Step 1: | Write the correct equation. Combined variation problems are solved using a combination of direct variation (y = kx), inverse variation , and joint variation (y = kxz) equations. When dealing with word problems, you should consider using variables other than x, y, and z, 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 combined 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 y varies directly as x and inversely as z, and y = 24 when x = 48 and z = 4, find x when y = 44 and z = 6.
Step 1: Write the correct equation. Combined variation problems are solved using a combination of variation equations. In this case we will combine the direct and inverse variation equations. | |
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 = 48, y = 24, and z = 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 x when y = 44 and z = 6. |
Example 2 – If f varies directly as g and inversely as the square of h, and f = 20 when g = 50 and h = 5, find f when g = 18 and h = 6.
Step 1: Write the correct equation. Combined variation problems are solved using a combination of variation equations. In this case, you will combine the direct and inverse variation equations, use f, g, and h instead of x, y, and z, and notice how the word “square” 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 f = 20, g = 50, and h = 5. | |
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 f when g = 18 and h = 6. |
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Example 3 – If y varies jointly as a and b and inversely as the square root of c, and y = 12 when a = 3, b = 2, and c = 64, find y when a = 5, b = 2, and c = 25.
Step 1: Write the correct equation. Combined variation problems are solved using a combination of variation equations. In this case, you will combine the joint and inverse variation equations, use y, a, b, and c instead of w, x, y, and z, 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 y = 12, a = 3, b = 2, and c = 64. | |
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 a = 5, b = 2, and c = 25. |
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Example 4 – The number of minutes needed to solve an exercise set of variation problems varies directly as the number of problems and inversely as the number of people working on the solutions. It takes 4 people 36 minutes to solve 18 problems. How many minutes will it take 6 people to solve 42 problems.
Step 1: Write the correct equation. Combined variation problems are solved using a combination of variation equations. In this case we will combine the direct and inverse variation equations and use m, p for problems, and w for working instead of x, y, and z. | |
Step 2: Use the information given in the problem to find the value of k. In this case, you need to find k when m = 36, p = 18, and w = 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 m when p = 42 and p = 6. |
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Example 5 – The electrical resistance of a wire varies directly as its length and inversely as the square of its diameter. A wire with a length of 200 inches and a diameter of one-quarter of an inch has a resistance of 20 ohms. Find the electrical resistance in a 500 inch wire with the same diameter.
Step 1: Write the correct equation. Combined variation problems are solved using a combination of variation equations. In this case, you will combine the direct and inverse variation equations, use r, l, and d instead of x, y, and z, and notice how the word “square” 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 = 20, l = 200, and d = 0.25. | |
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 l = 500 and d = 0.25. |