This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the direct and inverse proportion questions: 1. 6 pens cost £2.16. Calculate the cost of 12 pens. Step 1: Find the cost of one pen. Cost per pen = Total costNumber of pens = £2.166 = £0.36 Step 2: Calculate the cost of 12 pens. Cost of 12 pens = Cost per pen × 12 = £0.36 × 12 = £4.32 2. 8 water bottles cost £20. Calculate the cost of 13 water bottles. Step 1: Find the cost of one water bottle. Cost per bottle = Total costNumber of bottles = £208 = £2.50 Step 2: Calculate the cost of 13 water bottles. Cost of 13 bottles = Cost per bottle × 13 = £2.50 × 13 = £32.50 3. X is directly proportional to the square of Y. When X = 50, Y = 5. Find X when Y = 3. Step 1: Write the proportionality relationship. X Y^2 Introduce the constant of proportionality, k. X = kY^2 Step 2: Use the given values (X=50, Y=5) to find k. 50 = k(5^2) 50 = k(25) k = (50)/(25) k = 2 Step 3: Use the value of k and the new value of Y (Y=3) to find X. X = 2(3^2) X = 2(9) X = 18 4. C is directly proportional to the cube root of D. When C = 32, D = 8. Find D when C = 16. Step 1: Write the proportionality relationship. C [3]D Introduce the constant of proportionality, k. C = k[3]D Step 2: Use the given values (C=32, D=8) to find k. 32 = k[3]8 32 = k(2) k = (32)/(2) k = 16 Step 3: Use the value of k and the new value of C (C=16) to find D. 16 = 16[3]D (16)/(16) = [3]D 1 = [3]D To find D, cube both sides. D = 1^3 D = 1 5. X is inversely proportional to the square of Y. When X = 2, Y = 5. Find X when Y = sqrt(10). Step 1: Write the proportionality relationship. X (1)/(Y^2) Introduce the constant of proportionality, k. X = (k)/(Y^2) Step 2: Use the given values (X=2, Y=5) to find k. 2 = (k)/(5^2) 2 = (k)/(25) k = 2 × 25 k = 50 Step 3: Use the value of k and the new value of Y (Y=sqrt(10)) to find X. X = (50)/((sqrt(10))^2) X = (50)/(10) X = 5 Last free one today — make it count tomorrow, or type /upgrade for unlimited.