B2.12 solve problems involving ratios, including percents and rates, using appropriate tools and strategies
Skill: Solving Problems Involving Ratios, Including Percents and Rates, Using Appropriate Tools and Strategies
Multiplicative reasoning is a concept that requires the ability to deal with several ideas or quantities at once. The
idea is to see problems in relative rather than absolute values. Consider the following problem, "If one
dog's weight goes from 5 kg to 8 kg and another dog's weight goes from 3 kg to 6 kg, which dog gained more
weight?" If the student is approaching the problem from the perspective of absolute values or addition, they may
be inclined to answer that the 2 dogs have gained equal weight. However, based on relative values, the student may
argue that the second dog gained more weight since it doubled its starting weight as opposed to the1stdog, which would
have had to reach 10 kg for its relative weight gain to be equivalent. The following table visually illustrates the 2
answers to this problem. If it is true that the 2 answers can be defended, it is on the relative value (multiplicative
reasoning) that one must base oneself to apply a proportional reasoning.
Why is this important?
It is difficult to move students from additive to multiplicative reasoning, which is why it is important to start at
a young age. This is the basis of the Ontario mathematics curriculum for Grades 1 to 8, which includes important
interrelated ideas such as multiplication, division, fractions, decimals, ratios, percentages and linear functions. It
takes time, a variety of learning situations, and a variety of opportunities to build knowledge in different
ways.
Source: What is proportional reasoning, p. 5-6.
A ratio is a comparison between two quantities of the same unit.
A rate, like a ratio, is also a comparison between two quantities, but of different units.
Ratios are present in everyday life and in many mathematical situations, including in place values (for example, the
ratio of units to thousandths is 1 000 : 1), in fractions (such as \(\frac{2}{3}\) or \(2 \div 3\)), in similar figures (for
example an enlargement of \(1 \div 3\)), in decimal numbers (for example the ratio of hundredths to tenths is 10 : 1), in
metric units of measure (for example the ratio of metres to millimeters is 1 : 1 000).
Source: A Guide to Effective Instruction in Mathematics, Grades 4-6, p. 54.
Rates
Example 1
I buy 2 bags of rice for $10.
I can represent this rate in different ways:
- Using words: 2 bags of rice cost $10.
- By division: 2 bags of rice / $10.
- As a fraction: \(\frac{{2\;{\rm{bag\; of\; rice}}}}{10\;\$} \).
Using the multiplicative relationship, I can determine the rate of other quantities of rice, namely equivalent
rates.
For 4 bags of rice, the cost will be $20. 4 bags is twice as much as 2, so the price is twice as much as $10.
For 10 bags of rice, the cost will be $50. 10 bags is 5 times more than 2, so the price is 5 times more than $10.
I can also determine the unit rate, the cost per 1 bag of rice, using division.
For 1 bag of rice, the cost will be $5. 1 bag is 2 times less than 2, so the price is 2 times less than $10. So the
unit rate is $5/1 bag of rice.
Equivalent rates can be represented in a table of values with emphasis on the multiplicative, not additive, relationship.
Rates are part of our daily lives. Students can be led to find different examples related to measurements (for example, I run 12 km per hour, the electric bike goes 35 km per hour, etc.) and costs (for example, $3 for 12 apples, $15/hour, etc.).
Students can solve problems representing multiplicative relationships involving rates using familiar contexts.
Example 2
If you get $24 for 2 hours of work, how much money do you get for 4 hours? The student should therefore observe that
4 hours is double or twice as much as 2 hours so the amount of money will also double, so $48. If we divide by 2, we
can find the unit rate, which is the amount of money received for 1 hour of work ($12/hour).
These relationships can be represented in a table of values with emphasis on the multiplicative, not additive,
relationship.
In this table of values, we can also recognize that there is a proportional relationship between the number of hours
worked and the amount of money received, that is, × 12. We multiply the number of hours worked by 12 to get the amount
of money received.
A ratio table can also be constructed without listing the values in ascending order or without listing all the values. It is sometimes easier to find the solution to the problem by using the proportionality relationship as shown in the following example.
Example 3
Abdala buys cold cuts to make sandwiches for the school picnic. Each kilogram of meat costs $12 and will make 10
sandwiches. How much will the meat cost to make 25 sandwiches?
