Mathathon Sample Problem $4$ (Last One!)

Let $v$ be the number of vases and $f$ be the number of bunches of flowers.

$2v + 8f = \$15$ $(1)$

$20v + 5f = \$25$ $(2)$

Now, we are going to use the LCM method - find the LCM of the coefficients of the equations (i.e. the coefficients of $v$ and $f$ . (i.e. the numbers behind $v$ and $f$ .))

$(\text{LCM}(2, 8, 20, 5) = 40$

Using the fact that the LCM of the coefficients of $v$ and $f$ equals to $40$ , multiply equation $(1)$ by $10$ and equation $(2)$ by $4$ :

$20v + 80f = \$150$

$80v + 20f = \$100$

Add the two equations together:

$100v + 100f = \$250$

Simplify:

$100(v + f) = \$250$

Now, since we need to give our answer in the form $v + f$ , we will divide $\$250$ by $100$ :

$v + f =$ $\frac{\$250}{100}$

$v + f =$ $\frac{\$25}{10}$

$v + f =$ $\frac{\$2.5}{1}$

$v + f = \$2.50$

Answer: $\rightarrow \fbox{\$2.50}$

Answer (in word format): $\rightarrow \fbox{2 dollars and 50 cents}$

Although there are multiple ways to solve this problem, the fastest way is listed here.

First, let's set up a system of equations to represent the problem, with $v$ representing the cost of one vase and $f$ representing the cost of one flower bunch. With that, we have these equations:

$\begin{cases} 2v+8f=15 \\ 20v+5f=25 \end{cases}$

Now, let's divide both sides of the first equation by two, and divide both sides of the second equation by five.

$\begin{cases} 1v+4f=7.5 \\ 4v+1f=5 \end{cases}$

Next, let's add the two equations together and divide by five to get out answer!

$5v+5f=12.5$ $\boxed{v+f=\$2.50}$

Let the cost of one vase be $\$x$ and of one bunch of flowers be $\$y$ .

Then $\color{#D61F06}\text {2x+8y=15.20}$

$20x+5y=25$

Solving these two equations we get

$\color{#D61F06}\text {x=0.8266667},y=1.6933333$

So, $\color{#D61F06}\text {x}+y=2.52$

Hence the required answer is $\boxed {\$2.52}$ .