Answer:
Time taken by zeno is 20 minutes and time taken by lia is (20 – 8) = 12 minutes.
Solution:
Given, Zeno takes 8 min longer to decorate a cupcake than it takes lia.
And when they work together it takes 7.5 min.
We have to find how long would each take to do the job alone?
Now, let the time taken to work by the zeno alone be n then time taken by lia will be n - 8.
Then, work done by zeno in one minute [tex]=\frac{1}{n}[/tex]
And, work done by lia in one minute [tex]=\frac{1}{n-8}[/tex]
Together, the work done in one minute [tex]=\frac{1}{n}+\frac{1}{n-8}=\frac{2 n-8}{n(n-8)}[/tex]
And, they can complete the work in 7.5 min = [tex]\frac{2 n-8}{n(n-8)} \times 7.5 \rightarrow \frac{15 n-60}{n(n-8)}=1 \text { cup cake }[/tex]
[tex]\begin{array}{l}{\rightarrow 15 n-60=n(n-8) \rightarrow 15 n-60=n^{2}-8 n} \\ {\rightarrow n^{2}-8 n-15 n+60=0 \rightarrow n^{2}-23 n+60=0}\end{array}[/tex]
[tex]\begin{array}{l}{\rightarrow n^{2}-(20+3) n+20 \times 3=0 \rightarrow n^{2}-20 n-3 n+20 \times 3=0} \\ {\rightarrow n(n-20)-3(n-20)=0} \\ {\rightarrow(n-3)(n-20)=0}\end{array}[/tex]
So, n = 20 or 3
If we take n = 3, the time taken by lia will become negative which is not acceptable. So n = 20
Hence, time taken by zeno is 20 minutes and time taken by lia is (20 – 8) = 12 minutes.
