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Unit Notes U2 - Ratios.rtf

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Department
Business
Course
BUS 100
Professor
Marla Spergel
Semester
Fall

Description
U2 Full Chapter Chapter 5 Ratio Math Skills for Business- Full Chapters 5.1 Definition of 45 Ratio Ratio is a quantitative comparison of the relative size of two or more quantities with the same units of measurement. Businesses use ratios in a variety of situations and it is a powerful tool for making financial decisions. It is particularly important for analyzing business profitability and its ability to meet debt payment. As well, knowledge of ratio helps us make all kinds of comparisons in business. A common ratio in business is to compare the value of current assets to that of current liabilities. Another common ratio is to compare earnings after taxes to equity. Sometimes businesses compare the cost of raw materials to cost of labour. As well, in the manufacturing of confectionary products such as bread, candies and muffins, there are standard proportion of ingredients that each product should ratio can contain. For a ratio in standard form, all the terms are integers and there is no common divisor for all the terms. For a ratio with more than two terms, it is possible for two or more of the terms to have a common divisor but not for all of the terms. A ratio can always be expressed in standard form by multiplying and/or dividing each of the terms by the same number. Multiplying or dividing each of the term of a ratio by the same number except zero, do not change terms of a ratio, because they are relative comparison of their sizes. Notation Ratio of quantity A to A quantity B is written in mathematics as follows, A is to B, Bor A to B. A is called the first term and B the second term of the ratio. Therefore, A: B is called a two-term ratio. A: B: C is called a three-term ratio. Terms of a ratio can be reduced to lowest term just as we reduce fractions into lowest term. Example 1 : The daily sale at Bagra Enterprises is $1 500 and at Akua Enterprises $2 000. The ratio of daily sales at Bagra to daily sales at Akua Enterprises is written as; Daily Sales at Bagra Enterprises : Daily Sales at Akua Enterprises = $1 500 : $2 000. Each term can be divided by $500 (or by 100 and then by 5) to reduce them to lowest term: Daily Sales at Bagra Enterprises : Daily Sales at Akua Enterprises = (1500 ÷ 500) : (2000 ÷ 500) = 3:4. (What does this mean?) Interpretation: 1. For every 3 dollars in daily sale at Bagra, there is $4 in sale at Akua. 2. If the total daily sales for Bagra and Akua are divided into 7 (which is 3 + 4) equal parts, 3 of the parts are Bagra’s sales and 4 parts are Akua’s sales. 5.3 Finding Ratios of Quantities Example 2 The hourly wage of a sales associate is $9.50 at Balmort; $10.50 at Scheers; and $12 at Makola. Find the ratio of the hourly wage of sale associates at Balmot to that of Scheers and to Makola. Soluti Balmot : Scheers : Makola = $9.50 : $10.50 : $12 then (Multiply each term by 100 and divide by $) = 950 : 1050 : 120 (Divide each term by 50, which the common factor) Balmot: Scheers : Makola = 19 : 21 : 24 Exampl e 3 dimensions of a rectangular box are length 1m 35cm, width 1m 80cm, and depth 90cm. Find the ratio of the length to the width to the depth of the box. Length:Width : Depth = 1m 35cm : 1m 80cm : 90cm then (Change units to cm = 135cm: 180cm: 90cm then (Divide each term by 45cm)= 3 : 4 : 2 Exampl e 4 The ratio of 4:2 is not in standard form but 3: 4: 2 is in standard form. The ratio 4 : 2 is expressed in standard form by dividing both terms by the common divisor 2; Thus, ratio; A : B : k any number except zero. = m any number except zero. A change in the order of a ratio changes the ratio. The ratio of quantity A to quantityB is not the same as the ratio of quantity B to quantity A. Exampl e 5s example shows that we can use our understanding of ratio to calculate unknownquantity. This idea is very important to understanding proportion. It should be stated that proportion is an extension of ratio. Joe’s two-week wages is $800 for 80 hours of work. a) Given this ratio, how many hours does he have to work in order to earn$1600?b) If he works 245 hours how much will he earn? a) Given the ratio, 80 = 10:1 (After diving each term by 80) Wages : Hours 1600 : h Since for every $10, he has to work an hour, there are 160 tens in $1600. So, Joe has to work 160 hours in order to make $1600. Note that the ratio of wages to hours does not change, except when one of the terms of the original ratio changes. b) Again, he has to work 240 hour of an hour; This example seems very simple athat ishould provide you an insight into the nature of ratios in mathematics. However, other examples may not be so simple and they may require a more sophisticated procedure. Math Skills