Answer
$50$ Midnight Mango, $60$ Tropical Torrent, and $30$ Pineapple Power.
Work Step by Step
Step 1. Assume $x$ number of Midnight Mango, $y$ number of Tropical Torrent, and $z$ number of Pineapple Power were sold that day.
Step 2. Given the amounts of different juices used, we can set up the following equations:
$8x+6y+2z=820, 3x+5y+8z=690, 3x+3y+4z=450$
Step 3. Multiple the first equation by 4 to get $32x+24y+8z=3280$, multiply the last equation by 2 to get $6x+6y+8z=900$. Now subtract the middle equation from these two recent equations, we get
$29x+19y=2590$ and $3x+y=210$
Step 4. Rewrite the last equation as $y=210-3x$ and back-substitute in the equation before it, we have
$29x+19(210-3x)=2590$
Step 5. Solve the above equation to get $x=50$, thus $y=210-3\times50=60$ and $z=410-4x-3y=30$
Step 6. We conclude that the Juice Company sold $50$ Midnight Mango, $60$ Tropical Torrent, and $30$ Pineapple Power that day.
