Ratio, Proportion, and Percent

Learning Objectives Student will use ratios to solve allocation and equivalence problems Student will find percent and percent bases to solve business problems Student will use proportions and currency cross rate table to convert currency Student will use index numbers and the consumer price index to compute purchasing power of the dollar Student will use federal income tax brackets and tax rates to calculate federal income taxes.

Ratios A ratio compares values. A ratio says how much of one thing there is compared to another thing. 3:1 There are three orange squares to 1 blue square Ratios can be shown in different ways: -Using the “:” to separate the values: 3:1 -Use the word “to”: 3 to 1 -Write it as a fraction: 3/1 -Write it as a decimal: 3.00 -Write it as a percent: 300%

Terms of the ratio- The numbers appearing in the ratio. All terms in the ratio must be expressed in the same units, but the units are not usually included once the ratio is obtained. Perform a conversion if units are not the same E.g #1: 5 kg: 3 kg : 2 kg All of the same units, so we simply write: 5:3:2 E.g. #2: 5 kg to 1500g = 5 kg to 1.5 kg = 5 : 1.5

When rates are expressed as ratios, we also drop the units: E.g. 100 km/hr = 100:1 50 m in 5 s = 50:5 $1.49 for 2 items = 1.49: 2 Expressing Ratios In Lowest Terms -Just as fractions can be expressed in lowest/simplest terms, so too can ratios. *Remember a ratio can be expressed as a fraction* -When converting a ratio to lowest/simplest terms, we simply need to find the greatest common factor (GCF) of the terms in the ratio, and then divide each number in the ratio by the GCF. GCF the largest positive integer that divides evenly into all numbers with zero remainder. E.G: for the set of numbers 18 to 30 to 42 the GCF = 6 each number ÷ 6 is 3:5:7

A Little Hint… If you are having trouble finding the GCF: begin by dividing all terms by 2 (If possible), and then decide if the numbers may be divided by 2 again. e.g. 16: 28 : 12 All divided by 2 - 8:14:6 Then ask yourself if the numbers still have a common factor: YES! 2 can be used again! 8:4:6 all divided by 2 – 4:2:3 Since there are no further common factors, this ratio is in lowest/simplest terms.

Equivalent Ratios Just as ratios can be reduced into lowest/simplest terms, they may also be expressed as an Equivalent Ratio simply by multiplying all terms by the same number. (Reverse of simplifying ratios) E.g. 3:4 -both terms multiplied by 3 9:12 these are equivalent ratios. There is not limit on the number of equivalent ratios that correspond to any given ratio.

Ratios and Allocation Problems Allocate: to distribute/ share out for a particular purpose. E.g. allocate resources, time, money etc. Usually we allocate based on ratios or percentages. Example 1 I have a budget of $10,000 , I need to allocate this amount to the marketing department and the distribution department in the ratio of 5:3. How much do I give to each department?? Decide how many parts are available for allocation: 5 + 3 = 8 Create fractions that represent the amount going to each department: 5 and 3 8 8 Multiply these fractions by the amount to be allocated to determine how much money will go to each department. 5 x 10,000 = 0.625 x 10,000 = $6250 to the marketing department 8 3 x 10,000 = 0.375 x 10,000 = $3750 to the distribution department **Notice that these two amounts should add to give our total amount ($10,000) 

Example 2: Profits of a company must be divided between company owners according to the ratio 3:2:1. If the yearly profits are $81,000, how much money will each partner receive?? Step 1: Decide how many parts are available for allocation: 3 + 2 + 1= 6 parts Step 2: Create fractions that represent how much will go to each partner (out of the total number of parts available), and multiply these fractions by the total profits. 3 X 81,000= $40,500 6 2 X 81,000= $ 27,000 (Rounded) from $26,999.997 1 x 81,000 = $13,499.99

Example 3 A business suffered a fire loss of $224,640. It was covered by an insurance policy that stated that any claim was to be paid by three insurance companies in the ratio 1/3 : 3/8 : 5/12. What is the amount that each of the companies will pay? Step 1: Decide how many parts are available for allocation -In this case we must convert the fractions so that they have the same denominator. (these will be equivalent fractions) a) Find the lowest common multiple of the denominators 3, 8, 12 =LCM =24 , this will be our new denominator. 1 = 8 3 24 3 = 9 8 24 5 = 10 24 Step 2: Allocate according to the ratio formed by the numerators: number of parts = 8+9+10 = 27 8/27 x 224, 640 = $66,559.98 (Rounded) 9/27 x 224,640 = 74,880 10/27 x 224,640= 83,199.98 (Rounded)

Proportions is a mathematical comparison between two numbers. Often, these numbers can represent a comparison between things or people. For example, say you walked into a room full of people. You want to know how many boys there are in comparison to how many girls there are in the room. You would write that comparison in the form of a proportion. Two equivalent ratios form a proportion e.g. 3 = 6 4 8 We can use our knowledge of equivalent fractions/ratios to solve proportion problems, or solve for unknown quantities in proportion equations

Proportion Problems 2:1 = 4: x What is the value of x?? Step 1: change the proportion to fractional and/or decimal form. 2 = 4 1 X Step 2: Use cross multiplication! = 4 x 1 ÷ 2 = 2 X is equal to 2.

