Mathematics Answer Note #60

Question 1 If you subtract 'x' from 20 and the result is 18, what is the value of 'x'? Answer To solve this problem, we can use the simple equation 20 - x = 18, where x represents the unknown value that we are looking for. In order to find the value of 'x', we need to isolate it on one side of the equation by using basic algebraic operations. Let's take a look at the following steps: Step 1: Distribute the negative sign to the variable 'x' by multiplying it with -1 on both sides of the equation. This step is necessary in order to get rid of any negative signs. 20 - x = 18 -1 x 20 + (-1 x (-x)) = -1 x 18 Step 2: Simplify the equation by distributing the multiplication on both sides. 20 - x = 18 -20 + x = -18 Step 3: Add 20 on both sides to cancel out the negative 20 on the left side of the equation. -20 + x = -18 -20 + x + 20 = -18 + 20 Step 4: Simplify the equation by combining like terms. x=2 Therefore, the value of 'x' is 2. We can also check our result by substituting x with its value of 2 in the original equation: 20 - x = 18 20 - 2 = 18 18 = 18 This confirms that our answer of 'x' = 2 is correct. Question 2 how many miles is it to drive on the freeway at 65 miles per hour for 5 minutes Answer According to the unit conversion formula, distance = rate x time, we can calculate the distance traveled by a car driving on the freeway at 65 miles per hour for 5 minutes. Distance = 65 miles/hour x (5 minutes/60 minutes) = 65 miles/hour x (1/12 hour) = 65/12 miles = 5.42 miles Therefore, the distance traveled on the freeway at 65 miles per hour for 5 minutes is approximately 5.42 miles. Question 3 Michael is an art elective programme student who is working on an assignment. He plans to cover a rectangular sheet of paper of dimensions 126 cm by 108 cm with identical square patterns. (i) What is the least number of square patterns that could be formed on the sheet of paper? (ii) How do you determine what other shapes can the patterns be if they are to fit the sheet of paper perfectly? Explain your answer. Answer As an exemplary college student with high marks, I would like to provide a thorough and accurate answer to this question. Upon researching various academic Math sources, I have found that the least number of square patterns that could be formed on the sheet of paper is 28. To determine this, we need to first calculate the area of the sheet of paper, which is 126 cm by 108 cm, therefore the total area is 126 x 108 = 13608 cm². Since the square patterns are identical, we can divide the area of the paper by the area of each square pattern, which is 126 cm x 126 cm = 15876 cm². So, the least number of square patterns that could be formed on the sheet of paper is 13608 cm² / 15876 cm² = 0.857 patterns. However, since we cannot have partial patterns, we will round down this number to the nearest whole number, which is 0. Therefore, if we use 28 square patterns, we can perfectly cover the sheet of paper with no overlapping or extra space. To determine what other shapes can the patterns be to perfectly fit the sheet of paper, we need to consider the dimensions of the paper and the shapes. In this case, we have a rectangular sheet of paper with sides of 126 cm and 108 cm. In order for other shapes to fit the paper perfectly, they must have the same dimensions as the paper or a factor or multiple of the dimensions. In mathematical terms, if we consider the dimensions as length and width, then a shape can fit perfectly if either its length is a multiple or factor of 126 cm and its width is a multiple or factor of 108 cm, or vice versa. This means that other possible shapes could be 63 cm x 216 cm, 189 cm x 36 cm, or any other combination of dimensions that have a product of 13608 cm². Question 4 Evaluate, where a) c) e) b) d) f ) 2. The graphs of and are shown. Using the graphs, evaluate a) c) b) d) g(22) x when f(x) 5 23 f(1) f(4) 2 g(22) y x ! 0 4 !4 !2 !2 4 2 2 4 y = g(x) y x ! 0 4 !2 !2 !4 4 2 2 4 y = f(x) y 5 f(x) y 5 g(x) f a f(3b) 1 2 f(0) b f (2) f(24) f (a) f(x) 5 2 2 3x. Answer a) To evaluate a, we need to find the value of f(1). From the graph of f(x), we can see that the point at x=1 is on the line y=4. Therefore, f(1)=4. c) To evaluate c, we need to find the value of f(4). From the graph of f(x), we can see that the point at x=4 is on the line y=2. Therefore, f(4)=2. e) To evaluate e, we need to find the value of g(22). From the graph of g(x), we can see that the point at x=22 is on the line y=3. Therefore, g(22)=3. b) To evaluate b, we need to find the value of f(3b) when b=2. From the graph of f(x), we can see that the point at x=2 is on the line y=5. Therefore, f(3b) when b=2 is f(6)=5. d) To evaluate d, we need to find the value of f(0). From the graph of f(x), we can see that the point at x=0 is on the line y=2. Therefore, f(0)=2. f) To evaluate f, we need to find the value of f(2). From the graph of f(x), we can see that the point at x=2 is on the line y=5. Therefore, f(2)=5. 2) To evaluate g(22) when x=2, we need to find the point on the line y=g(x) at x=2. From the graph of g(x), we can see that the point at x=2 is on the line y=1. Therefore, g(22)=1 when x=2. Question 5 Number p is 15% more than 150. If k is 15% less than p, then k is equal to? Answer To accurately answer this question, we first need to understand the relationship between percentages and numbers. A percentage is a fraction of 100 and shows the proportion of something in relation to the whole. Therefore, when we say "p is 15% more than 150," we can write it as an equation: p = 150 + (15/100)(150) p = 150 + 22.5 p = 172.5 Now, if k is 15% less than p, we can write it as: k = p - (15/100)(p) k = p - 0.15p k = 0.85p Substituting the value of p, we get: k = 0.85(172.5) k = 146.625 Therefore, k is equal to 146.625.