On a bus, 1/10 of the total number of passengers get off at one spot, then 1/3 of the rest...

Question

On a bus, 1/10 of the total number of passengers get off at one spot, then 1/3 of the rest (the passengers left on the bus) get off at another point. There are 42 passengers left on the bus. What's the total number of passengers who were originally on the bus?

I was sent this question and I got 140 as the total.

I used the calculation as such:

Total - $\frac{1}{10}$ Total = NewTotal

$\frac{1}{3}$NewTotal = 42

NewTotal = 126

Total - $\frac{1}{10}$ Total=126

Total$(1-\frac{1}{10})$ = 126

Total = $\frac{1260}{9}$

Total = 140

Now, I was then sent another solution that went like this

$x - \frac{1}{10}x-\frac{1}{3}*\frac{9}{10}x=42$

$\frac{18}{30}x=42$

$x = \frac{1260}{18}$

$x=70$

Now, I maintain that the second solution used two-thirds of the nine tenths left one the bus rather than one third. But, I may very well be experiencing some blindness due to my desire to be right. Given that my answer is twice the answer I was given. I feel a little embarrassed to be writing this, but I claim that it is less to do with my mathematical abilities and more to do with my ability to see what I like to see when I want to be correct! So, can you all clear this up for me?

Answer 1

$\frac{\color{red}2}3 \cdot \text{New Total}=42$ as that is the leftover on the bus after the second point.

$\text{New Total} = \frac{126}{2}$

$\text{Total} - \frac{1}{10}\cdot \text{Total }=\frac{126}{2}$

Total$(1-\frac{1}{10}) = \frac{126}2$

Total = $\frac{1260}{9}\cdot \frac12$

Total $= 70$

Answer 2

You have misread the question.

• At the first stop, one-tenth get off, so nine-tenths of the original are left in the bus

• At the second stop, one-third of those get off, so two-thirds of those are left on the bus.

Further, both solutions you mention and those of other answers are needlessly complicated.

$R$ is the original number of people on the bus, and 42 is the number of people left on the bus after both stops, so we may write $R\space\times\frac{9}{10}\space\times\frac{2}{3}=42$. One can then divide by the fractions to obtain $R=70$.