# Ratios

Purpose

In this unit students convert between the metric units for capacity, within the context of drinks. Students engage in proportional thinking as they use their knowledge of fraction, decimal and percentage conversions, to explore, compare and combine ratios, and to calculate rates.

Achievement Objectives
GM5-2: Convert between metric units, using decimals.
NA5-1: Reason with linear proportions.
Specific Learning Outcomes
• Convert between millilitres and litres.
• Convert ratios to fraction and percentage expressions.
• Compare ratios.
• Combine ratios.
• Solve problems involving rates.
Description of Mathematics

A ratio describes the part-to-part or part-to-whole relationship between two or more measures. In much of this unit the ratio describes the relationship between the quantities of liquid that make up a mixed drink, for example the amount of concentrate to water, or fizz to juice. Ratios can be expressed as fractions and percentages, for example a 1:4 ratio of concentrate to water can be expressed as 1/5 concentrate or 20% concentrate. Proportional thinking is required to co-ordinate pairs of numbers and calculate the multiplicative operative between the numbers to find missing values and to make comparisons. For example, if the ratio of concentrate to water is 2:3 how many millilitres of concentrate is needed to make 500mL of drink? A rate describes a relationship between two different measures. In this unit one of the examples we use is the cost per quantity.

This unit links to Stage 8 of the Number Framework where students are required to use proportional thinking to solve problems.

Required Resource Materials
• Drink concentrate
• Measuring cups, jugs or cylinders
• Flavoured drink eg cordial, powdered juice, juice, lemonade
• Bottles or jugs for making drink in
• Cups
• Counters
Activity

#### Prior to this unit

Some experience with rates may be helpful prior to this unit.

#### Session 1

In this session we introduce students to the concept of a ratio and ways that it can be expressed.

1. Show the students a bottle of drink concentrate and read the instructions on the label on how to make the drink. For example some bottle labels say "Makes 5 litres" or mix 1 part concentrate to 4 parts water.
2. Write the ratio as 1:4 and explain to the students that the 1 refers to 1 part of concentrate and the 4 refers to 4 parts of water. So for every one measure of concentrate you need to add four measures of water. Ask the students why they think we use the term “parts” rather than giving the amount in millilitres
3. To demonstrate the reason for using the term “parts” divide the students in to groups and give each group some concentrate mixture and measuring devices. Give each group a different sized bottle and ask all the groups to make the juice using the same ratio.
4. Discuss with the students the strategies they used to make up the juice. Allow the students to taste the drink that different groups make. They should conclude the taste is similar because the ratio of concentrate to water was the same in each.
5. Explain to the students that ratios can also be expressed as fractions or percentages. Using the ratio of 1:4 make a litre of drink. Ask the students how many parts of liquid are in the mixture? (5) and how many parts of the mixture are concentrate (1), what is this as a fraction? (1/5).
6. Give the students other ratios to convert to fractions including examples such as 2:3. Students at this level should also be able to convert the fractions to percentages. Students may find it easier to draw the ratios for example: the ratio of 3:5

#### Session 2

In this session students compare ratios by converting between ratios, fractions and percentages.

1. Give groups of students drink concentrate, measuring devices, bottles/jugs and cups. Ask each group to make a different ratio of drink, for example, 2:3, 2:4, 3:2, 2:1, 3:1, 2:8.
2. Ask the students to convert the ratios to fractions or percentages and to order the drinks from strongest flavoured (highest proportion of concentrate) to weakest flavour. Students may wish to draw diagrams as they did in session one.
3. Students can taste the drinks to test their order.
4. Students can continue to compare ratios by converting them to fractions and percentages. Here are some examples of problem structures you can use to develop problems to ask students:
• What drink would taste the strongest, a ratio of 4:6 or 1:2?
• If a drink is made to a ratio of 1:4 what percentage of the drink is concentrate?
• If a drink is made to a ratio of 2:3 what fraction of the drink is concentrate?

#### Session 3

In this session students solve problems that involve using the ratios to find the proportions in millilitres and litres. Students also investigate making changes to quantities and ratios.

1. Ask the students how many millilitres of concentrate you would need to make a 500mL drink using a ratio of 1:4?
Ask the students what fraction of the drink is concentrate? (1/5)
Ask the students what is 1/5 of 500mL? (100ml)
2. Give the students problems to solve that involve using the ratios to find the proportions in millilitres and litres. Here are some examples of problem structures you can use to develop problems to ask students:
• If the ratio is 2:3 and you are using 60mL of concentrate how much water should you add?
• If there is 20% concentrate in a 1.25L drink how many millilitres of concentrate is that?
• If a drink is made to a ratio of 1:4 how many millilitres of concentrate would you expect in a 250mL glass of drink?
3. Show the students a 300mL of drink made using a ratio of 1:4. Ask the students how many millilitres of concentrate are in the bottle? It may be useful to draw a diagram of this

