Ratio in the Real World
Task Description
The students explore which ratio produces a stronger cranberry taste between a 4:3 or
3:2 mix of cranberry juice to apple juice. The students use various strategies to answer
this question and apply these to a second task that asks, ‘Which rectangle is ‘more square’
a rectangle of 35 x 39 or one of 22 x 25?’
Length of Task
60 - 80 minutes
Materials

Calculator
Using the Activity
Introductory
The teacher poses the following problem to the class.
Eva Brick makes and sells her own cranberry-apple juice. In jug A, she mixed 4
cranberry flavoured cubes and 3 apple flavoured cubes with some water. In
jug B, she mixed 3 cranberry and 2 apple flavoured cubes in the same amount
of water.
If you ask for a drink that has the stronger cranberry taste, from which jug
should she pour your drink? Please explain.
The teacher encourages the students to work on the problem individually or talk it through
with a partner. The teacher shares that there multiple possible approaches for answering
this question.
Whole class: The teacher draws the students together to share their approaches and
responses to the question. Some students may have commenced with changing the ratios
into equivalent fractions 4/3 and 3/2 and then calculated the equivalent decimal 1.33333
and 1.5 respectively. The larger response indicates the greater level of cranberry juice to
apple juice. Therefore the ratio of 3:2 is stronger.
Another possible approach is converting the ratios to fractions and finding the common
denominator for the fractions. So 4/3 and 3/2 would become 8/6 and 9/6. Therefore with
6 equal parts of apple juice in each mixture there would be 8 parts and 9 parts of
cranberry respectively, making the second mixture stronger.
Main Activity
This activity is intended for use by teachers for research purposes only, as part of the Task Types and Mathematics Learning (TTML) project at
Monash University. No authority is granted for persons to use these activities beyond the scope of this project, without express permission of the
TTML Project Leader.
Ratio in the Real World
Building on the strategies used in the first problem the teacher poses the following
problem to the students.
Which rectangle is ‘more square’ a rectangle of 35 x 39 or one of 22 x 25?
The students are seeking a rectangle that is closest to a 1:1 ratio.
A common response to this task is for students to convert these figures into fractions
35/39 and 22/25 and then calculate the fractions into equivalent decimals 0.89 and .088.
The decimal closer to 1 is the ‘more square’ rectangle, 35 x 39 is the correct response.
Alternatively, the students may reverse the fractions to 39/35 and 25/22 then convert to
decimals 1.11 and 1.13. Again the number closest to 1 is correct.
Whole class: The teacher encourages students to share any difficulties they have
encountered with the task and their methods for overcoming them. A discussion of
incorrect responses or misconceptions will assist in developing the students’ understanding
of ratio. The teacher invites the students to share any similarities or differences they found
with the two problems.
Key Mathematical Concepts

Comparison of ratios through conversion to fractions and decimals.
Prerequisite Knowledge

Understanding of the relationship between ratios, fractions, decimals and percentages.
Links to VELS
Dimension
Number (Level 4)
Working mathematically
(Level 4)
Working mathematically
(Level 4)
Standard
Students use decimals, ratios and percentages to find
equivalent representations of common fractions (for example,
3/4 = 9/12 = 0.75 = 75% =3 : 4 = 6 : 8).
Students use the mathematical structure of problems to
choose strategies for solutions. They explain their reasoning
and procedures and interpret solutions. They create new
problems based on familiar problem structures.
Students engage in investigations involving mathematical
modelling. They use calculators and computers to investigate
and implement algorithms, explore number facts and puzzles,
generate simulations, and transform shapes and solids.
This activity is intended for use by teachers for research purposes only, as part of the Task Types and Mathematics Learning (TTML) project at
Monash University. No authority is granted for persons to use these activities beyond the scope of this project, without express permission of the
TTML Project Leader.
Ratio in the Real World
Assessment
To



