Number & Algebra
Rational Numbers, Ratio, and Rate
Grade 7 · Grade 8 · Grade 9
- ✓By the end of this lesson students will be able to define rational numbers and identify different forms.
- ✓By the end of this lesson students will be able to perform all four basic operations (addition, subtraction, multiplication, division) with rational numbers.
- ✓By the end of this lesson students will be able to represent ratios in various forms and simplify them.
- ✓By the end of this lesson students will be able to calculate unit rates and solve problems involving rates.
- ✓By the end of this lesson students will be able to solve problems using proportional reasoning.
Key concepts
A rational number is any number that can be expressed as a fraction a/b, where 'a' and 'b' are integers and 'b' is not zero. This includes all integers (e.g., -3 = -3/1), fractions (e.g., 1/2, -3/4), terminating decimals (e.g., 0.5 = 1/2), and repeating decimals (e.g., 0.333... = 1/3).
To add or subtract rational numbers expressed as fractions, you must first find a common denominator. Once the denominators are the same, add or subtract the numerators and keep the common denominator. For decimals, align the decimal points and add or subtract as with whole numbers.
To multiply rational numbers expressed as fractions, multiply the numerators together and multiply the denominators together. Simplify the resulting fraction to its lowest terms if possible. For decimals, multiply as with whole numbers and then place the decimal point based on the total number of decimal places in the factors.
To divide rational numbers expressed as fractions, multiply the first fraction by the reciprocal of the second fraction (the divisor). The reciprocal of a fraction is found by flipping the numerator and the denominator. For decimals, you can convert to fractions or move the decimal point in both the divisor and dividend until the divisor is a whole number.
A ratio is a comparison of two or more quantities that have the same units. Ratios can be written in several ways: using a colon (a:b), using the word 'to' (a to b), or as a fraction (a/b). Ratios should always be simplified to their lowest terms.
A rate is a comparison of two quantities that have different units. A unit rate is a rate where the second quantity is one unit. For example, kilometres per hour (km/h) or dollars per kilogram ($/kg) are common unit rates.
A proportion is an equation that states that two ratios or two rates are equal. Proportions are often used to find an unknown quantity when you know that two ratios are equivalent. The cross-multiplication method is a common way to solve proportions.
Proportional reasoning is the ability to understand and solve problems involving ratios, rates, and proportions. It involves scaling quantities up or down while maintaining the same relationship between them. This skill is essential for solving real-world problems in various contexts.
Key facts to remember
- 1Rational numbers can be written as a fraction a/b, where 'a' and 'b' are integers and b ≠ 0.
- 2To add or subtract fractions, you must find a common denominator.
- 3To multiply fractions, multiply the numerators and multiply the denominators.
- 4To divide fractions, multiply the first fraction by the reciprocal of the second fraction.
- 5Ratios compare quantities with the same units; rates compare quantities with different units.
- 6A proportion is an equation stating that two ratios or rates are equal.
- 7Cross-multiplication (ad = bc) is a reliable method for solving proportions.
Worked examples
Example 1
Evaluate: (1/3 + 1/6) ÷ 3/4
Answer
2/3
Always perform operations in parentheses first, and remember to simplify fractions at each appropriate step.
Example 2
A baker uses 3 cups of sugar for every 5 cups of flour. a) Write the ratio of sugar to flour in simplest form. b) If the baker uses 15 cups of flour, how much sugar is needed?
Answer
a) 3:5 b) 9 cups of sugar
Ensure the order of quantities in your ratio and proportion matches the problem statement.
Example 3
A cyclist travels 120 km in 3 hours. a) What is the cyclist's average speed (unit rate) in km/h? b) How far will the cyclist travel in 5 hours at the same speed?
Answer
a) 40 km/h b) 200 km
Unit rates simplify comparisons and calculations for different time periods or quantities.
Common mistakes
- ✗Forgetting to find a common denominator when adding or subtracting fractions.
- ✗Dividing fractions by simply dividing the numerators and denominators, instead of multiplying by the reciprocal.
- ✗Not simplifying ratios or fractions to their lowest terms.
- ✗Mixing up the order of quantities when setting up ratios or proportions (e.g., flour to sugar vs. sugar to flour).
- ✗Incorrectly applying cross-multiplication or making calculation errors during the process.
Exam tips
- ★Always show all your steps clearly, especially when working with fractions and solving proportions, to earn full marks.
- ★Check if your answer makes sense in the context of the problem; for example, a distance or quantity should usually be positive.
- ★Pay close attention to the units involved in rate problems and ensure your final answer includes the correct units.
- ★Simplify fractions and ratios as you go to keep numbers manageable and reduce the chance of errors.
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