Question

A carpenter was hired to build $192$ window frames. The first day he made five frames and each day thereafter he made two more frames than he made the day before. How many days did it take him to finish his job?

Answer 1

For solving this question, we will note down the number of frames for some days and will find that it is a series in A.P. So, we will get some values like first term, difference between two consecutive terms and sum of the series as $5$, $2$ and $192$ respectively. Then, we will use the formula of sum of series in A.P. as $S=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$ to find the number of days.

Complete step by step answer:
We will start from writing the number of frames made by carpenter for some days by using the given condition in the question that he made two more frames than he made the day before as:
$\Rightarrow First\text{ }day=5$
$\Rightarrow Second\text{ }day=5+2=7$
$\Rightarrow Third\text{ }day=7+2=9$
$\Rightarrow Fourth\text{ }day=9+2=11$
$\Rightarrow Fifth\text{ }day=11+2=13$
As we can clearly observe that the number of frames for different consecutive days shows a series in A.P. as:
$\Rightarrow 5,7,9,11,13,...$
So, we will use the formula of sum of series to calculate the total number of days he worked.
$\Rightarrow S=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$
Where, $S=total\text{ }number\text{ }of\text{ }frames$, $a\text{ }=\text{ }number\text{ }of\text{ }frames\text{ }is\text{ }made\text{ }first\text{ }day$, $d=\text{ }difference\text{ }of\text{ }frames\text{ }between\text{ }con\sec utive\text{ }days$and$n=\text{ }total\text{ }number\text{ }of\text{ }days$.
Now, we will substitute the corresponding values in the mentioned formula as:
$\Rightarrow 192=\dfrac{n}{2}\left[ 2\times 5+\left( n-1 \right)2 \right]$
Here, we will solve the bracketed terms by using multiplication as:
$\Rightarrow 192=\dfrac{n}{2}\left[ 10+\left( 2n-2 \right) \right]$
Now, we will open the small bracket as:
$\Rightarrow 192=\dfrac{n}{2}\left[ 10+2n-2 \right]$
Here, we will subtract $2$ from $10$ and will get $8$ as:
$\Rightarrow 192=\dfrac{n}{2}\left[ 8+2n \right]$
After opening the bracket, we will have the above step as:
$\Rightarrow 192=\dfrac{n}{2}\times 8+\dfrac{n}{2}\times 2n$
After simplification of above step, we will get:
$\Rightarrow 192=4n+{{n}^{2}}$
Now, we will write the above equation as:
$\Rightarrow {{n}^{2}}+4n-192=0$
By factorization, we can write the above quadratic equation as:

& \Rightarrow {{n}^{2}}+16n-12n-192=0 \\\ & \Rightarrow n\left( n+16 \right)-12\left( n+16 \right)=0 \\\ & \Rightarrow \left( n+16 \right)\left( n-12 \right)=0 \\\ \end{aligned}$$ Now, we will have two value of $n$ by equation $0$ as: $\Rightarrow n=-16,12$ Since, the number of days can be negative. So, we will chose: $$\Rightarrow n=12$$ Hence, the carpenter will finish his work in $12$ days. **Note:** A.P. (Arithmetic Progression) is a series of numbers in which the difference between two consecutive numbers is the same and a constant number. For example, we can say the series of odd numbers or even numbers. Quadratic equation is an equation in the form of $a{{x}^{2}}+bx+c=0$, where, $a,b,c\ne 0$.