MHB Solve Math Problem: Number of Passengers on a Bus

[anemone]

Hi all, this is another math problem that one can use the Singapore model method to solve. All are welcome to post a solution using the method you would like, if this problem interests you. :)

There were some passengers on a bus. At the first stop, $\dfrac{1}{5}$ of the passengers alighted and 80 boarded the bus. At the second stop, 240 passengers got off and 60 passengers boarded the bus. The bus now had $\dfrac{5}{8}$ of the number of passengers when it left the first stop. How many passengers were on the bus before the first stop?

 

[HOI]

Hi all, this is another math problem that one can use the Singapore model method to solve. All are welcome to post a solution using the method you would like, if this problem interests you. :)

Standard algebra:

There were some passengers on a bus.

Let the number passengers initially be x.

At the first stop, $\dfrac{1}{5}$ of the passengers alighted and 80 boarded the bus.

Now there are $x- \dfrac{1}{5}x+ 80= \dfrac{4}{5}x+ 80$ passengers on the bus.

At the second stop, 240 passengers got off and 60 passengers boarded the bus.

Now there are $\dfrac{4}{5}x+ 80- 240+ 60= \dfrac{4}{5}x- 100$ passengers on the bus.

The bus now had $\dfrac{5}{8}$ of the number of passengers when it left the first stop. How many passengers were on the bus before the first stop?

So $\dfrac{4}{5}x- 100= \dfrac{5}{8}\left(\dfrac{4}{5}x+ 80\right)= \dfrac{1}{2}x+ 50$.

Solve the equation $\dfrac{4}{5}x- 100= \dfrac{1}{2}x+ 50$ for x.

Subtract $\dfrac{1}{2}x$ from both sides- $\dfrac{4}{5}- \dfrac{1}{2}= \dfrac{8}{10}- \dfrac{5}{10}= \dfrac{3}{10}$.
$\dfrac{3}{10}x- 100= 50$.

Add 100 to both sides. $\dfrac{3}{10}x= 150$.

Multiply both sides by $\dfrac{10}{3}$.
$x= 500$.

There were originally 500 passengers on the bus.
(That's a large bus!)
Check:
(When I first did this problem, I made a trivial arithmetic mistake. Always check!)

At the first stop, $\dfrac{1}{5}$ of the passengers alighted and 80 boarded the bus.

$\dfrac{1}{5}$ of 500 is 100. 100 of the 500 passengers got off, leaving 500- 100= 400 passengers and another 80 boarded so there are now 480 passengers on the bus.

At the second stop, 240 passengers got off and 60 passengers boarded the bus.

480- 240= 240. 240+ 60= 300.

The bus now had $\frac{5}{8}$ of the number of passengers when it left the first stop.

When it left the first stop it had 480 passengers. $\frac{5}{8}$ of that is 5(60)= 300. Yes, that is what we got.

Before the first stop the bus had 500 passengers,

Now, how would the "Singapore model method" solve this problem?

 

[kaliprasad]

Hi all, this is another math problem that one can use the Singapore model method to solve. All are welcome to post a solution using the method you would like, if this problem interests you. :)

There were some passengers on a bus. At the first stop, $\dfrac{1}{5}$ of the passengers alighted and 80 boarded the bus. At the second stop, 240 passengers got off and 60 passengers boarded the bus. The bus now had $\dfrac{5}{8}$ of the number of passengers when it left the first stop. How many passengers were on the bus before the first stop?