4. Ask the students what they would do to make the flavour taste twice as strong? Explain to the students that to double the flavour the number of concentrate parts needs to double to 2 parts giving a ratio of 2:3. Students should see that the total number of parts remains the same.
5. Give the students problems to solve that involve changing the ratios Here are some examples of problem structures you can use to develop problems to ask students:
• If one litre of juice was made using the ratio 2:3, how much concentrate is needed to make a 1 litre of drink that is half the strength? (Answer= ratio 1:4, 200mL concentrate)
• The label says to use 125mL of concentrate to make 500mL of juice, what would the ratio be if you double the amount of concentrate? (Answer= 2:2 or 1:1)
• The label says to use a ratio of 1:3, there was only 50mL of concentrate left in the bottle if you made a 250mL drink will it taste stronger or weaker than the label recommended? (Answer = weaker)

#### Session 4

In this session students solve problems that involve combining ratios. The important concept for students to be aware of is that when two ratios represent different quantities the combined ratio needs to worked out from the combined quantities.

1. Explain the following problem to students and work with them to solve it. Two classes were surveyed about which drink they prefer, juice or flavoured milk.
1. One class of 30 students voted 10 for juice and 20 for flavoured milk. Use coloured counters to represent the votes (orange for juice and yellow for flavoured milk). Ask the students how this could be expressed as a ratio. (1:2)
2. The second class of 25 students voted 10 for juice and 15 for flavoured milk. Use coloured counters to represent the votes (orange for juice and yellow for flavoured milk). Ask the students how this could be expressed as a ratio. (2:3)
3. Combine the votes for juice (20) by moving the orange counters together and combine the votes for flavoured milk (35) by moving the yellow counters together. Ask the students what is the ratio of juice to flavoured milk preference? (Answer = 20:35 or 4:7). Group the counters to show the simplified ratio.
4. Explain to the students that because the two classes were of different sizes the ratios of 1:2 and 2:3 can not be added together to give the correct ratio.
2. Give the students problems to solve that involve combining ratios. Here are some examples of problem structures you can use to develop problems to ask students:
• John and Mike bought bottles of fizz and juice. John bought 5 bottles, 2 fizz and 3 juice. Mike bought 10 bottles in the ratio of 3:2. If John and Mike combined their bottles together what is the new ratio? (Answer = 8:7)
• Lucy made fruit punch using ginger ale and juice in the ratio of 3:2. Amy made her fruit punch in the ratio of 1:4 but she made twice as much. When the girls poured their punch into punch bowl what is the ratio of ginger ale to juice? (Answer = 1:2)
• For camping trip one family brought 8 bottles of soft drink, 1/4 of the bottles were lemonade and the rest were cola. The second family brought 7 bottles, 3 of them were lemonade and 4 were cola. When they combined what is the ratio of lemonade to cola? (Answer = 1:2)

#### Session 5

In this session students solve problems that involve rates. A rate describes a relationship between two different measures.

1. Set a problem and work with the students to solve it:
What would be a better buy, 3 litres of juice for \$4 or 4 litres of juice for \$5? To solve this problem students will need to find the cost per litre and compare them.
Ask the students what the cost per litre of the first bottle is? \$4 ÷ 3 = \$1.33 per litre
Ask the students what the cost per litre of the second bottle is? \$5 ÷ 4 = \$1.25 per litre
Ask the students to compare the cost per litre to find the cheapest. (Second bottle is cheaper)
2. Show the students a 2 litre bottle of fizz and a 3 litre bottle of juice. Tell the students the supermarket has this 2 litre fizz for \$2 and this 3 litre juice for \$4. Work through the following problem with the students.
If you decided to mix fizz and juice in a ratio of 2:3 what does it cost to make 5 litres of the mix? The ratio of the drink mix is exactly one bottle of each fizz and juice, so the cost is \$2 + \$4 = \$6. What does 250mL glass cost to make? For this question students need to work out what proportion 250mL is of 5 litres, it is easier to firstly find the cost of 1 litre. 1 litre is 6/5 = \$1.20 so 250mL is 30c.
What is the cost of a 250mL glass of fizz? What is the cost of a 250mL glass of juice?
How can you adjust the ratio to decrease the cost of making the drink?
3. Set a problem and work with the students to solve it:
If you decided to make the drink mix in a ratio of 1:1 what would it cost to make 6 litres of the drink? Discuss with the students that 6 litres in a 1:1 ratio is 3 litres of fizz and 3 litres of juice. 3 litres of fizz is \$3 and 3 litres of orange is \$4, so the mix costs \$7.
4. Give the students more problems to solve that involve rates. Here are some examples of problem structures you can use to develop problems to ask students:
• The 4 litre bottle of drink costs \$5, what is the cost per litre?
• What is a better buy 3 litres for \$4 or 4 litres for \$4.80?
• The drink costs \$4.20 for 1.5 litres, what is the cost for a 250mL glass
• The drink costs \$3 for 375mL what is the cost per litre?
• If 2 litres of fizz costs \$2 and 3 litres of juice costs \$5, what is the cost of making 6 litres using a 1:1 ratio of fizz to juice?