Frank wants to make a fruit punch drink for a family party. The recipe for his fruit punch drink is:
\(500\text{ mL}\) cranberry juice
\(1.8\text{ L}\) orange juice
\(1450\text{ mL}\) pineapple juice
\(350\text{ mL}\) ginger ale
\(2.4\text{ L}\) water
Frank has three different-sized containers: a \(6\text{ L}\) jug, a \(5700\text{ mL}\) jug, and a \(6.8\text{ L}\) jug. Which jug(s) could Frank use to hold his fruit punch?
If the costs of his four flavoured drinks average to \(\$4.56\) per litre, approximately how much will it cost Frank to make his fruit punch? Assume Frank does not need to pay for water.
There will be \(31\) family members at the party and Frank wants to make enough fruit punch so that each person can drink one cup (\(250\text{ mL}\)). Currently his recipe will not make enough fruit punch for everyone. Adjust the amounts of each ingredient in the recipe so that Frank has enough of his fruit punch drink for everyone, but not too much extra.
Since there are \(1000\text{ mL}\) in one litre, we can convert all volumes in the recipe to litres. This gives \(0.5\text{ L}\) of cranberry juice, \(1.45\text{ L}\) of pineapple juice, and \(0.35\text{ L}\) of ginger ale. Now we can find the total volume of the fruit punch by adding up the volumes of each of the five ingredients. This gives \(0.5+1.8+1.45+0.35+2.4=6.5\text{ L}\).
In litres, the capacity of the jugs are \(6\text{ L}\), \(5.7\text{ L}\), and \(6.8\text{ L}\). Therefore, the only jug large enough to hold \(6.5\text{ L}\) of fruit punch is the \(6.8\text{ L}\) jug.
First we find the total volume of the four flavoured drinks. We could add up the volumes of the four flavoured drinks, or simply subtract the volume of water from the total we found in part (a). This gives \(6.5-2.4=4.1\text{ L}\). Since the costs of the four flavoured drinks average to \(\$4.56\) per litre, and their total volume is \(4.1\text{ L}\), then it will cost Frank approximately \(\$4.56 \times 4.1 \approx \$18.70\) to make the fruit punch.
One recipe makes a total of \(6.5\text{ L}\) or \(6500\text{ mL}\) of fruit punch. In order to make enough for everybody, Frank should make \(31 \times 250 = 7750\text{ mL}\). So his recipe is short by \(7750-6500=1250\text{ mL}\).
We could just double the recipe, but since the difference is only \(1250\text{ mL}\), that would make much more fruit punch than Frank actually wants. Notice that \(7750 \div 6500 \approx 1.1923\). So if we multiply each of the volumes in the recipe by \(1.2\), then there will be enough fruit punch, without having too much extra. This gives the following volumes, in \(\text{mL}\):
Cranberry juice: \(500 \text{ mL} \times 1.2 = 600\text{ mL}\)
Orange juice: \(1800 \text{ mL} \times 1.2 = 2160\text{ mL}\)
Pineapple juice: \(1450 \text{ mL} \times 1.2 = 1740\text{ mL}\)
Ginger ale: \(350 \text{ mL} \times 1.2 = 420\text{ mL}\)
Water: \(2400 \text{ mL} \times 1.2 = 2880\text{ mL}\)
The total volume of the fruit punch will then be \(600 + 2160 + 1740 + 420 + 2880 = 7800\text{ mL}\), which is enough for everyone, with only \(7800 - 7750 = 50\text{ mL}\) extra.