My Attempt

After if left the 1st stop $\dfrac{5}{8}$ remained so $\dfrac{3}{8}$ was the number of more persons alighted than boarded

The number is 180

so $\dfrac{3}{8}$ of number of persons when it left the $1^{st}$ stop is 180

so number of persons when it left $1^{st}$ stop =$\dfrac{180}{{\frac{3}{8}}} = 480$

That was the number after 80 boarded

So the number before 80 boarded was 480-80 = 400

after $\dfrac{1}{5}$ got of remaining was $\dfrac{4}{5}$ which is 400

so number started with $\dfrac{400}{{\frac{4}{5}}} = 500$

 

[anemone]

[TIKZ]
\filldraw [fill=yellow,thick] foreach \i in {-4,...,-1} { ({\i*2.4},0) rectangle ({(\i+2)*2.4},1) };
\filldraw [fill=green, thick, dotted] foreach \i in {3,...,9} { ({\i*0.8},0) rectangle ({(\i+2)*0.8},1) };
\node at (-8.5,1.4) {\small 1 unit = 2 parts};
\draw [<->] (-9.5, 1.2) -- (-7.2, 1.2);
\node at (-8.4,0.6) {\tiny alighted passengers};
\node at (-8.4,0.2) {\tiny at 1st stop};
\draw[thick, dotted] (-7.2, 1) -- (-9.5, 0);
\draw[thick, dotted] (-7.8, 1) -- (-9.6, 0.2);
\draw[thick, dotted] (-8.4, 1) -- (-9.65, 0.4);
\draw[thick, dotted] (-7.2, 0.7) -- (-9, 0);
\draw[thick, dotted] (-7.2, 0.4) -- (-8.2, 0);
\draw [<->] (-7.2, 1.2) -- (-6, 1.2);
\node at (-6.6,1.4) {\small 1 part};
\draw [<->] (-7.2, 2) -- (8.8, 2);
\node at (0,2.2) {\small Total passengers on the bus after 1st stop and before 2nd stop};
\draw [<->] (-1.2, 1.2) -- (4.8, 1.2);
\node at (2,1.5) {\small Number of passengers got off the bus at 2nd stop = 240 - 60 = 180};
\draw[thick, dotted] (4.8, 0.2) -- (3.6, 0);
\draw[thick, dotted] (4.8, 0.4) -- (2.4, 0);
\draw[thick, dotted] (4.8, 0.6) -- (1.2, 0);
\draw[thick, dotted] (4.8, 0.8) -- (-0, 0);
\draw[thick, dotted] (4.8, 1) -- (-1.2, 0);
\draw[thick, dotted] (3.6, 1) -- (-1.2, 0.2);
\draw[thick, dotted] (2.4, 1) -- (-1.2, 0.4);
\draw[thick, dotted] (1.2, 1) -- (-1.2, 0.6);
\draw[thick, dotted] (0, 1) -- (-1.2, 0.8);
\draw[thick, dotted] (-6,0) -- (-6,1);
\draw[thick, dotted] (-3.6,0) -- (-3.6,1);
\draw[thick, dotted] (-1.2,0) -- (-1.2,1);
\draw[thick, dotted] (1.2,0) -- (1.2,1);
\node at (2.8,0.5) {\small 10};
\node at (3.6,0.5) {\small 10};
\node at (4.4,0.5) {\small 10};
\node at (5.2,0.5) {\small 10};
\node at (6,0.5) {\small 10};
\node at (6.8,0.5) {\small 10};
\node at (7.6,0.5) {\small 10};
\node at (8.4,0.5) {\small 10};
[/TIKZ]

Represent the given information in the model diagram above, the question wanted us to find the value for 10 parts.

From the model diagram,

$\begin{align*}3 \text{ parts} + 30 &=180\\3 \text{ parts}&=150\\ 1\text{ part}&=50 \\ \therefore 10\text{ parts}&=500\end{align*}$

 

[castor28]

Hi anemone,

Is there information available somewhere about the Singapore method ?

 

[anemone]

Hi castor28!

Thanks for showing your interest in Singapore model method in solving primary math word problems.

I just found a few insightful articles that discuss about the effectiveness to use model method to solve word problems, I hope you or anyone who are interested (@Fantini) find those articles meaningful to you!

https://bsrlm.org.uk/wp-content/uploads/2016/02/BSRLM-IP-35-3-20.pdf
http://people.math.harvard.edu/~engelwar/MathS305/Singapore Model Method Text.pdf
https://files.eric.ed.gov/fulltext/EJ1115069.pdf