Ratios & Proportional Relationships
Ratios and Rates
Grade 6
- ✓By the end of this lesson students will be able to understand the concept of a ratio and use ratio language to describe a relationship between two quantities.
- ✓By the end of this lesson students will be able to understand the concept of a unit rate associated with a ratio and use rate language in the context of a ratio relationship.
- ✓By the end of this lesson students will be able to solve problems involving finding the whole, given a part and the percent.
- ✓By the end of this lesson students will be able to use ratio tables to solve real-world and mathematical problems.
Key concepts
A ratio is a comparison of two quantities using division. It can be written in three ways: 'a to b', 'a:b', or as a fraction 'a/b'. The order of the quantities in a ratio is important.
A rate is a special type of ratio that compares two quantities with different units. For example, 'miles per hour' or 'dollars per pound'.
A unit rate is a rate where the second quantity (the denominator when written as a fraction) is 1 unit. It tells you 'how much per one unit' of the second quantity. To find a unit rate, divide the first quantity by the second quantity.
A percent is a ratio that compares a number to 100. The word 'percent' means 'per hundred' or 'out of 100'. It is often represented by the symbol '%'.
A ratio table is a table used to organize and display equivalent ratios. Equivalent ratios are ratios that have the same value. You can find equivalent ratios by multiplying or dividing both quantities in a ratio by the same non-zero number.
Key facts to remember
- 1A ratio compares two quantities.
- 2A rate compares two quantities with different units.
- 3A unit rate has a denominator of 1, meaning it tells you 'how much per one unit'.
- 4Percent means 'out of 100'.
- 5Equivalent ratios can be found by multiplying or dividing both quantities in a ratio by the same non-zero number.
- 6Ratio tables are useful tools for organizing and finding equivalent ratios.
Worked examples
Example 1
A car travels 280 miles in 4 hours. What is its average speed in miles per hour?
Answer
The car's average speed is 70 miles per hour.
The unit rate tells you how many miles are traveled in 1 hour.
Example 2
In a class of 30 students, 18 are boys. What percent of the students are boys?
Answer
60% of the students are boys.
Remember that 'percent' means 'out of 100'.
Example 3
A baker uses 3 cups of sugar for every 5 cups of flour. Complete the ratio table to find out how many cups of sugar are needed for 20 cups of flour.
Answer
12 cups of sugar are needed for 20 cups of flour.
When using a ratio table, ensure you apply the same multiplication or division factor to both quantities to maintain an equivalent ratio.
Common mistakes
- ✗Confusing the order of quantities in a ratio (e.g., comparing apples to oranges instead of oranges to apples as requested).
- ✗Incorrectly setting up the division for unit rates (e.g., dividing the unit by the quantity instead of the quantity by the unit).
- ✗Forgetting to multiply by 100 when converting a fraction or decimal to a percent.
- ✗Not applying the same operation (multiplication or division) to both quantities in a ratio table when finding equivalent ratios.
- ✗Misinterpreting 'percent of' problems, such as finding the part instead of the whole, or vice-versa.
Exam tips
- ★Always label your units, especially when working with rates and unit rates, to ensure your answer makes sense.
- ★Read the problem carefully to identify what quantities are being compared and in what order the ratio should be written.
- ★When working with percents, remember that 'percent' literally means 'per 100', which can help you set up proportions correctly.
- ★Use ratio tables to visualize and organize equivalent ratios; this can make it easier to spot patterns and solve problems.
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