Here are two different ways to use the proportionality relationship in a ratio table to solve this problem.
This use of the ratio table in a proportionality situation does not necessarily have to be taught since
students, in problem-solving situations, can note it on their own.
The double number line also highlights relationships that can be used to solve a problem. It can, for example, be
used instead of a ratio table to solve the previous problem.
Students then locate ratios on the right that are equivalent to the one given in order to solve the problem. Students can choose ratios based on their needs and understanding of the problem. Here are two different ways to solve the problem using a double number line.
The difference between using a ratio table and the double number line to represent a proportionality situation is the
order in which the values that represent ratios are placed. On a double number line, the numbers are placed in
ascending order at constant intervals, whereas in the ratio table, they are placed in the order that corresponds to
the reasoning used to solve the situation.
Source: A Guide to Effective Instruction Grades 4-6Mathematics, p 52-53.
Percentage
In Grade 6, since students are exposed to the concept of ratio, they learn that a percentage represents a ratio of
100 (for example, 30% represents the ratio 30:100).
Source: A Guide to
Effective Instruction in Mathematics, Grades 4 to 6, p. 35.
Any ratio can be expressed as a percentage and once a result is expressed as a percentage, it can be rewritten as a
decimal fraction and then represented by an equivalent fraction.
Source: Curriculum de l’Ontario, Programme-cadre de mathématiques de la 1re à la 8e année, 2020, Ministère
de l’Éducation de l’Ontario.
It is important to note that a result expressed as a percentage does not mean that the quantity in question is
necessarily composed of 100 parts, as explained in the following table.
| Representation | Percentage | Instructional Notes |
|---|---|---|
  |
The ratio of green circles to the total number of circles is 3:4 or \(\frac{3}{4}\) and this can be interpreted as "75% of the circles are green." » | Even if 75% of the circles are green, it does not mean that there are 100 circles in the whole. However, if there were 100 circles, there would be 75 green circles. Also, the fraction of circles that are green is equivalent to \(\frac{{75}}{{100}}\)(for example, \(\frac{3}{4}\; = \;\frac{75}{{100}}\) and \(\frac{{150}}{{200}}\; = \;\frac{75}{100}\). |
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 35.
In the junior grades, proportions are taught formally. Initially, the focus is on building students' knowledge and understanding of equivalent fractions. Students who have mastered the concept of equivalent fractions can use multiplicative relationships to solve problems involving proportions when the answer is a whole number (e.g., "2 = 40").
Proportional relationships can be used to solve a variety of everyday problems using simple reasoning that is accessible to junior students. For example, if 30% of the 500 students in a school like couscous, it is possible to determine in various ways that 150 students like couscous. Here are some examples.
Example 1
Use a semi-concrete representation.
Example 2:
Build a report table.
Example 3
Determine equivalent fractions.
\(\frac{{30}}{{100}} = \frac{60}{200} = \frac{90}{300} = \frac{120}{400} = \frac{150}{500}\)
Example 4
Establish a proportion by multiplication.
\(\begin{array}{l}\frac{{30}}{{100}}\; = \;\frac{?}{{500}}\\\frac{{30\; \times \;5}}{{100\; \times \;5}}\; =
\;\frac{{150}}{{500}}\end{array}\)
Source: A Guide to
Effective Instruction in Mathematics, Grades 4-6, p 55-56.
Reports
There is a proportional relationship between two quantities when these quantities can increase or decrease simultaneously by the same factor.
Example 1
To enable students to complete an activity, teachers distribute straws as follows:
- the student who works alone receives 4 straws;
- a team of 2 students receives 8 straws;
- a team of 3 students receives 12 straws.
A study of the pattern in this relationship allows us to recognize that the number of students and the number of straws increase by the same factor (factor of 2 for 2 students, factor of 3 for 3 students). It then becomes easy to determine that a group of 6 students will receive 24 straws (factor of 6). The proportional relationship between the number of students and the number of straws can be represented by the equality between 2 of the ratios (for example, "="). Such an equality between 2 ratios is called a proportion.
The following situation, on the other hand, does not have a proportional relationship.
Example 2
To enable students to complete an activity, teachers distribute straws as follows:
- 1 student working alone receives 5 straws;
- a team of 2 students receives 9 straws;
- a team of 3 students receives 13 straws.
