Write three ratios that are equivalent to $14: 21$.
Answer (2)
Simplify the given ratio 14 : 21 by dividing both sides by their GCD, which is 7, resulting in 2 : 3 .
Multiply the simplified ratio 2 : 3 by 2 to get the equivalent ratio 4 : 6 .
Multiply the simplified ratio 2 : 3 by 3 to get the equivalent ratio 6 : 9 .
Multiply the simplified ratio 2 : 3 by 4 to get the equivalent ratio 8 : 12 . The three equivalent ratios are 4 : 6 , 6 : 9 , 8 : 12 .
Explanation
Understanding the Problem We are given the ratio 14 : 21 and we need to find three equivalent ratios. To do this, we can first simplify the given ratio by dividing both sides by their greatest common divisor (GCD).
Simplifying the Ratio The greatest common divisor (GCD) of 14 and 21 is 7. Dividing both sides of the ratio by 7, we get: 7 14 : 7 21 = 2 : 3
Finding Equivalent Ratios Now, we can find equivalent ratios by multiplying both sides of the simplified ratio 2 : 3 by any three different integers greater than 1. Let's multiply by 2, 3, and 4:
Multiplying by 2: 2 × 2 : 2 × 3 = 4 : 6
Multiplying by 3: 3 × 2 : 3 × 3 = 6 : 9
Multiplying by 4: 4 × 2 : 4 × 3 = 8 : 12
Final Answer Therefore, three ratios equivalent to 14 : 21 are 4 : 6 , 6 : 9 , and 8 : 12 .
Examples
Ratios are used in everyday life, for example, when cooking. If a recipe calls for a ratio of 1 cup of flour to 2 cups of water, we can express this as 1 : 2 . If we want to make a larger batch of the recipe, we need to maintain this ratio. For instance, to double the recipe, we would use 2 cups of flour and 4 cups of water, maintaining the 1 : 2 ratio as 2 : 4 , which is equivalent. Understanding ratios helps us scale recipes, mix ingredients correctly, and ensure consistent results in cooking and baking.
Three ratios equivalent to 14 : 21 are 4 : 6 , 6 : 9 , and 8 : 12 . This is found by simplifying the ratio to 2 : 3 and multiplying by different integers. Each equivalent ratio maintains the same relationship between its two parts.
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