be working at Level 4, students should be able to:
Develop an appropriate strategy for comparing ratios.
Use ratio to compare the relationship between quantities.
Use ratios to find equivalent representations of common fractions.
Extension Suggestions
For students who would benefit from additional challenges:
 The IXL website has a range of comparison word problems that some students may
wish to explore. Students select the multiple choice responses and the site provides
feedback if the answer is incorrect. http://www.ixl.com/math/practice/grade-6compare-percents-and-fractions-word-problems Word problems from this site may be
extended for whole class problems.
Teacher Advice and Feedback
Some of the students attempted to draw both rectangles to make the comparison. It was
noted that to fit the rectangles on a piece of paper students were reducing the ratio terms
by the same amount and therefore not keeping these numbers in proportion to the
original ratio. For example some students reduced 35:39 by 3 for each term to 32:36. This
method produces rectangles of different ratios. The teacher may illustrate the impact of
reducing terms in this manner by referring the students to the ratio table in the ‘Making
cordial’ task and asking what the effect this would have on the cordial and ratio.
Many students found the second task challenging and did not transfer their knowledge of
the strategies used in the first task to the second task without prompting from the
teacher.
Potential Student Difficulties
In the case of the cranberry-apple juice task some students developed an appropriate
strategy for approaching the task however had difficulties in interpreting their results.
They were unsure whether a larger decimal or one closer to 1 was the best result.
Students may benefit from creating simplified ratios of cranberry to apple juice that they
know in advance gives the higher concentrate of cranberry. For example without
converting the ratios 5:1 compared to 2:1 to fractions or decimals most students would
understand that a 5:1 ratio gives a stronger cranberry taste. The students can reproduce
the steps to the task again using these new simple ratios to assist in determining the
correct response.
Students who are experiencing difficulties may be given alternative ratios to compare with
two equal terms. For example the problem below offers equal parts of apple juice in each
jug (3:1 and 2:1).
This activity is intended for use by teachers for research purposes only, as part of the Task Types and Mathematics Learning (TTML) project at
Monash University. No authority is granted for persons to use these activities beyond the scope of this project, without express permission of the
TTML Project Leader.
Ratio in the Real World
Eva Brick makes and sells her own cranberry-apple juice. In jug A, she mixed 3
cranberry flavoured cubes and 1 apple flavoured cubes with some water. In
jug B, she used 2 cranberry and 1 apple flavoured cubes in the same amount
of water.
If you ask for a drink that has the stronger cranberry taste, from which jug
should she pour your drink? Please explain.
This slightly more difficult question offers equal parts of cranberry juice to unequal parts
of apple juice (5:3 and 5:4).
Eva Brick makes and sells her own cranberry-apple juice. In jug A, she mixed 5
cranberry flavoured cubes and 3 apple flavoured cubes with some water. In
jug B, she used 5 cranberry and 4 apple flavoured cubes in the same amount
of water.
If you ask for a drink that has the stronger cranberry taste, from which jug
should she pour your drink? Please explain.
References / Acknowledgements
Thank you to the teachers and students from Lloyd Street PS, for providing valuable
feedback on the use of this activity.
This activity is intended for use by teachers for research purposes only, as part of the Task Types and Mathematics Learning (TTML) project at
Monash University. No authority is granted for persons to use these activities beyond the scope of this project, without express permission of the
TTML Project Leader.
Ratio in the Real World
Student work samples
Example 1: Working at Level 3-4
When comparing the proportions of two ratios, this student appears to hold the
misconception that if the difference between the two terms of each ratio is the same then
the ratio is the same proportion. For example the difference between the ratio of 4:3 and
3:2 is one part in each. 4 parts subtract 3 parts is 1 part. 3 parts subtract 2 parts is also 1
part. However, this student was able to represent both ratios as an equivalent decimal
correctly. The student makes a real-life observation that although you may want to drink
the juice from the 3:2 jug you probably won’t be able to taste the difference between the
mixtures.
This activity is intended for use by teachers for research purposes only, as part of the Task Types and Mathematics Learning (TTML) project at
Monash University. No authority is granted for persons to use these activities beyond the scope of this project, without express permission of the
TTML Project Leader.
Ratio in the Real World
This activity is intended for use by teachers for research purposes only, as part of the Task Types and Mathematics Learning (TTML) project at
Monash University. No authority is granted for persons to use these activities beyond the scope of this project, without express permission of the
TTML Project Leader.
Ratio in the Real World
Example 2: Working at Level 4
This student converted the ratios to fractions and calculated the result as a decimal. The
student accurately determined that the number closest to 1 is closest to the 1:1 ratio of a
square.
This activity is intended for use by teachers for research purposes only, as part of the Task Types and Mathematics Learning (TTML) project at
Monash University. No authority is granted for persons to use these activities beyond the scope of this project, without express permission of the
TTML Project Leader.