Determining how many sibling a person has

Question

Ask a fellow student you know to write down on a piece of paper the number of brothers he or she has (if this number is less than $10$) and multiply the result by $2$. Now add $3$ to the result and multiply the new number by $5$. Add to this the number of sisters this person has (again, only if this is less than $10$),and multiply the number by $10$. Finally, tell him or her to add the number of courses he or she taking this semester and then tell you the result. you will be able to tell how many brothers, sisters and courses he/she has.

Lets say, the student has $5$ brothers, $3$ sisters and $5$ courses.

$5 * 2=10$,

$10+3=13$,

$13*5=65$

$65+3=68$

$68*10=680$

$680+5=685$

Lets say, the student has $3$ brothers, $2$ sisters and $4$ courses.

$3*2=6$,

$6+3=9$,

$9*5=45$

$45+2=47$

$47*10=470$

$470+4=474$

So, I am able to see that the last number must represent the courses, but I dont see the trick that says how many sisters or brothers a person has.

Anyone sees it?

Answer 1

Let $B$ be the number of brothers, $S$ the number of sisters. It is easy to follow the steps and the final result is $$100B+150 +10S + \#courses$$

As you remarked, so long as the number of courses is $<10$ it just occupies the last digit and can be ignored. We will assume this to be the case.

Note that $B$ has no effect on the $10s$ place. Thus, knowing that place tells you $S$. For example, if the $10s$ place is $8$ then $S=3$. If it is $t$ then $S\equiv t-5\;mod(10)$.

Once you know $S$, you can get $B$ by looking at the $100s$ place (and the $1000s$ place if needs be). Let us say that $h$ is the entry in the $100s$ place (and by abuse of notation we'll include the $1000s$ place in $h$, so that if, for example, the final number was $1061$ we'd say $h=10$). If $S<5$ then $B=h-1$. If $S≥5$ then $B=h-2$.