Question 14/20: Set 5 - Try Out - Mathematical Reasoning (SNBT)

:00

Angga bought $3$ packs of soy sauce, $1$ pack of sauce, and $2$ packs of pepper, paying $\text{Rp20,000.00}$ . Bimo bought $1$ pack of soy sauce, $2$ packs of sauce, and $1$ pack of pepper, paying $\text{Rp12,500.00}$ . Meanwhile, Cinta bought $2$ packs of soy sauce, $1$ pack of sauce, and $2$ packs of pepper, paying $\text{Rp16,000.00}$ . If Rizal buys one pack each of soy sauce, sauce, and pepper, he must pay...

$\text{Rp10,500.00}$

$\text{Rp10,000.00}$

$\text{Rp9,500.00}$

$\text{Rp9,000.00}$

$\text{Rp8,500.00}$

Explanation

This problem can be solved by modeling it into a System of Linear Equations with Three Variables. Let's define the variables first for clarity:

\begin{aligned} x &= \text{price of 1 pack of soy sauce} \\ y &= \text{price of 1 pack of sauce} \\ z &= \text{price of 1 pack of pepper} \end{aligned}

Based on the shopping information of Angga, Bimo, and Cinta, we can set up three equations as follows:

\begin{cases} 3x + y + 2z = 20,000 & \text{(Angga)} \\ x + 2y + z = 12,500 & \text{(Bimo)} \\ 2x + y + 2z = 16,000 & \text{(Cinta)} \end{cases}

Our target is to find out how much Rizal has to pay if he buys one of each item, which means we are looking for the value of $x + y + z$ .

Notice Angga's and Cinta's equations. Both have the exact same amount of sauce ( $y$ ) and pepper ( $2z$ ). This is a shortcut to finding the value of $x$ very quickly! We simply subtract Cinta's equation from Angga's equation:

\begin{aligned} (3x + y + 2z) - (2x + y + 2z) &= 20,000 - 16,000 \\ x &= 4,000 \end{aligned}

Now we know the price of soy sauce is $\text{Rp4,000.00}$ . Let's substitute this value into Bimo's and Cinta's equations to simplify the system:

\text{From Bimo: } 4,000 + 2y + z = 12,500 \implies 2y + z = 8,500

\text{From Cinta: } 2(4,000) + y + 2z = 16,000 \implies y + 2z = 8,000

We get a new system with two variables. From the equation $y + 2z = 8,000$ , we can rewrite it as $y = 8,000 - 2z$ . Substitute this into the equation $2y + z = 8,500$ :

\begin{aligned} 2(8,000 - 2z) + z &= 8,500 \\ 16,000 - 4z + z &= 8,500 \\ 16,000 - 3z &= 8,500 \\ 3z &= 7,500 \\ z &= 2,500 \end{aligned}

After getting $z = 2,500$ , we can find $y$ :

y = 8,000 - 2(2,500) = 8,000 - 5,000 = 3,000

Now all unit prices are known: $x = 4,000$ (soy sauce), $y = 3,000$ (sauce), and $z = 2,500$ (pepper).

Thus, the total amount Rizal must pay is:

\text{Total} = x + y + z

\text{Total} = 4,000 + 3,000 + 2,500

\text{Total} = 9,500

So, Rizal must pay $\text{Rp9,500.00}$ .

Command Palette

Number 14

Explanation