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.
ScanSolve AI Answer
Here are the solutions to the problems: 2. When a certain number is increased by five, the result is six times the same number decreased by ten. What is the number? Step 1: Let the number be x. Translate the problem into an equation. "A certain number is increased by five" can be written as x + 5. "Six times the same number decreased by ten" can be written as 6(x - 10). The problem states that these two expressions are equal. x + 5 = 6(x - 10) Step 2: Distribute the 6 on the right side of the equation. x + 5 = 6x - 60 Step 3: Gather the x terms on one side and the constant terms on the other side. Subtract x from both sides: 5 = 5x - 60 Add 60 to both sides: 5 + 60 = 5x 65 = 5x Step 4: Solve for x by dividing both sides by 5. (65)/(5) = x x = 13 The number is 13. 3. If (3)/(5) of a numbers is 4 more than (1)/(2) the number, what is the number? Ans 40 Step 1: Let the number be x. Translate the problem into an equation. "(3)/(5) of a number" can be written as (3)/(5)x. "4 more than (1)/(2) the number" can be written as (1)/(2)x + 4. The problem states that these two expressions are equal. (3)/(5)x = (1)/(2)x + 4 Step 2: To eliminate the fractions, multiply the entire equation by the least common multiple (LCM) of the denominators (5 and 2), which is 10. 10 · ((3)/(5)x) = 10 · ((1)/(2)x + 4) 6x = 5x + 40 Step 3: Gather the x terms on one side. Subtract 5x from both sides: 6x - 5x = 40 x = 40 The number is 40. This matches the given answer. 4. I am thinking of a certain number. If I multiply by 6, add -18 to the product and multiply by one - third of the result, I get -2. What number am I thinking of? Step 1: Let the number be x. Translate the problem into an equation. "Multiply by 6": 6x "Add -18 to the product": 6x - 18 "Multiply by one-third of the result": (1)/(3)(6x - 18) "I get -2": means the expression equals -2. (1)/(3)(6x - 18) = -2 Step 2: Distribute (1)/(3) into the parenthesis. (1)/(3)(6x) - (1)/(3)(18) = -2 2x - 6 = -2 Step 3: Isolate the term with x. Add 6 to both sides of the equation: 2x = -2 + 6 2x = 4 Step 4: Solve for x by dividing both sides by 2. x = (4)/(2) x = 2 The number is 2. Drop the next question! 📸