Mathematics Answer Note #51

Question 1 Mother bought four mangoes at php. 38.00. At the same rate. How much will a dozen and half of the mangoes cost? Answer According to the given information, the cost of four mangoes is at php. 38.00. In order to determine the cost of a dozen and half mangoes, we first need to convert the given information into a ratio. Since four mangoes cost php. 38.00, then one mango would cost php. 9.50 (38.00 divided by 4). Therefore, the ratio of the cost of one mango to the cost of four mangoes is 1:4. Now, to determine the cost of a dozen and half mangoes, we can use the given ratio of 1:4 and apply it to the given quantity of a dozen and half (18). This would result in a ratio of 1:4 x 18 = 18:72. Therefore, the cost of a dozen and half mangoes would be 18 x php. 9.50 = php. 171.00. In summary, if Mother bought a dozen and half of mangoes at the same rate of php. 38.00 for four mangoes, the cost would be php. 171.00. Alternatively, we can also directly calculate the cost by dividing the total cost of four mangoes by the quantity of four mangoes (php. 38.00/4 = php. 9.50) and then multiplying it by the quantity of a dozen and half mangoes (php. 9.50 x 18 = php. 171.00), as shown in the step-by-step breakdown below. Step-by-Step Breakdown: 1. Divide the total cost of four mangoes (php. 38.00) by the quantity of four mangoes (38.00/4 = 9.50). 2. Multiply the result (9.50) by the quantity of a dozen and half mangoes (9.50 x 18 = 171.00). 3. Therefore, the cost of a dozen and half mangoes at the same rate would be php. 171.00. In conclusion, the cost of a dozen and half mangoes at the same rate as four mangoes at php. 38.00 would be php. 171.00. Question 2 show that the product of four consecutive integers is divisible by 8 Answer To show that the product of four consecutive integers is divisible by 8, we can make use of the Fundamental Theorem of Arithmetic which states that every positive integer greater than 1 can be expressed as a unique product of prime numbers. Let the four consecutive integers be n, n+1, n+2, and n+3. Then, their product can be expressed as n(n+1)(n+2)(n+3). Since n is one of the integers, we can express n as a product of its prime factors, say p1^a1 * p2^a2 * … * pn^an. Similarly, we can express (n+1) and (n+2) as products of their prime factors. However, since (n+3) is consecutive to (n+2), it will share at least one of its prime factors with (n+2), say pi. Hence, the product of these four consecutive integers can be expressed as p1^a1 * p2^a2 * … * pn^an * pi. Now, we need to prove that this product is divisible by 8. Since pi is shared by (n+2) and (n+3), it will be present twice in the product. Therefore, we can express the product as p1^a1 * p2^a2 * … * pn^an * pi * pi. We know that any number raised to the power of 2 will be divisible by 4. Hence, pi * pi will be divisible by 4. Moreover, we also know that p1^a1 * p2^a2 * … * pn^an is a product of prime factors and we have already proved that any such product is divisible by 2. Therefore, the product of four consecutive integers, n(n+1)(n+2)(n+3), is divisible by 4 * 2 = 8. Question 3 The entrance to the great pyramid is 17 m above ground. What is the vertical distance from the entrance to the base of the pyramid which is 31 m underground? Answer To accurately answer this question, we first need to understand the concept of vertical distance and how it relates to the entrance and base of the great pyramid. The vertical distance is the measurement of the vertical or up-down distance between two points. In this case, we are measuring the distance from the entrance to the base of the pyramid which is located 31 m underground. To determine the vertical distance, we will first need to calculate the total height of the pyramid. According to research, the height of the great pyramid is approximately 146.5 m (480.6 ft). This includes the 31 m underground and the 17 m above ground entrance. Next, we need to subtract the height of the entrance (17 m) from the total height of the pyramid (146.5 m). This will give us the height from the entrance to the top of the pyramid, which is 129.5 m. Since we are interested in the distance from the entrance to the base of the pyramid, we need to subtract the height of the pyramid itself (146.5 m) from the distance from the entrance to the top of the pyramid (129.5 m). This will give us the vertical distance from the entrance to the base, which is 17 m. Therefore, the vertical distance from the entrance to the base of the pyramid which is 31 m underground is 148 m (basically the total height of the pyramid). Question 4 At noon, there are 0.3 grams of mold. At 6:00 AM, there were 0.1 grams of mold. If this trend continues, how many grams of mold will be present at 10:00 PM on the same day? Answer To accurately answer this question, we will need to apply some basic mathematical calculations to determine the amount of mold present at 10:00 PM on the same day. Firstly, let's establish the rate at which the mold is increasing. From the information given, we can see that the mold has increased by 0.2 grams (0.3 - 0.1) in a period of 12 hours (from 6:00 AM to noon). Therefore, the mold is increasing at a rate of 0.2 grams per 12 hours. Now, we can use this information to find out how much the mold will increase in a span of 16 hours (from noon to 10:00 PM). To do this, we simply need to calculate the ratio of the increase in mold (0.2 grams) to the time period (12 hours) and then multiply it by the time period we are interested in (16 hours). 0.2 grams/12 hours = 0.01667 grams per hour 0.01667 grams per hour * 16 hours = 0.2667 grams Therefore, at 10:00 PM on the same day, there will be a total of 0.3 grams + 0.2667 grams = 0.5667 grams of mold present. Question 5 every hour the amount of water in a tank decreases by 17% . At 1pm the tank contains 1620 litres of water. How many litres will drain from the tank between 5pm and 6pm Answer To answer this question accurately, we first need to understand the concept of percentage decrease and how it affects the amount of water in the tank. In simple terms, percentage decrease refers to the reduction of a certain value by a certain percent. In this case, the amount of water in the tank decreases by 17% every hour. To find the amount of water that will drain from the tank between 5pm and 6pm, we need to determine how many hours have passed since 1pm. From 1pm to 5pm, a total of 4 hours have passed. Therefore, the amount of water in the tank at 5pm will be 1620 litres minus 17% for every hour (4 x 17% = 68%). This means the tank will contain 52% (100% - 68%) of 1620 litres, which is equivalent to 842.4 litres. Now, to find the amount of water that drains between 5pm and 6pm, we will use the same concept. In this case, only 1 hour has passed since 5pm. So, the amount of water in the tank at 6pm will be 842.4 litres minus 17% for 1 hour, which is equivalent to 140.3 litres. Therefore, the amount of water that will drain from the tank between 5pm and 6pm is 842.4 litres minus 140.3 litres, which is equal to 702.1 litres.