CHAP 4 Rates and Variations Rate Rate: is the ratio that compares two different units of quantities. - - It is a constant relation between the sizes or amounts of two quantities. A rate is found by dividing one quantity by another. For example a rate of pay consists of the money paid divided by the time worked. If a man receives 1,000 shillings for two hours work, his rate of pay is 1000/2 = 500 shillings per hour. From the above example, we find out that: Example 1. A man is paid T.sh 16 000 for 8 hours’ work. (a) What is the rate of payment? (b) At this rate, how much would he receive for 20 hours of work? (c) At this rate, for how long must he work in order to receive T.sh 30 000? Solution (a) To find the rate of payment, divide T.sh 16 000 by 8 hours, that is, R = M/T = T.sh 16 000/8 hours = T.sh 2 000 per hour. Therefore, the rate of payment is T.sh 2 000 per hour. (b) Multiply T.sh 2 000 per hour by 20 hours to obtain the money he would receive by working for 20 hours. Thus, M = R x T = T.sh 2 000 per hour × 20 hours = T.sh 40 000. Therefore, for 20 hours work, he would receive T.sh 40 000. (c) Divide T.sh 30 000 by T.sh 2 000 per hour. T = M/R = T.sh 30 000 / T.sh 2 000 per hour = 15 hours Therefore, he must work for 15 hours to receive T.sh 30 000. Exchange Rates - When money from country A is to be used in country B, it is necessary to exchange the currency of country A to the currency of country B. Various currencies in the world are linked together by exchange rates. This enables smooth transfer of money and payments to take place between countries. Table 4.1 shows foreign mean exchange rates as supplied by one of the commercial banks in Tanzania on 15th March, 2021. These are mean rates in Tanzanian shillings per unit of a foreign currency. Example 4.3 1. A tourist from Sweden wished to exchange 1 000 Kronor into Tanzanian shillings. How many Tanzanian shillings did the tourist receive if the mean exchange rate was as shown in Table 4.1? Solution Let x be the amount of money in Tanzanian shillings the tourist received. Table 4.1 shows that 1 Swedish Kronor was equivalent to T.sh 282.56, that is, 1 Kronor = T.sh 282.56 1,000 Kronor = x; Cross multiplication gives: x = (1,000 Kronor × T.sh 282. 56) / 1 Kronor = T.sh 282.56 × 1,000 = T.sh 282,560 Therefore, the tourist received T.sh 282,560. Exercise 1. A student had two plant seedlings. She measured the rate at which the seedlings were growing. Seedling A grew 5 cm in 10 days and seedling B grew 8 cm in 12 days. Which seedling was growing more quickly? 2. How many US Dollars are worth T.sh 600 000 on 15th March, 2021 as far as exchange rates in Table 4.1 is concerned? Variations There are basically three types of variations, namely; direct, inverse, and joint variations. Direct variations - - - Some quantities are related in such a way that they increase or decrease together at the same rate. Quantities with this relationship are directly proportional, or vary directly. If a car travels at a constant speed, the distance it covers is directly proportional to the time taken. If Y is directly proportional to X, mathematically, it is written as Y ∝ X, where ∝ is a symbol of proportionality. This expression can be written as an equation by introducing equal sign and a constant of proportionality instead of the proportionality sign. For instance, if Y varies directly as the square of X, then it can be written as Y ∝ X2 and the corresponding mathematical equation is Y = kX2, where k is the constant of proportionality. For any two pairs of quantities X and Y, say, (X1, Y1) and (X2, Y2), two equations Y1 = kX12 and Y2 = kX22 are obtained. This implies that: k = Y1/X12 =Y2/X22. So, it is said that x and y vary directly if the ratios of the values of y to the values of x are proportional. Example 1. If y varies directly as x and x = 15 when y = 4, find the value of y when x =12. Solution Given that y ∝ x. This implies that y = kx, where k is the constant of proportionality. Making k the subject of the equation gives: k = y/x. But, for any two pairs of quantities, k =y1/x1, and also k = y2/x2 - - - If x and y represent variables such that y ∝ x, then y = kx. The form of the equation y = kx is a straight line passing through the origin, k being the gradient (slope) of the line. The graph in Figure 4.1 shows the relation y ∝ x for k = 1 and Figure 4.2 shows a graph of y ∝ (x + 1) for k = 1. Exercise 1. If x varies directly as the square of y, and x = 4 when y = 2, find the value of x when y = 8. 2. A car travels 60 kilometers using 5 liters of diesel. How many liters of diesel are needed to travel 150 kilometers? Inverse variations - In some cases, one quantity increases at the same rate as another quantity decreases. In this case, the quantities vary inversely, or they are in inverse proportion. Inverse proportion is sometimes called indirect proportion. For example, the number of men employed to cultivate a farm and time it takes for them to complete the work are inversely related. Example 1. If x varies inversely as y and x = 2 when y = 3, find the value of y when x = 18. Exercise 1. If y varies inversely as x and if y = 3 when x = 4, find y when x = 6. 2. If x varies inversely as the square of y and if x = 3 when y = 4, what is the value of x when y = 8? 3. If x varies inversely as y and if x = 4 when y = 3/2, then find the value of y when x = 8. Joint variations - A given quantity can relate directly or inversely with two or more quantities. This kind of relationship is called joint variation. Joint variation can involve variables which both vary either directly or inversely. Also, a joint variation can involve a variable which varies directly with some other variables as well as inversely with other variables. - - Example Suppose y varies directly as x and z. Given x = 4, z = 2 and y = 24, find: (a) The variation equation connecting x, y, and z. (b) The value of y when x = 5 and z = 6. (b) With y = 3xz, if x = 5 and z = 6, then y = 3 × 5 × 6 = 90 Therefore, the value of y is 90. Exercise 1. If x varies directly as y and inversely as z, and x = 8 when y = 12 and z = 6, find the value of x when y = 16 and z = 4. 2. Three tailors can sew 15 clothes in 5 days. How long will it take 5 tailors working at the same speed to sew 20 clothes?
