Algebraic problem - Equation and mean value

Question

"The price of a train-ticket is 110 dollars for grown-ups and 90 dollars for children. To a train, 120 tickets were sold for a total of 11640 dollars. How many grown-ups bought a ticket to the train?"

So here's my thought process:

$11640/120 = 97$

So now I have the mean value, but I have no idea how to get the number of grown-ups that bought a ticket and are on the train. Help is extremely appreciated!

Answer 1

A grown-up ticket costs $\$20$ more than a child ticket. Every ticket costs at least $\$90$, so $120$ tickets cost at least $\$10800$. The difference between the total price, $\$11640$, and this minimum cost, $\$10800$, represents how much more is paid to "upgrade" to grown-up tickets.

The difference $\$11640 - \$10800 = \$840$ can be used to upgrade $42$ tickets, so there are $42$ grown-up tickets bought.

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In equations, $$\begin{align*} &&110g+90c &= 11640&(1)\\ &&g+c&=120&(2)\\ &(1)-90\times(2),&110g+90c-90(g+c)&= 11640-90\times120\\ &&20g&=840\\ &&g&= 42 \end{align*}$$

Answer 2

We can set up a system of equations from the information provided: $$ \begin{align} g + c &= 120 \\ 110g + 90c &= 11640 \end{align} $$ The first equation is concerning the total number of tickest (where $g$ and $c$ represent the number of grown-up tickets and child tickets sold respectively). The second equation concerns the price of tickets. We have $\$110$ times the number of grown-up tickets plus $\$90$ times the number of child tickets will sum to our total amount of $\$11640$.

From here we just need to manipulate these two equation and "solve for" $g$ and $c$. That is one of the useful traits of mathematics is that you can translate real-world scenarios into mathematical equations and then just forget about what is "really happening" but instead just manipulate the symbols.

The first of these two equations is a bit simple, so we will solve it for $g$. $$ g + c = 120 \quad\implies\quad g = 120 - c $$ But then this $g$ is the same as the $g$ in the second equation, so we can substitute this expression into the second equation in place of $g$. $$ \begin{align} 110g + 90c &= 11640 \\ 110(120 -c) + 90c &= 11640 \end{align} $$ Now we have an expression with just $c$ in it and we can solve for $c$. $$ \begin{align} 110(120 -c) + 90c &= 11640 \\ 13200 -110c + 90c &= 11640 \\ -20c &= -1560 \\ c &= 78 \end{align} $$

Now that we have $c$, we have the number of child tickets that were purchased. We see that the question is asking for the number of grown-up tickets, so we can plug our value of $c$ back into the first of our original two equations and solve for $g$. $$ \begin{align} g + c &= 120 \\ g + (78) &= 120 \\ g &= 42 \end{align} $$

Answer 3

You're going about it the wrong way. Use elimination in a system of equations.

                                g + c = 120
                                110g + 90c = 11640

Then, you can turn the top equation negative, and simply forget about the reality of it.

                              -g - c = -120

Then, you multiply it by either 90 or 110, trying to remove one of the variables.

                           -90g - 90c = -10800

or

                           -110g - 110c = 13200  

I'm going to use the first instance in this case(-90g - 90c = -10800) and add it to 110g + 90c = 11640, and I get 20g = 840. Then, you can divide by 20, giving you g = 42 or 42 grownups bought tickets for the train. You can then input it into one of your earlier equations, g + c = 120

                              g + c = 120
                              c = 120 - g

Then, lastly, you input grownups, and subtract, giving you your children, which is 78.

So, in the end, you get 42 grownups and 78 children riding your train.