How to solve the work and time problems mentally?

Question

A can do 50% of the job in 16 days, B can do 1/4th of the job in 24 days. In how many days can they do 3/4 th of the job working together?

My approach.

A can complete the job in 32 days alone

B can complete the job in 96 days alone.

Together they can complete the job in ?

A rate of building something is 3 times the rate of B assuming 96 houses need to be build.

A + B in 1 day is going to build - 4 houses.

total work assumed here is 96 so .

96/4 = 24days for total work. But, we need to 3/4 of the work only so the answer is 24 * 3 /4 = 18days.

Answer 1

$A$ takes $32$ days to do it fully. So, $A$ does $\frac{1}{32}$ of the work in a single day. Similarly, $B$ does $\frac{1}{96}$ of the work in a single day. Together they do $\frac{1}{32}+\frac{1}{96}=\frac{1}{24}$ of work in single day. So together they take $24$ days to do complete work and hence $18$ days to do $75\%$ of the work.

Answer 2

Mentally, to see the problem more clearly, you could start by eliminating the fractions of the job since they differ between $A$ and $B$. I.e. answer the questions for $A$ and $B$: in how many days do they do the whole job?

For $A$ it's $\dfrac{16}{0.5} = 32$ days, for $B$ it's $\dfrac{24}{0.25} = 96$ days.

That means in one day $A$ does $\dfrac{1}{32}$ of the job, and $B$ does $\dfrac{1}{96}$.

Together they do $\dfrac{1}{32} + \dfrac{1}{96} = \dfrac{4}{96} = \dfrac{1}{24}$ of the job. And after two days, $\dfrac{2}{24}$ etc...

So basically by multiplying $\dfrac{1}{24}$ by a number of days, you get how much work was done. Now you want $0.75$ of the job done. By how much do you multiply $\dfrac{1}{24}$ to get $0.75$? $$\dfrac{1}{24}\times d = \dfrac{3}{4}$$

Together they do $\dfrac{3}{4}$ of the job in $18$ days.