Step 1: | Write the correct equation. Inverse variation problems are solved using the equation . When dealing with word problems, you should consider using variables other than x and y, you should use variables that are relevant to the problem being solved. Also read the problem carefully to determine if there are any other changes in the inverse variation equation, such as squares, cubes, or square roots. |
Step 2: | Use the information given in the problem to find the value of k, called the constant of variation or the constant of proportionality. |
Step 3: | Rewrite the equation from step 1 substituting in the value of k found in step 2. |
Step 4: | Use the equation found in step 3 and the remaining information given in the problem to answer the question asked. When solving word problems, remember to include units in your final answer. |
Example 1 – If x varies inverse as y, and x = 7 when y = 3, find y when x = 9.
Step 1: Write the correct equation. Inverse variation problems are solved using the equation . | |
Step 2: Use the information given in the problem to find the value of k. In this case, you need to find k when x = 7 and y = 3. | |
Step 3: Rewrite the equation from step 1 substituting in the value of k found in step 2. | |
Step 4: Use the equation found in step 3 and the remaining information given in the problem to answer the question asked. In this case, you need to find y when x = 9. |
Example 2 – If r varies directly as the cube of s, and r = 5 when s = 3, find r when s = 2.
Step 1: Write the correct equation. Inverse variation problems are solved using the equation . In this case, you should use r and s instead of x and y and notice how the word “cube” changes the equation. | |
Step 2: Use the information given in the problem to find the value of k. In this case, you need to find k when r = 5 and s = 3. | |
Step 3: Rewrite the equation from step 1 substituting in the value of k found in step 2. | |
Step 4: Use the equation found in step 3 and the remaining information given in the problem to answer the question asked. In this case, you need to find r when s = 2. |
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Example 3 – If a varies inversely as square root of b, and a = 7 when b = 36, find a when b = 289.
Step 1: Write the correct equation. Inverse variation problems are solved using the equation . In this case, you should use a and b instead of x and y and notice how the word “square root” changes the equation. | |
Step 2: Use the information given in the problem to find the value of k. In this case, you need to find k when a = 7 and b = 36. | |
Step 3: Rewrite the equation from step 1 substituting in the value of k found in step 2. | |
Step 4: Use the equation found in step 3 and the remaining information given in the problem to answer the question asked. In this case, you need to find a when b = 289. |
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Example 4 – The time it takes you to get to campus varies inversely as your driving speed. Averaging 20 miles per hour in bad traffic, it takes you 1.5 hours to get to campus. How long would the trip take averaging 50 miles per hour?
Step 1: Write the correct equation. Inverse variation problems are solved using the equation . In this case, you should use t and s instead of x and y. | |
Step 2: Use the information given in the problem to find the value of k. In this case, you need to find k when t = 1.5 and s = 20. | |
Step 3: Rewrite the equation from step 1 substituting in the value of k found in step 2. | |
Step 4: Use the equation found in step 3 and the remaining information given in the problem to answer the question asked. In this case, you need to find t when s = 50. |
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Example 5 – The volume of gas in a container at a constant temperature varies inversely as the pressure. If the volume is 32 cubic centimeters at a pressure of 8 pounds, find the pressure when the volume is 60 cubic centimeters.
Step 1: Write the correct equation. Inverse variation problems are solved using the equation . In this case, you should use v and p instead of x and y. | |
Step 2: Use the information given in the problem to find the value of k. In this case, you need to find k when v = 32 and p = 4. | |
Step 3: Rewrite the equation from step 1 substituting in the value of k found in step 2. | |
Step 4: Use the equation found in step 3 and the remaining information given in the problem to answer the question asked. In this case, you need to find p when v = 60. |