Grade 5 → Ratios and Proportions ↓
Equivalent Ratios
Equivalent ratios are a fundamental concept in mathematics that show the relationship between two quantities. Ratios can describe anything from recipes to maps, comparing two different quantities to show how they relate to each other. When two ratios are equivalent, they show the same relationship between the numbers, meaning they have the same value when simplified.
What is the ratio?
Ratios compare two numbers or quantities and are often used to show how much of one value contains or is included in another. Ratios can be written in several forms: as a fraction, with a colon, or using the word "to." For example, if you have 2 apples and 3 oranges, the ratio of apples to oranges can be written like this:
2:3Or
2/3Or
"2 to 3."
Understanding equivalent ratios
Equivalent ratios are different comparisons between numbers that express the same relationship. For example, if you have two ratios such as:
2/3And
4/6These two ratios are equivalent. This means that the relationship between the first numbers in both ratios is the same.
Visual example
To visually demonstrate equivalent ratios, consider a rectangle divided into parts. Imagine that the rectangle has a total of 6 parts - 2 parts are blue, and the remaining 4 parts are red. The ratio of the blue and red parts can be expressed as:
2:4This is how it looks:
When we multiply both sides of a ratio by the same number, we get an equivalent ratio. If we need to double each side of the ratio, we do this:
2 * 2 = 44 * 2 = 8This will give us:
4:8In both cases, if we divide the parts into smaller parts, it still shows the same proportional relationship between the blue and red parts.
How to find equivalent ratios
Finding equivalent ratios is simple. Multiply or divide both sides of the original ratio by the same number. Be sure not to get fractions when dividing, unless you are specifically working with them.
Example with multiplication
You start with the ratio:
3:6Multiply both numbers by 2:
3 * 2 = 66 * 2 = 12Hence, another equivalent ratio is:
6:12Example with division
Start with the proportions:
6:9Divide both numbers by 3:
6 ÷ 3 = 29 ÷ 3 = 3you received:
2:3Real life examples using equivalent ratios
Equivalent ratios are widely used in many fields. They are especially beneficial when adjusting portions for recipes, creating accurate models using a scale, and solving real-world proportional problems.
Cooking example
If a pasta recipe calls for 2 cups of sauce for every 3 cups of pasta (ratio 2:3) and you want to make twice as much sauce, you would use:
2 * 2 = 43 * 2 = 6So, you will need 4 cups of sauce and 6 cups of pasta, keeping the ratio at 2:3.
Conclusion
Understanding equivalent ratios is vitally helpful in understanding proportional relationships in everything from simple mathematical problems to more complex real-world scenarios. Being able to identify and create equivalent ratios allows us to scale quantities up or down, ensuring that we maintain the same relationship in different scenarios.
Practice problems
- Find the equivalent ratio for
By multiplying both the numbers by 3.5:10 - Simplify the ratio
in its smallest equivalent ratio.9:12 - If the ratio of cats and dogs is
, find a similar ratio with 21 cats.7:4 - A car travels 60 miles in 2 hours. What is the equivalent ratio of miles and hours?
Answer
And5 * 3 = 15
, so the equivalent ratio is10 * 3 = 30
,15:30
And9 ÷ 3 = 3
, so the simplified ratio is12 ÷ 3 = 4
,3:4- If you multiply 7 by 3, you get 21. So, multiply both parts of the ratio:
And7 * 3 = 21
, so the equivalent ratio is4 * 3 = 12
,21:12 - the ratio is
In simplified form, it is60:2
,30:1
With practice and understanding, equivalent ratios become a powerful tool for solving many mathematical and real-world problems, reinforcing fundamental math skills in measurement, proportion, and scaling.
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