Step 1: Convert to fraction and/or decimal format 2.5 : x = 5.5: 38.5 Step 2: Cross multiply (2.5 x 38.5)÷ 5.5 = 17.5 or

Proportion Word Problems Step 1: Write a sentence using a variable to represent the unknown quantity. If 10 lobsters cost $15.00, how much would 5 lobsters be? 10: $15.00 = 5: x Step 2: change to fraction and/or decimal and step 3: cross multiply (5 x 15) ÷ 10 = $7.50 Practice: Exercise 3.2, pg 104. Answers to odd exercises in the back of the text

Using Proportions: Currency Conversions Set up proportion equations to solve these problems 1st currency = Amount ($) in 1st currency 2nd currency Amount in ($) 2nd currency ** notice currency types are lined up horizontally

e. g. If 1 American dollar cost 1 e.g. If 1 American dollar cost 1.07 Canadian to buy, how much would it cost to purchase $550.00 American dollars? Step 1: fill in known quantities Step 2: solve for unknown =(1.07*550) ÷ 1 = $588.50

Percentages A percentage is understood to be a specified amount out of 100. =50% = 50 out of 100 = 0.50 fifty hundredths * You must be able to convert between these forms

Convert the following decimals into percentages: 0.52 = 52 = 52% 100 b) 0.02 = 2 = 2% c)0.025 = 2.5 = 2.5 %

Percentage Problems General formula: % = New Number Original Number/Whole amount OR % = Place known quantities into formula, and solve for the unknown quantity Finding the percentage of a number: E.g. What is 15.5% of $232.00?? Step 1: Fill formula in with “known” quantities: Step 2: Multiply both sides by 232 to isolate unknown

Finding what percentage one number is of another: e.g. What percentage is 5 out of 50? Step 1: Fill in formula with known quantities %= = % = 5 50 Step 2: Solve for the unknown quantity: in this case, %. 5 = 0.10 0.10= 10%

60% of what number is 42?? Step 1: Fill in formula with known quantities: % = 0.60 = 42 x Multiply both sides by x: 0.60x =42 Divide both sides by 0.60 to isolate x/get it alone. Practice: pg. 112-113 in the text. Odd answers are in the back of the Text.

% Increase and Decrease Problems Formula: Original number = New number ** Note the increase/ decrease is the original number * %

36 increased by 25% gives what number? Step 1: Fill in known quantities 36 + (36*0.25) = New number Step 2: solve for unknown quantity 36 + 9= New number 45=New number

How much is 160 increased by 250%?? Step 1: Fill in known quantities 160 + (160*2.50)= New number Step 2: solve for unknown quantity 160 + 400 = x 560 = x

A share price has increased by 25% giving it a new value of $230 A share price has increased by 25% giving it a new value of $230. What was the original share price? Step 1: fill in known quantities x + (x*0.25)= 230 Step 2: solve for unknown quantity x + 0.25x = 230 1.25x = 230

The Consumer Price Index “the CPI is the most widely accepted indicator of changes in overall price level of goods and services” Certain goods and services prices are monitored and the change in cost of these goods and services is reported on by Statistics Canada. You can use the CPI to determine the purchasing power of the Canadian dollar and to compute income. Purchasing power of the dollar = 1 x (100) CPI

CPI and Purchasing Power The CPI was 111.5 for 2007 and 114.1 for 2008. Determine the purchasing power of the Canadian dollar for the two years, and interpret the meaning of the results. Purchasing power of the dollar for 2007: =$1 x (100)=0.896861 111.5 Purchasing power of the dollar for 2008: =$1 x (100)= 0.876424 114.1 This means the dollar in 2007 could purchase only 89.7% of what it could purchase in 2002 (the base year). In 2008, the dollar could purchase even less (about 87.6% of what it could purchase in 2002).

CPI, Nominal income and Real income Nominal Income: Income stated in current dollars Real income: Income stated in base-period dollars e.g. James income was $50,000 in 2002, $53,000 in 2006, and $56,000 in 2009. The Canadian CPI was 109.1 in 2006 and 114.4 in 2009. The CPI base year is 2002. Determine James real income in 2006 and 2009. Real income 2006= nominal income x 100 = $53,000 x 100 = $48,579.29 CPI 2006 109.1 Real income 2009= nominal income = $56,000 x 100 = $48,951.05 CPI 2009 114.4

We need to compare all years to properly compare real and nominal income: 2002 2006 2008 Nominal Income $50,000 $53,000 $56,000 Simple Price Index $50,000 x 100= $100 $53,000 =$106 $56,000 =$112 Absolute $ Increase $3,000 $6,000 Relative % Increase 6% 12% Real Income $48,579.29 $48,951.05 $50,000 x 100= 100 $48,579.29x100= 97.16 $50,000 $48,951.05x100= 97.90 $50,000 Absolute $ Increase or decrease -$1420.71 -$1048.95 Relative % increase or decrease -2.8% -2.1%