In this situation, it is impossible to establish an equality between two ratios (for example, \(\frac{1}{5}\; \ne \;\frac{3}{{13}}\)). The analysis of proportionality relations is carried out by applying proportional reasoning. This reasoning occurs when comparing two relationships with each other and recognizing a multiplicative relationship. Note that multiplicative relations include the operation of division, since any division can be transformed into multiplication (for example, dividing by 2 is the equivalent of multiplying by \(\frac{1}{2}\)).
The ability to use proportional reasoning develops throughout the learning of mathematics. For example, teachers ask primary students to determine the number of pieces in 3 chocolate bars if a bar contains 8 pieces. This is a multiplicative relationship since the number of pieces is 8 times greater than the number of bars (ratio of 8 to 1). However, to solve this kind of problem, students will first use repeated addition \(\left( {8\; + \;8\; + \;8} \right)\). Later, when they have been exposed to the concept of multiplication, they can solve it by multiplying \(\left( {8\; \times \;3} \right)\), which is a first step towards using proportional reasoning.
The terms ratio and proportion, and their related notations (for example, 2:3), are part of the junior mathematics curriculum. In addition, they develop strategies for solving a problem involving proportionality algebraically. Beginning in Grade 4, the study of proportional relationships focuses on the recognition and description of the multiplicative relationship in a variety of problem-solving situations. Students intuitively use proportional reasoning to solve problems involving 2 quantities that are in a ratio of 1 to many (for example, 1 bar for 8 pieces), many to 1 (for example, 3 people per table) or many to many (for example, 2 litres of juice for 5 people). They also use concrete materials and various models such as illustrations, tables of values or number lines.
Example 3
For track and field day, students in Ms. Guerin's class prepare juice for the runners. For each container of juice concentrate, they need to add 5 containers of water. How many containers of water will they need to add to 4 containers of juice concentrate?
Solution using illustrations
This will require 12 containers of water (\(4\; \times \;3\) containers of water).
Solution using a table of values
In this table of values, which represents a situation of proportionality, the ratios between the corresponding quantities are equivalent. In the previous example, the multiplicative relationship by 3 (the proportionality coefficient) between the number of containers of juice and the number of containers of water is easily recognized. In addition, this table of values allows proportions to be established (for example, \(\frac{1}{3} = \frac{4}{12}\)).
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p 49-51.
Knowledge: Ratios
Ratios are used to compare one part to a whole, two parts to each other, or two quantities to each other. A ratio is written symbolically with a colon.
Example
The ratio of blue marbles to red marbles is 10:15 which is called "10 to 15".
A ratio relationship can also be described using fractions, decimal numbers and percentages.
Source : The Ontario Curriculum, Mathematics, Grades 1-8, Ontario Ministry of Education, 2020.
Knowledge: Percent
A percentage is a special way of presenting a fraction. It is often used in everyday life. A numeric expression like 30% (which reads "thirty percent") is actually another notation for the number thirty hundredths, either \(\frac{{30}}{{100}}\) or 0.30. To facilitate understanding of the concept of percent, students must first be introduced to the relationship between the percent and the fraction with a denominator of 100, using concrete or semi-concrete materials. The percentage also represents 1: 100.
Example:
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 34.
Knowledge: Rate
A rate describes the multiplicative relationship between two quantities expressed with different units (eg, bananas per dollar; cereal bars per child; kilometres per hour).
A rate can be expressed in words such as 150 kilometres per 3 hours.
A rate can be expressed as a division such as 50 km/h.
There are many applications of rates in everyday life.
Note
Like ratios, rates make comparisons based on multiplication and division; however, rates compare two related, but different, measures or quantities. For example, if 12 cookies were eaten by 4 people, the rate would be 12 cookies per 4 people. An equivalent rate would be 6 cookies per 2 people. A unit rate would be 3 cookies per person.
Source : The Ontario Curriculum, Mathematics, Grades 1-8, Ontario Ministry of Education, 2020..
Knowledge: Unit Rate
Rate where the second term of the ratio is 1 (for example, cost of $0.35/mg).
Source: En avant, les maths! Grade 6, CM, Nombres, p. 2.