Orange juice - mix with 3 cans of cold water
Lemonade - add 4 1/3 cans of cold water (one-third!)
Hawaiian Punch - mix in a 5-to-1 ratio with cold water
(I once drank a whole bottle of Hawaiian Punch concentrate! I can't stand the sight of it now.)
Why is there a difference in how much water you add to reconstitute juice?
It turns out to be a complicated question related to the viscosity of frozen concentrate, the freezing point of fruit juice from which most of the water has been removed, price of the juice, the amount of sugar in the mix, etc.
Freshly squeezed juices are evaporated in a vacuum chamber before being frozen. The process reduces essences and oils (readily noticed by our taste buds). When the juice is reconstituted, it may be refreshed with those essences and oils. Other things are added, such as citrus pulp, calcium and vitamins. Water is mixed back in until frozen orange concentrate is about three times strength of fresh juice. Lemonade and Limeade are slightly different because they need extra sugar to be palatable.
If the subject of fruit juice intrigues you, and you have enough math under your belt, you could do a research paper like this one, which investigated pomegranate juice concentrate:
Fourier transform infrared spectroscopy and chemometric techniques were used to detect adulteration of pomegranate juice concentrate (PJC) with grape juice (GJC). The differences between PJC and GJC infrared spectra occurred in the 1780–1685 mm region. Analysis of the spectra was used to: (1) differentiate pure PJC from GJC and (2) classify adulterated (containing 2–14% vol/vol GJC) and pure PJC samples. Two components explained 99% of the variability in each application. Partial least square analysis of spectra could also predict % acidity and solids in PJC with correlation coefficients of 0.9114 and 0.9916, respectively. Conclusion? FTIR and chemometrics provide a useful approach for authenticating pomegranate juice concentrate.
In other words, it's possible to identify and catch a supplier of concentrate who is "stretching it" with cheaper grape juice. These photos show unadulterated pomegranate juice, straight out of the juicer!
ISO Analytical is a small company with 6 employees. They can tell you if your juice comes from concentrate or is fresh from the fruit; or if it has been artificially sweetened. Often your taste buds make you suspicious, but because taste is so subjective, it's hard for taste buds to convince a jury!
CONCENTRATING: Freshly-squeezed juice commands higher status and price than juice from concentrate. Most fruit juices are made from concentrate, which involves removing water from the juice prior to shipment. On arrival in the consumer region, concentrate is diluted with local water to its original strength. Juice made from concentrate (but claimed to be freshly squeezed) can easily be detected by isotope analysis.
EXTENDING: There are economic incentives to adulterate apple or orange juice with less expensive sugar solutions, especially when a poor harvests result in a fruit shortfall. The simplest fraudulent method of extending fruit juice is to add inexpensive sugars and dilute with water to rebalance the sweetness. Luckily, the addition of inexpensive corn syrup or cane sugar to apple or orange juice can be detected by carbon-13 analysis.
ADULTERATION: Maple Syrup is the concentrated sap of certain species of maple tree. Pure maple syrup is a traditional sweetener, renowned for its unique taste and flavour. Maple syrup can be adulterated by adding cane sugar - the taste of a little cane sugar or corn syrup is virtually undetectable. The temptation to fraud can be strong, but the carbon-13 signatures of corn syrup (-11.29 ‰) and cane sugar (-11.85 ‰) are very different from maple syrup (-24.27 ‰). Thus adulterated maple syrup is easily detected with carbon-13 analysis.
This is a useful application of mathematics!