for Business- Full Chapters 48 5.3 Comparing RAn important idea in mathematics is how to compare ratios. It gives us an idea about the multiplicative relationship between quantities. For example, what is the relationship between 1/2 and 2/4? Do you think that 1/2 and 2/4 equivalent fractions or ratios? With a little drawing and intuition, it becomes clear that 1/2 and 2/4 are equivalent fractions or ratios. And that 1/5 and 20/100 are also equivalent. Ratios are equivalent or equal if their cross-products are the same, or if they can be written as equal fractions. To compare ratios do the following: a) Write the ratios as fractions b) Calculate their cross-products c) Then compare them. If the products are the same, the ratios are equal or equivalent. Are the ratios 3 to 4 and 6/8 equal? The ratios . Their cross- e 5 of 3 × 8= 6 × 4 are obviously equal. Since both products are equal, the ratios are equal. Exampl Are the ratios 1:5 and 4/20 equivalent? Yes, they are equivalent. The fraction 4/20 canbe reduced to the lowest by dividing the numerator by 4 (4÷ 4 = 1) and the denominator by 4 (20 ÷ 4 = 5). This gives us 1/5, which is equal to 1/5. Their cross-products give, 1 × 20 = 5 × 4. Exampl e 7 Compare 13:17 and 5/7. Are they equal or equivalent ratios? No, they are not equivalent because their cross-products are not equal: 13 × 7 ≠ 17 × 5. 5.4 Proportional Division This is the division of a quantity in a given ratio. That is, the given quantity is divided into piles such that the ratio of quantities in the piles is equal to the given ratio. eMaya and Nina’s uncle gave them an amount of $150,000 in his will. The money is to be divided between them in the ratio of 3 to 2. How much does each receive? Math Skills for Business- Full Chapters 49 Soluti Since when the $150,000 is divided between them it should correspond to the given ratio, we can try a few pairs of numbers. Maya’s share : Nina’s share $100,000 : 50,000 2 :1 After dividing each term by 50,000, you 2: 1 does nocan see that to 3:2. So, we try another pair: 90,000: 60,000 = 3: 2 After dividing each term by 30,000 Thus, Maya’s share is $90,000 and Nina’s share is $60,000. This is correct because the divided quantity corresponds to the given ratio. Exampl Mohammed , Ben , and Rachel earned some money by selling toys and lemonade at a festival stand. They agreed to share any profit in the ratio of their ages. Mohammed is 9 years old, Rachel is 10 years old, and Ben is 7 years old. Ben’s share came to $28. Find the share of Mohammed and that of Rachel. The solution to this problem is not so easy, as example one. This is because the amount of profit they made is not given. First, let us find the profit they made. Let p stand for the profit they made. The total ratio is 26 (9 + 10 + 7). This is how Ben’s share was obtained = = = x p = 28 2 = 26 = 28 x We multiply each side by 26 in order to clear the fractions. We now have 7p = 728 The 728 was obtained after multiplying 28 by 26. P = 104 After dividing each side of the equation by 7. Mohammed ’ s share = Rachel’s share = x 104 = $40 104 = The ratio is 36: $36 40: 28 = 9: 10: 7 after dividing each term by 4. 5.5 Review, Exercises and 1. Give two examples of situations in which a business owner may use ratio for the purpose of comparison. 2. Reduce to the lowest terms a) 12to24 b) 84to56 c) 15to24to39 d) 32: 100 e) 30: 240 f) 50: 49 g) 10.5:3. 5 3. Set up a ratio for each of the following and reduce to lowest terms.5 quarters b) 6 minutes for 50 metres c) $45 per day for 12 employees for 20 day d) $2000 sales for 5 days 4. The cost of a unit of a product consists of $4.25 direct materials cost, $2.75 direct labour cost, and $3.25 overhead cost. What is the ratio existing between these three elements of cost? 5. Mariam applied for residential mortgage from Seaway Financial Services. Toligibility for the mortgage, the financial officer used the following ratio: Personal debts: personal annual income. a) Mariam’s personal debts were $40,000 and personal annual income was $50,500. What is her ratio of personal debt to personal annual income? the financial officer, would you approve the mortgage for Mariam? Why? 6. a) The gross sales to the number of sales persons ratio for a retail company is $20,000:5 for the first quarter. What does this mean to the retailer? What advice would you give to the retailer, if the total wage bill for the quarter is $15,000?r a grocery store is 600:1 for the third quarter. Explain what this means for the retail operations. 7. Mike received $100 a week for distributing flyers to houses. He spent $40 at the movies and $25 on comic books. After buying $10 worth of ice cream, he saved the rest. a) What is the ratio of the amount of money he spent on buying comic books and movies to the amount he saved? Firm Name 2000 2001
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