a}{b}\] of the painting. Thus, we have,
\[1\text{ day}\to \dfrac{a}{b}\text{ of the painting}\]
Now, we know that the value of a is \[\dfrac{1}{4}\] and b is 8. Thus, the painting printed by Tarun in one day will be
\[1\text{ day}\to \dfrac{\left( \dfrac{1}{4} \right)}{8}\text{ of the painting}\]
\[\Rightarrow 1\text{ day}\to \dfrac{1}{4\times 8}\text{ of the painting}\]
\[\Rightarrow 1\text{ day}\to \dfrac{1}{32}\text{ of the painting}\]
Thus, in one day, Tarun completes \[\dfrac{1}{32}\] of the full painting. Now, we have to determine the time taken by Tarun to complete the full painting. Let us assume that Tarun takes x days to complete y parts of the painting where y is the complete painting. Now, y is a complete painting, so it will have \[\dfrac{32}{32}\] parts of the painting. So, we have,
\[\Rightarrow x\text{ days}\to y\text{ parts of the painting}\]
\[\Rightarrow x\text{ days}\to \dfrac{32}{32}\text{ parts of the painting}\]
\[\Rightarrow x\times 1\text{ day}\to \dfrac{32}{32}\text{ parts of the painting}\]
\[\Rightarrow \dfrac{x}{32}=\dfrac{32}{32}\]
\[\Rightarrow x=32\text{ days}\]
Hence, Tarun takes a total of 32 days to print the entire painting.

Note: The question can also be solved using the following alternate method. The amount of work done by Tarun is proportional to the number of days he has printed the painting. Thus, we can say that,
\[\dfrac{1}{4}\text{ of the painting}\propto 8\text{ days}\]
\[\Rightarrow \dfrac{1}{4}\text{ of the painting}=8k......\left( i \right)\]
Similarly, we can say that
\[\dfrac{4}{4}\text{ of the painting}\propto x\text{ days}\]
\[\Rightarrow \dfrac{4}{4}\text{ of the painting}=xk\]
\[\Rightarrow 4\times \dfrac{1}{4}\text{ of the painting}=kx\]
\[\Rightarrow 4\times 8k=kx\]
\[\Rightarrow 32k=kx\]
\[\Rightarrow x=32\]
Hence, Tarun takes a total of 32 days to print the entire painting.