RJ's fabulous juice bar

In this activity, students use equivalent ratios in the context of mixing juice cocktails.

In this context, a ratio is a short, symbolic way of describing the relative amounts of the different juices used to make a cocktail. A cocktail made with 1 part of A, 2 parts of B, and 3 parts of C is described by the ratio 1:2:3. All parts of a ratio are measured using the same unit (in this case; millilitres [ml]), but the ratio works regardless of the unit.

It is important to distinguish between ratios, which involve part-to-part comparisons, and proportions, which involve part-to-whole comparisons. You could discuss the difference between the two with reference to this 1:2:3 ratio:

Only 1 of the 6 parts is black, so the part-to-whole comparison for black is , while for grey it is 2/6 or 1/3, and for white it is 3/6 or 1/2.

In an equivalent ratio, the amounts of each component part of the ratio are increased or decreased without changing these part-to-part and part-to-whole comparisons. Have the students imagine the 1:2:3 ratio doubled:

The ratio is now 2:4:6, and the part-to-whole comparisons are 2/12 or 1/6 black, 4/12 or 1/3 grey, and 6/12 or 1/2 white. The students should note that the comparisons remain preserved when a ratio is duplicated or split into equal amounts. This fact, known as conservation of ratio, is not always an easy one for students to grasp.

Strip diagrams, ratio tables, and double number lines have been shown to be excellent tools for assisting students to think through problems involving ratios and proportions. This activity provides a good context for developing students’ understanding and skill with these tools.

Here is question 1a represented as a strip diagram:

The diagram shows clearly that the amount of orange is 1/5 of 300 (60 mL) and that the amount of pineapple and mango is 2/3 of 300 (120 mL) each.

Here is the question represented with interlocking cubes of three patterns:

The students could illustrate the problem using a double number line:

Alternatively, the information could be set out in a ratio table like this:

Note that the strip diagram is the least abstract of these representations.

Help the students to model question 1a as above but encourage them to draw their own diagrams to represent the other problems. Be aware that some may be uncertain of measurement conversions, for example, 1 l = 1 000 ml.

Here is one way of using strip diagrams to visualise the comparison in question 1b:

The 2 parts of pineapple in a medium Hawaiian amount to 2 x 120 = 240 ml. 1 part of pineapple in the medium Pineapple Punch is 200 mL, so 2 parts is 400 ml. The 400 ml in the second drink is clearly more than the 240 ml in the first. Alternatively, 2/5 is less than 2/3, so a Pineapple Punch needs more pineapple juice than a Hawaiian.

Here is a strip diagram for question 1c:

Any number less than 300 ml (the size of a small drink) is a possibility for the tiny drink called for in question 3a. For convenience, the number should be a multiple of 5 so that it can be easily divided into 5 parts as required by the ratio. Better still, if the number is a multiple of 50, each part will be a multiple of 10 ml, which will give sensible and convenient amounts for each of the three component juices. If 150 ml is chosen for the tiny size, the information can be represented in this way:

1:2:2 is equivalent to 30:60:60, so a tiny Hawaiian will have 30 mL of orange and 60 mL of both pineapple and mango.

For the giant drink in question 3b, any size greater than 1 L and less (perhaps) than 1.5 L is possible. If the size is a multiple of 40, each part will be a multiple of 10 mL.

1.

a. 120 ml. (300 ÷ 5 = 60. 60 x 2 = 120)

b.

• A medium Pineapple Punch. Both cocktails require 2 parts of pineapple juice, but the pineapple juice in the Pineapple Punch is 2 parts of 3 (that is, 2/3 of 600) and in the Hawaiian is 2 parts of 5 (that is, 2/5 of 600).
• A medium Hawaiian needs 600 ÷ 5 x 2 = 240 ml of pineapple juice.
• A medium Pineapple Punch needs 600 ÷ 3 x 2 = 400 ml of pineapple juice.

c. 250 ml. ( of 1 l [1 000 ml])

2.

75 mL coconut milk (300 ÷ 4), 75 ml orange juice (300 ÷ 4), 150 ml mango juice (300 ÷ 4 x 2 or 300 ÷ 2)

3.

Amounts will vary. For example, a tiny cocktail could be 150 ml or even 75 ml. A giant cocktail could be 2 or 3 times as big as a large cocktail. For both new sizes, the quantities need to be kept in proportion.

a. Answers will vary. For example, for a 150 ml tiny cocktail, 1 part for a tiny Hawaiian would be 30 ml (150 ÷ 5), and the drink would be 30 ml orange, 60 mL pineapple, and 60 ml mango. A 75 ml tiny Hawaiian would be half of these quantities.

b. Answers will vary. For a giant Tropical that was twice the large size, you would have 500 ml coconut milk, 1 l mango, and 500 ml orange; for 3 times the large size: 750 ml coconut milk, 1.5 l mango, and 750 ml orange.

4.

Problems will vary. You may have chosen different ml for the size of your drink, along with different flavours and ingredients.